Local defect correction for time-dependent problems

Transcription

1 Loca defect correction for time-dependent probems Minero, R. DOI: /IR Pubished: 01/01/2006 Document Version Pubisher s PDF, aso known as Version of Record (incudes fina page, issue and voume numbers) Pease check the document version of this pubication: A submitted manuscript is the author's version of the artice upon submission and before peer-review. There can be important differences between the submitted version and the officia pubished version of record. Peope interested in the research are advised to contact the author for the fina version of the pubication, or visit the DOI to the pubisher's website. The fina author version and the gaey proof are versions of the pubication after peer review. The fina pubished version features the fina ayout of the paper incuding the voume, issue and page numbers. Link to pubication Citation for pubished version (APA): Minero, R. (2006). Loca defect correction for time-dependent probems Eindhoven: Technische Universiteit Eindhoven DOI: /IR Genera rights Copyright and mora rights for the pubications made accessibe in the pubic porta are retained by the authors and/or other copyright owners and it is a condition of accessing pubications that users recognise and abide by the ega requirements associated with these rights. Users may downoad and print one copy of any pubication from the pubic porta for the purpose of private study or research. You may not further distribute the materia or use it for any profit-making activity or commercia gain You may freey distribute the URL identifying the pubication in the pubic porta? Take down poicy If you beieve that this document breaches copyright pease contact us providing detais, and we wi remove access to the work immediatey and investigate your caim. Downoad date: 06. Dec. 2018

2 Loca Defect Correction for Time-Dependent Probems

3 Copyright c 2006 by Remo Minero, Eindhoven, The Netherands. A rights are reserved. No part of this pubication may be reproduced, stored in a retrieva system, or transmitted, in any form or by any means, eectronic, mechanica, photocopying, recording or otherwise, without prior permission of the author. CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Minero, Remo Loca defect correction for time-dependent probems / door Remo Minero. - Eindhoven: Technische Universiteit Eindhoven, Proefschrift. - ISBN NUR 919 Subject headings: paraboic differentia equations; numerica methods / initia vaue probems; numerica methods 2000 Mathematics Subject Cassification: 65M50, 65M06, 76F25

4 Loca Defect Correction for Time-Dependent Probems PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het Coege voor Promoties in het openbaar te verdedigen op donderdag 1 juni 2006 om uur door Remo Minero geboren te Borgosesia, Itaië

5 Dit proefschrift is goedgekeurd door de promotor: prof.dr. R.M.M. Mattheij Copromotor: dr.ir. M.J.H. Anthonissen This project was funded by NWO (Netherands Organisation for Scientific Research) under the research program Computationa Science, grant number

6 To Lidia

7

8 Contents 1 Introduction Transport of passive tracers Adaptive grid methods for time-dependent probems The LDC method for eiptic probems Outine of this thesis The LDC method for paraboic partia differentia equations Performing one time step with LDC A composite grid soution at time t n The defect correction Properties of the LDC time step The LDC agorithm incuding regridding Numerica experiments Exampe 1: a 2D convection-diffusion probem Exampe 2: comparison between LDC and LUGR Convergence properties of LDC The iteration matrix A one-dimensiona mode probem Bounds for the M 1 infinity norm The expression of the M 2 infinity norm Iteration matrix norm asymptotics The imit for sma vaues of the time step t The stationary case imit Pots of the iteration matrix norm One-dimensiona numerica experiments Two-dimensiona numerica experiments A finite voume adapted LDC method Probem formuation and initiaization of LDC Composite grid definition Computation of a composite grid soution

9 viii Contents 4.2 The finite voume adapted defect term Approximation of the defect The finite voume adapted LDC agorithm Practica considerations on the defect term Conservation properties of LDC Numerica experiments Exampe 1: comparison with a uniform grid sover Exampe 2: comparison with the standard LDC method Generaizations of the LDC method A conservative regridding strategy Providing initia vaues on the new composite grid A numerica exampe LDC agorithm with mutipe eves of refinements Soving transport probems using LDC Mathematica mode Numerica method Impementation Numerica resuts Comparison between LDC and the spectra method Concusions and recommendations 113 Bibiography 115 Index 121 Summary 123 Samenvatting (Summary in Dutch) 125 Acknowedgements 127 Curricuum vitae 129

10 Chapter 1 Introduction 1.1 Transport of passive tracers In everyday ife we often see the dispersion in the atmosphere of exhaust gases from a chimney, or the dispersion of the smoke produced by a cigarette. These are exampes of tracer transport in given veocity fieds. The fow fied is given by the movement of the air in the atmosphere, whie the tracer is represented by the mixture of heated gas and suspended partices that are injected in the air. The tracer is caed passive when its concentration is so ow that the main fow dynamics is not infuenced by the presence of the externa partices. An environmenta exampe of passive tracer transport is the dispersion of dye or poutants in a river or in the sea. The same kind of phenomenon occurs in many engineering appications; exampes are the transport of fy ashes in burners or the turbuent mixing in chemica reactors. In order to efficienty design this type of machines, it is important to unrave the infuence of the turbuent fow on the transport mechanism. In the case of chemica reactors, in fact, turbuence can positivey affect mixing of chemica species, and hence faciitate their reaction. Mathematicay the transport process is modeed by a time-dependent advectiondiffusion equation. In turbuent fows, passive scaar transport is characterized by a ength scae that can be significanty smaer than the size of the turbuent eddies in the main fow [26]. From a computationa point of view this means that even a higher resoution is needed to simuate the transport process than required for the turbuence itsef. This impies that a proper description of the transport phenomena is computationay more expensive than running Direct Numerica Simuations (DNS) of turbuence. However, in many practica appications fiaments of tracer materia are mainy confined in a imited part of the computationa domain. In other words, the soution of the advection-diffusion equation that modes the

11 2 Chapter 1 - Introduction transport process is often characterized by oca regions of high activity, i.e. regions where spatia gradients are quite arge compared to those in the rest of the domain, where the soution presents a reativey smooth behavior. Exampes of soutions of Partia Differentia Equations (PDEs) exhibiting oca regions of high activity are frequenty encountered aso in many other areas, ike shock hydrodynamics, combustion, etc. An efficient numerica soution of this kind of probems requires the usage of adaptive grid techniques. In adaptive grid methods, a fine grid spacing and possiby a sma time step are adopted ony where the arge variations occur, so that the computationa effort and the memory requirements are minimized Adaptive grid methods for time-dependent probems A arge number of adaptive grid methods for time-dependent probems have been proposed in the iterature. A first category incudes the moving-grid or dynamicregridding methods. In this approach, nodes are moving continuousy in the spacetime domain, ike in cassica Lagrangian methods, and the discretization of the PDEs is couped with the motion of the grid. Methods in this category differ in how the motion of the grid is governed. The grid is anyhow aways nonuniform and the number of nodes remains constant in time. The nonuniformity of the grid impies that programming these methods often invoves quite compicated data structures. An exampe of a dynamic-regridding technique is the moving mesh strategy (MMPDE) introduced in [36] for one-dimensiona probems and then extended to 2D in [37]. In MMPDE the physica PDE is soved together with a partia differentia equation that describes the movement of the mesh. Practica issues about MMPDE are addressed in [35]; these incude a proper contro of the mesh concentration by means of monitor functions. In [8, 9] it is shown that the monitor functions strongy infuence the accuracy of the computation; moreover agorithms in which the physica PDE is decouped from the mesh equation are proposed. In [68] an adaptive moving mesh technique for reaction-diffusion equation is described; in this case the movement of the mesh is determined by minimizing an energy functiona, the so-caed mesh-energy integra, which is defined from certain monitor functions. In [58] a conservative-interpoation formua is introduced to guarantee discrete mass conservation in the numerica soution when the mesh is moved. Another type of adaptive grid techniques is represented by static-regridding methods. Here, the idea is to adapt the grid at each time step by adding grid points where a high activity occurs and removing them where they are no onger needed. This process is controed by error estimates or methods based on the measure of some characteristics of the soution (e.g. gradients, sope, etc.). In this kind of methods the number of grid points is not constant in time. An exampe of a static-regridding strategy is the Adaptive Mesh Refinement (AMR) technique introduced in [12, 14] for the soution of hyperboic PDEs. In AMR a goba coarse grid covers the whoe domain, and finer and finer grids are added ocay to resove the high variations of

12 1.1 Transport of passive tracers 3 the soution ti the required eve of accuracy is reached. In [14] the rectanguar subgrids may be skewed with respect to the goba rectanguar domain axes: this is to aow aignment of the oca grids with the steep regions of the soution. In [12] the focus is on discrete conservation and the boundaries of the nested grids coincide with grid ines of the underying coarse mesh; time integration is performed expicity. More recent variants and appications of AMR can be found in [13, 31, 50]. Another exampe of a static-regridding technique is the Loca Uniform Grid Refinement (LUGR) method, described and anayzed in [60, 62 64]. LUGR is proposed for the soution of paraboic PDEs and in this method time integration is generay performed impicity. LUGR is appied to the soution of transport probems in heterogeneous porous media in [61], whie an LUGR based strategy for eectrochemica appications is introduced in [15]. Impementation and agorithmic issues of LUGR are discussed in [16]. Briefy, the method works as foows: at each time step the PDE is first integrated on a goba uniform coarse grid. The coarse grid soution at the new time step provides artificia boundary conditions on a oca uniform fine grid and the probem is then soved ocay with a smaer time step than the one used on the goba grid. At this point, the fine grid vaues are used to repace the coarse grid vaues in the region of refinement. The procedure can be repeated recursivey to incude more eves of refinement. The technique reies on the fact that the coarse grid soution provides artificia boundary conditions for the oca probem that are accurate enough. The main advantage of LUGR is the possibiity of working with uniform grids and uniform grid sovers ony. A drawback of static-regridding methods against moving-grid methods is that they generay use more nodes to obtain a given accuracy. However, in moving-grid methods specia care must be taken in order to prevent grid distortion; this occurs for exampe when vertex anges become too cose to 0 or 180 in trianguar meshes. This phenomenon can reduce the accuracy of the computations consideraby. As a consequence moving-grid methods require more parameter tuning than static-regridding techniques. This thesis focuses on a new static-regridding technique for time-dependent probems: Loca Defect Correction (LDC). LDC shares with LUGR the possibiity and the advantage of working with uniform grids and uniform grid sovers ony. In LDC, however, the fine grid soution at the new time step is used not ony to repace the coarse grid vaues in the area of refinement, but to overa improve the coarse grid approximation. This can be achieved through a defect correction, in which the fine grid soution is used to approximate the coarse grid oca discretization error. The improved coarse grid approximation defines new artificia boundary conditions for a new oca probem, which in turn can correct the soution gobay. In this way, LDC does not have to rey on the accuracy of the artificia boundary condition provided by the first coarse grid approximation, turning out to be a more robust technique than LUGR. The method we introduce in this thesis is a generaization of the oca defect correction method deveoped for the efficient soution of eiptic PDEs.

13 4 Chapter 1 - Introduction 1.2 The LDC method for eiptic probems As a technique for soving eiptic probems with highy ocaized properties, oca defect correction was initiay presented in [34]. LDC is an iterative procedure in which the eiptic probem is first soved on a goba coarse grid (grid size H). This first coarse grid approximation provides, via interpoation, artificia Dirichet boundary conditions to a oca fine grid (grid size h < H); the fine grid is ocated where the high activity in the soution occurs. A discrete boundary vaue probem is now soved ocay. The oca soution is then pugged into the coarse grid discretization scheme in order to get an approximation of the coarse grid oca discretization error or defect. The defect, added to the right hand side of the coarse grid probem, eads to determining a more accurate coarse grid approximation. This provides updated boundary conditions for a new oca probem, whose soution wi be used to further improve the approximation gobay. The entire procedure can be repeated unti a fixed point in the iteration is reached. The scheme of the LDC iteration is iustrated in Figure 1.1. At convergence a discrete soution of the continuous probem is avaiabe in a the points of the composite grid, union of coarse and fine grid, see Figure 1.2. In this thesis we extend the origina idea of oca defect correction to the time variabe: when LDC is appied to a time-dependent PDE, the oca probem is soved not ony with a finer grid size, but aso with a smaer time step than the one adopted gobay. In this way the defect term can be used to improve the goba approximation not ony in space, but aso in time. The LDC method for eiptic PDEs introduced in [34] is cosey reated to two other iterative procedures that compute the soution of boundary vaue probems on composite grids: the Fast Adaptive Composite grid (FAC) agorithm proposed in [41, 42], and the Muti-Leve Adaptive Technique (MLAT) studied in [18]. In [28] it is expained that in certain situations the three methods produce the same iterates. However, FAC requires an a priori given discretization on the composite grid, whie LDC needs separate discretizations on the goba and oca grid ony. One crucia difference between LDC and MLAT is that MLAT, based on mutigrid principes, Boundary conditions Coarse grid approximation Fine grid approximation Defect correction Figure 1.1: Scheme of the LDC iteration.

14 1.2 The LDC method for eiptic probems 5 Figure 1.2: An exampe of a composite grid, union of a goba coarse and a oca fine grid. soves at each iteration the discrete oca probem ony approximatey, not exacty ike in LDC. Moreover, the factor of grid refinement H/h is sma in MLAT (typicay 2), whie it can be significanty arger in LDC. With reference to finite differences discretizations, the properties of the LDC method for eiptic probems are anayzed in [29, 30]. In [29] it is proved that, at convergence, the coarse and the fine grid soution coincide at the common points between coarse and fine grid. Moreover it is shown that, combining coarse and fine grid discretizations, LDC impicity soves a composite grid discretization; in [29] the system of equations that the imit of the LDC iteration satisfies on the composite grid is derived. In Chapter 2 of this thesis we show that simiar properties hod when LDC is appied to paraboic probems: aso for the time-dependent case it can be proved that, at convergence, the computed coarse and fine grid soution are the same in the common points between the two grids. Aso, we derive the system soved by the composite grid soution at a generic time eve. In [30] error bounds for the composite grid soution are given. For a two-dimensiona Poisson s equation discretized by second-order centered differences both gobay and ocay, it is shown that the goba discretization error of the LDC composite grid soution can be bounded by C 1 h 2 + C 2 H 2 + C 3 H j. These three terms correspond to the discretization errors on the oca fine grid, on the part of the coarse grid outside the area of high activity, and to the interpoation error at the interface respectivey. In the sum above h and H indicate the fine and coarse grid size; C 1, C 2 and C 3 are positive constants independent of h and H, whie the exponent j depends on the interpoation method used to provide the artificia boundary conditions to the oca grid: one has j = 1 for piecewise inear

15 6 Chapter 1 - Introduction interpoation, j = 2 for piecewise quadratic interpoation. The terms C 1, C 2 and C 3 measure the continuous soution s smoothness in the area of refinement, outside the area of refinement and aong the interface between coarse and fine grid respectivey. Since the soution of the Poisson s equation is assumed to have a high activity in the area of refinement, one typicay has C 1 C 2 ; this can be baanced by choosing h 2 H 2. Experimenta evidence (see again [30] and aso [28]) shows that C 3 is generay much smaer than C 1 and C 2. The convergence behavior of the LDC method for eiptic probems is anayzed in [4]. The LDC iteration is expressed in terms of an iteration matrix, whose properties are studied anayticay and experimentay for the two-dimensiona Poisson s equation discretized by finite differences. In genera, it is observed that LDC converges very fast and that iteration errors are reduced by severa orders of magnitude at each iteration step. In Chapter 3 of this thesis we use a simiar approach to study the convergence properties of the LDC method for paraboic probems. Aso for the timedependent case we find that LDC converges very fast and that, in practice, ony very few iterations per time step are required for convergence. In LDC one is not forced to use finite difference discretizations. In [66], for instance, oca defect correction is combined with the finite eement method, whie a finite voume adapted LDC agorithm that guarantees discrete conservation on the composite grid is proposed in [6]. The idea in [6] is to write the defect term in such a way that coarse and fine grid fuxes baance across the interface between coarse and fine grid. In Chapter 4 of this thesis we extend that idea, and we appy it to paraboic probems that are integrated on the goba and on the oca grid with different time steps. The LDC method for eiptic probems has aso been empoyed with different grid types. This is particuary convenient when the soution has a specia symmetry [47] or the oca high activity is not aigned with goba domain axes [3, 32, 33]. In [47] the goba coarse grid is Cartesian, whie the oca domain has a circuar shape and poar coordinates are used ocay. In [33] the oca fine grid is in a santwise direction. In [32] LDC is appied to sove combustion probems and curviinear coordinates are adopted ocay in order to foow the shape of the fame front. In [3] simiar appications are considered, and the cassica LDC agorithm is extended to incude mutipe eves of refinement, domain decomposition and regridding. In a recent paper [54] oca defect correction is finay combined with high order compact finite difference schemes for the soution of both one- and two-dimensiona boundary vaue probems. To summarize, LDC has been proved to be an efficient method for soving eiptic probems on composite grids. It has very good convergence properties, and it can be appied for a wide range of physica probems with severa discretization techniques. In this thesis we extend the LDC principes to the soution of paraboic PDEs.

16 1.3 Outine of this thesis Outine of this thesis In Chapter 2 of this thesis the LDC method for paraboic probems is presented. At each time step the PDE is integrated both on a goba and on a oca grid, and the two soutions at the new time eve are iterativey combined to utimatey give a soution on the composite grid. In the method we propose, time integration on the oca grid is performed with a smaer time step than the one adopted gobay. In this way, the oca soution is used to correct not ony the errors in the goba approximation due to the spatia discretization, but aso due to the tempora discretization. In practice the method is such that a fine grid spacing and a sma time step are adopted to resove the oca high activity ony, whie a arger grid spacing and time step are chosen gobay. This impies that the goba approximation at the new time eve is interpoated not ony in space, but aso in time, in order to compute the artificia boundary conditions for the oca probem. In the same chapter we aso iustrate some properties of the LDC time step. First, we show that the converged coarse and fine grid soution at the new time step coincide at the common points between the two grids. Secondy, we derive the system of equations that the converged LDC soution impicity satisfies on the composite grid. Ceary, these properties generaize the theorems proved in [29] for stationary LDC. Furthermore, our method takes into account that the oca high activity might move and be ocated in different parts of the goba domain at different times. Therefore the LDC agorithm for time-dependent probems incudes a regridding strategy: the first coarse grid soution at the new time eve indicates where the high variations at that time eve occur, and the oca grid is chosen accordingy. The agorithm is tested in some concrete exampes that iustrate its accuracy, efficiency and robustness. Chapter 3 of this thesis is devoted to the anaysis of the convergence behavior of the LDC agorithm introduced in Chapter 2. The LDC iteration that takes pace at a generic time step is expressed in terms of an iteration matrix, which is a generaization of the expression derived in [4] for stationary cases. For one-dimensiona diffusion probems, the properties of the iteration matrix are studied anayticay, whie for one- and two-dimensiona convection-diffusion probems the anaysis is carried out experimentay. In genera we observe that LDC converges for any choice of the discretization parameters and that iteration errors are reduced by severa orders of magnitude at each iteration step. When the new and fast converging LDC method for paraboic probems is appied in combination with the finite voume method, the resuting numerica soution is not necessariy conservative on the composite grid. In Chapter 4 of this thesis we present a finite voume adapted LDC agorithm for paraboic probems that guarantees baance of coarse and fine grid fuxes across the interface between the goba and oca grid at each time step. In this way the computed numerica soution satisfies a discrete system of conservation aws on the composite grid. This system

17 8 Chapter 1 - Introduction of equations is expicity derived in Chapter 4, and the features of the method are tested experimentay in two exampes: the first one iustrates the superior efficiency of the new method with respect to a uniform goba fine grid sover, and the second demonstrates discrete conservation. Our method generaizes the technique introduced in [6] for eiptic probems. One imitation of the finite voume adapted LDC agorithm introduced in Chapter 4 is that it can ony hande fixed grids. In practice it is of interest ony for those physica probems in which the high activity remains confined in the same imited part of the goba domain at a time eves. In Chapter 5 we overcome this imitation and we incorporate a conservative regridding strategy in the finite voume adapted LDC agorithm. Aso, we extend the agorithm to incude mutipe eves of refinement. The oca defect correction iteration is initiay appied between a goba coarse grid (eve 0) and a oca fine grid (eve 1); when this iteration has converged, a new iteration takes pace between eve 1 and a finer grid at eve 2, and so on. The time marching strategy is such that time integration at the finer eves can be performed with smaer time steps. At the end (Chapter 6), the new, mutieve, conservative and fast converging LDC method is appied to sove reaistic time-dependent probems. In particuar, we consider the transport of passive tracers in transient fow fieds. Resuts of numerica simuations show that LDC produces accurate resuts and that it is a very promising too to sove aso more genera physica probems characterized by highy ocaized properties. In Chapter 7 we finay draw some concusions and give recommendations for future research.

18 Chapter 2 The LDC method for paraboic partia differentia equations In this chapter we introduce the oca defect correction method for soving paraboic differentia equations with highy ocaized properties. The method is a generaization of the LDC technique for eiptic probems described and anayzed in [1, 29, 30, 34]. In a time-dependent setting LDC works as foows: first a time step is performed on a goba coarse grid. The goba soution at the new time eve provides artificia boundary conditions on a oca fine grid, which is adaptivey paced where the high activity occurs. A soution is then computed ocay, possiby with a smaer time step than the one adopted gobay. At this point the oca approximation provides an estimate for the coarse grid oca discretization error or defect. The defect, added to the right hand side of the coarse grid probem, eads to determining a more accurate (both in space and time) goba approximation of the soution. This can now be used to update the boundary conditions ocay and the entire procedure can be repeated again unti convergence. In practice, at each time step a goba and a oca approximation progressivey improve each other to utimatey compute a soution on the composite grid, union of coarse and fine grid. In comparison with other strategies for grid refinement, one of the main advantages of the method is that the goba and the oca grid can aways be uniform structured grids. With this respect LDC is simiar to the Loca Uniform Grid Refinement (LUGR) method presented and anayzed in [60, 62 64]. LDC, however, differs from LUGR because in LUGR the oca soution does not improve the soution gobay through the defect correction. In LUGR the oca soution is ony used to repace the coarse grid vaues in the area of refinement, whie in LDC the oca soution improves the coarse grid approximation overa. In addition, LUGR reies on the fact that the boundary conditions provided by the coarse grid approximation are accurate enough, whie in LDC aso the artificia boundary conditions are progres-

19 10 Chapter 2 - The LDC method for paraboic partia differentia equations sivey improved. For these reasons it turns out that LDC is a more robust technique than LUGR. This chapter is based on the research resuts previousy presented in [43, 44]. 2.1 Performing one time step with LDC In this section we describe the LDC time step for a paraboic probem. We assume a soution to be known on the composite grid at a certain time eve and we expain how to compute a soution on the composite grid at the next time eve by means of LDC. We consider the foowing two-dimensiona probem u(x, t) = Lu(x, t) + f(x, t), in Ω Θ, t u(x, t) = ψ(x, t), on Ω Θ, u(x, 0) = ϕ 0 (x), in Ω, (2.1) where Ω is a spatia domain, Ω its boundary and Ω := Ω Ω. Moreover, Θ is the time interva (0, t end ], L a inear eiptic operator, f a source term, ψ a Dirichet boundary condition and ϕ 0 a given initia condition. For ease of presentation we ony consider Dirichet boundary conditions. However this is not restrictive and the impementation of other types of boundary conditions (e.g. Neumann or Robin) is straightforward. We assume that u has a region of high activity that covers a sma part of Ω. Probem (2.1) is discretized in space and time in order to be soved numericay. For that, in Ω we introduce the goba uniform coarse grid (grid size H) Ω H and the time step t. Grid points Ω H are paced on Ω too and we define Ω H := Ω H Ω H. Because of the high activity of the soution, at time t n := n t a coarse grid approximation computed with a time step t might be not adequate enough to represent u(x, t n ). In order to better capture the oca high activity, we aso sove the probem on the oca domain Ω n Ω. We denote the boundary of Ω n by Ω n and we define Ω n := Ωn Ωn. In Ωn the superscript n refers to the time eve. This is needed because the area of refinement is updated at each time step to adaptivey foow the behavior of the soution. This is discussed in Section 2.2. Here we just assume the oca domain to be given and paced where the area of high activity at time t n occurs. On the oca domain we introduce a uniform fine grid (grid size h < H), which we denote by Ω h,n. Grid points Ω h,n are aso paced on Ω n and we define Ω h,n := Ω h,n Ω h,n. The region of refinement and the fine grid spacing h are chosen is such a way that coarse grid points that ie in the area of refinement beong to the fine grid too. On Ω h,n the time integration is performed

20 2.1 Performing one time step with LDC 11 Figure 2.1: Exampe of composite grid Ω H,h,n. using a time step δt = t/τ, with τ an integer 1. In LDC, the oca soution wi be used to improve the goba approximation through a defect correction. In the remainder of this section we wi assume that a soution u H,h,n 1 is known at a generic time t n 1 on the composite grid Ω H,h,n := Ω H Ω h,n, see Figure 2.1. If n = 1, u H,h,n 1 is given by a proper discretization of the initia condition ϕ 0, otherwise it may be the resut of the computation of the previous time step. Our goa is to compute an approximation of the soution on Ω H,h,n at the new time eve t n using LDC A composite grid soution at time t n The first step in the LDC method is the computation of a coarse grid approximation at t n. We ca it u H,n 0 and we compute it by appying the backward Euer method to the partia differentia equation in (2.1). Whie other impicit time integration schemes coud aso be adopted, the usage of expicit time integrators on the goba grid is not of interest in LDC; this is discussed in Section 2.2. We obtain (I t L H )u H,n 0 = u H,h,n 1 Ω H + f H,n t, (2.2) where L H is some spatia discretization of the eiptic operator L. In (2.2), the discretized source term f H,n aso incudes the Dirichet boundary conditions. By G(Ω H ) we indicate the space of grid functions that operate on Ω H ; simiar notation is used for the other sets. With M H := I t L H, we rewrite (2.2) as M H u H,n 0 = u H,h,n 1 Ω H + f H,n t. (2.3)

21 12 Chapter 2 - The LDC method for paraboic partia differentia equations Figure 2.2: Partition of the points of the composite grid. We assume M H : G(Ω H ) G(Ω H ) to be invertibe. We denote by Γ n the interface between Ω n and Ω \ Ω n. For convenience we partition the coarse grid points as foows Ω H = Ω H,n Γ H,n Ω H,n c, (2.4) where Ω H,n := Ω H Ω n, Γ H,n := Ω H Γ n, Ω H,n c := Ω H \ (Ω H,n Γ H,n ). (2.5) In Figure 2.2 the coarse grid points Ω H,n are marked with circes, whie trianges and squares denote points in Γ H,n and Ω H,n c respectivey. Using the partitioning above, we set u H,n w =: u H,n,w u H,n Γ,w u H,n c,w. (2.6) In (2.6) the subscript w is used to number the coarse grid approximations at time t n. At the moment we are describing how to compute a first coarse grid soution at the new time eve, hence we have w = 0, cf. (2.2) and (2.3). Assuming that the spatia discretization on the coarse grid is such that the stenci at grid point (x, y) invoves at most function vaues at (x + ih, y + jh), with i, j { 1, 0, 1}, we rewrite (2.3) as M H B H,Γ 0 B H Γ, M H Γ B H Γ,c 0 B H c,γ M H c u H,n,w u H,n Γ,w u H,n c,w = u H,h,n 1 Ω H,n u H,h,n 1 Γ H,n u H,h,n 1 Ω H,n c + f H,n f H,n Γ f H,n c t t. (2.7) The coarse grid soution u H,n w is used to prescribe artificia boundary conditions on the interface Γ n. Boundary conditions on Γ n are needed to define a discrete fine grid probem that eads to determining u h,n,w, a oca more accurate (both in space and t

22 2.1 Performing one time step with LDC 13 time) approximation of u(t n ). We can prescribe artificia Dirichet boundary conditions at t n by appying an interpoation operator in space P h,h : G(Γ H,n ) G(Γ h,n ) to u H,n w ; by Γh,n we denoted the set of fine grid points that ie on the interface Γ n. In Figure 2.2 the points Γ h,n are marked with sma diamonds. In LDC the operator P h,h generay performs piecewise inear or piecewise quadratic interpoation (cf. [30]). If we want to perform time integration with a time step δt = t/τ, we aso need to provide boundary conditions on Γ h,n at a the intermediate time eves t n 1+k/τ, with k = 1, 2,..., τ 1. Therefore we perform inear time interpoation between u H,h,n 1 Γ h,n and P h,h u H,n w. Note that in order to sove the probem ocay on Ω n we have to specify boundary conditions not ony on Γh,n Ω h,n, but on the whoe Ω h,n. However, for the fine grid points Ω h,n \ Γ h,n we can use a proper discretization of the boundary condition for the continuous probem (2.1). We et L h be a oca fine grid discretization of the operator L and we introduce M h := I δt Lh. A first fine grid approximation (w = 0) at time t n can thus be computed soving M h uh,n 1+k/τ,w = u h,n 1+(k 1)/τ,w ( k B h,γ τ Ph,H u H,n Γ,w + τ k τ + f h,n 1+k/τ δt uh,h,n 1 Γ h,n ), for k = 1, 2,..., τ. (2.8) In (2.8) the fine grid discretized source term f h,n 1+k/τ aso incudes the boundary conditions for the fine grid points Ω h,n \ Γ h,n. The procedure (2.8) is initiaized using u h,n 1,w = u H,h,n 1 Ω h,n. (2.9) We combine a the equations in (2.8) to express the fine grid approximation u h,n,w directy in terms of u H,h,n 1. We obtain ( M h ) τ u h,n or ( M h ) τu h,n,w = u H,h,n 1 Ω h,n Ω h,n + τ k=1 ( ) M h k 1 h,n 1+k/τ f δt τ ( ( ) M h k 1 B h k,γ τ Ph,H u H,n k=1,w = u H,h,n 1 Ω h,n Γ,w + τ k τ uh,h,n 1 Γ h,n ), (2.10) + F h,n δt W,ΓP n h,h u H,n Γ,w + Z n,γu H,h,n 1 Γ h,n. (2.11) In (2.11) F h,n depends ony on the source term and on the fine grid operator M h, whie W n,γ and Zn,Γ ony depend on Mh and Bh,Γ The defect correction The crucia part of the LDC method is how the oca soution u h,n,w is used to improve the goba approximation u H,n w through an approximation of the coarse grid oca

23 14 Chapter 2 - The LDC method for paraboic partia differentia equations discretization error or defect. The defect d H,n is defined as d H,n := M H u(t n ) Ω H u(t n 1 ) Ω H f H,n t. (2.12) In (2.12) we substituted the projection on Ω H of the continuous soution u into the discretization scheme (2.3). If we woud know the vaues of the defect d H,n, we coud use them to find a better approximation of u n on the coarse grid. This coud be achieved by adding d H,n to the right hand side of (2.3). However, since we do not know the exact soution of our partia differentia equation, we cannot compute the vaues of d H,n. What we can do, though, is to use the more accurate oca approximation u h,n to get a oca estimate d H,n of d H,n. The oca estimate of the defect is computed on Ω H pugging the fine grid soution into the coarse grid discretization scheme in that region. We obtain (cf. the first equation in (2.7)) d H,n,w 1 := MH RH,h u h,n,w 1 + BH,Γ uh,n Γ,w 1 uh,h,n 1 Ω H f H,n t, (2.13) where R H,h : G(Ω h,n ) G(Ω H,n ) is a restriction operator from the fine to the coarse grid, such that (R H,h u h,n,w 1 )(x, y) = uh,n,w 1 (x, y), (x, y) ΩH,n. (2.14) The defect d H,n,w 1 is now added to the right hand side of (2.7). A more accurate coarse grid approximation is thus computed soving M H u H,n w = = u H,h,n 1 Ω H,n u H,h,n 1 Γ H,n u H,h,n 1 0 u H,h,n 1 Ω H,n c Γ H,n u H,h,n 1 Ω H,n c + f H,n t + d H,n f H,n Γ f H,n c t t M H RH,h u h,n +,w 1,w 1 + BH,Γ uh,n Γ,w 1 f H,n Γ t f H,n c t. (2.15) The new coarse grid soution can be used to update the boundary conditions for a new oca probem on Ω h,n, which in turn wi correct the coarse grid approximation. This defines the LDC iteration process, cf. Figure 1.1. We note that, as discussed in [4, 27, 34, 66] for LDC in stationary probems, sometimes it might be convenient to compute d H,n,w 1 not at a points of ΩH,n, but in a subset Ω H def ony. In particuar, points ying cose to the interface Γn shoud be excuded. In this way points of Γ n and points of Ω H def are separated by a so-caed safety region. Figure 2.3 shows an exampe of a composite grid without a safety region (a) and with a safety region (b). In the figure, points Ω H def are marked with a back circe. The main advantage of using a safety region is to speed up the convergence of the LDC iteration. This is discussed in Chapter 3. With the introduction of the

24 2.1 Performing one time step with LDC 15 (a) (b) Figure 2.3: Composite grid without (a) and with (b) safety region. safety region, we can rewrite (2.15) as 0 M H u H,n Γ H,n w = u H,h,n 1 u H,h,n 1 Ω H,n c (I X H )fh,n + where the operator X H : G(ΩH,n + X H (MH RH,h u h,n,w 1 + BH,Γ uh,n f H,n Γ t t, f H,n c ) G(Ω H,n ) is defined by Γ,w 1 ), (2.16) { ( X H u H,n ) u H,n (x, y), (x, y) Ω H (x, y) := def, 0, (x, y) Ω H \ ΩH def. (2.17) Note that if no safety region is used, then Ω H def ΩH and XH function. reduces to the identity Properties of the LDC time step In this section we discuss some properties of the LDC time step. The foowing emma shows that once the coarse grid approximations at t n do not change on the interface Γ n, the LDC method converges and a fixed point of the iteration has been reached.

25 16 Chapter 2 - The LDC method for paraboic partia differentia equations Lemma 2.1 If u H,n Γ,w = uh,n Γ,w 1 for a certain index w, then the LDC iteration converges and for a q = w, w + 1,... u H,n q = u H,n w, uh,n q = u h,n w, (2.18) Proof. Assume that u H,n Γ,w = uh,n Γ,w 1 for a certain index w. From (2.11), we have that, and hence, from (2.16), u h,n w = uh,n w 1 M H u H,n w+1 = = 0 u H,h,n 1 Γ H,n u H,h,n 1 Ω H,n c 0 u H,h,n 1 u H,h,n 1 + Γ H,n Ω H,n c (I X H )fh,n (I X H )fh,n + +X H (MH RH,h u h,n f H,n Γ t t, f H,n c + X H (MH RH,h u h,n,w + BH,Γ uh,n f H,n Γ t t, f H,n c,w 1 +BH,Γ uh,n Γ,w 1 ) Γ,w ) =M H u H,n w. Because we have assumed M H to be invertibe, we have u H,n w+1 = uh,n w, for a grid points in Ω H. Since Γ H,n Ω H, we have u H,n Γ,w+1 = uh,n Γ,w. By induction, we find u H,n q = u H,n w and u h,n q = u h,n w, for a q = w, w + 1,... Lemma 2.1 is the time-dependent equivaent of [1, Lemma 3.2]. Using a matrix notation, equations (2.7) and (2.11) can be combined as foows ( ) M h τ 0 W,Γ n Ph,H 0 0 M H B H,Γ 0 0 B H Γ, M H Γ B H Γ,c 0 0 B H c,γ M H c X = MH RH,h 0 X H BH,Γ u h,n,w u H,n,w u H,n Γ,w u H,n c,w u h,n,w 1 u H,n,w 1 u H,n Γ,w 1 u H,n c,w 1 F h,n t (I X + H )fh,n f H,n Γ δt + δt f H,n c u H,h,n 1 Ω h,n + (I X H )uh,h,n 1 u H,h,n 1 Γ H,n u H,h,n 1 Ω H,n c Z n,γ uh,h,n 1 Γ h,n Ω H,n. (2.19) Equation (2.19) is an expression for the LDC iteration at time t n. It can aso be written using the short notation M H,h u H,h,n w = S H,h u H,h,n w 1 + ũ H,h,n 1 + f H,h,n + z H,h,n 1. (2.20)

26 2.1 Performing one time step with LDC 17 The imit of the LDC iteration at time t n is indicated by u H,h,n := u h,n u H,n u H,n Γ u H,n c. (2.21) In (2.21) we removed the subscript w that numbers the LDC iterations. Since u H,h,n is the fixed point, one has M H,h u H,h,n = S H,h u H,h,n + ũ H,h,n 1 + f H,h,n + z H,h,n 1. (2.22) With reference to the LDC method with no safety region, the foowing theorem states that, if the LDC iteration converges, the fine and the coarse grid approximation coincide at the common points between fine and coarse grid. Theorem 2.2 Consider an LDC time step and compute the defect term using no safety region. Assume that the LDC iteration converges. Then the fixed point u H,h,n is such that R H,h u h,n = u H. (2.23) Proof. If no safety region is used, then X H point can be written as ( ) M h τ 0 W,Γ n 0 M H RH,h M H B H Γ, M H Γ B H Γ,c 0 0 B H c,γ M H c = u H,h,n 1 0 u H,h,n 1 u H,h,n 1 Ω h,n Γ H,n Ω H,n c u h,n u H,n u H,n Γ u H,n c + The second equation of the system reads M H R H,h u h,n F h,n = I and equation (2.22) for the fixed t 0, δt + δt f H,n Γ f H,n c Z n,γ uh,h,n 1 Γ h,n (2.24) + M H u H,n = 0, (2.25) which gives (2.23), since we supposed M H (and hence M H ) to be invertibe. We finay write the system of equations that the imit of the LDC iteration satisfies at time t n.

27 18 Chapter 2 - The LDC method for paraboic partia differentia equations Theorem 2.3 Consider an LDC time step and compute the defect term using no safety region. Assume that the LDC iteration converges. Then u h,n, u H,n Γ and u H,n c satisfy the foowing system of equations ( ) M h τ W n,γ 0 B H Γ, RH,h M H Γ B H Γ,c 0 B H c,γ M H c = u h,n,w u H,n Γ,w u H,n c,w u H,h,n 1 Ω h,n u H,h,n 1 Γ H,n u H,h,n 1 Ω H,n c + F h,n f H,n Γ f H,n c t δt + δt Z n,γ uh,h,n 1 Γ H,n 0 0. (2.26) Proof. Eimination of u H,n from (2.24) gives (2.26). Theorems 2.2 and 2.3 are the generaization to time-dependent probems of [1, Theorem 3.3]. Note that (2.26) impies a discretization on the composite grid, whie, for soving that system, we have ony used uniform grids and uniform grid sovers. 2.2 The LDC agorithm incuding regridding In the previous section we expained how to perform the time step from t n 1 to t n using LDC. For that we assumed to have a oca grid Ω h,n suitabe to cover the soution s high activity at t n. Moreover we assumed to know a soution u H,h,n 1 in a the points of the composite grid Ω H,h,n = Ω H Ω h,n at time t n 1. Using this information, we coud compute a composite grid soution at t n on Ω H,h,n := Ω H Ω h,n. Incuding the boundary conditions for the goba and the oca probem into the soution vector, we have an approximation of u at t n in a the points of Ω H,h,n. Here we note that in a time-dependent probem it is ikey that the soution s high activity moves and changes its size as time proceeds. As a consequence, the oca grid Ω h,n used to perform the time step from t n 1 to t n might not be adequate to cover the soution s high activity aso during the foowing time step. In genera, we expect thus to have Ω h,n Ω h,n+1, where Ω h,n+1 is the oca grid used for computing a soution at t n+1. We note that, in order to perform the time step from t n to t n+1, an approximation u H,h,n must be avaiabe at t n in a the points of the composite grid Ω H,h,n+1 = Ω H Ω h,n+1. However, since the soution at t n was computed on another composite grid, namey Ω H,h,n = Ω H Ω h,n, a oca approximation of u at t n is directy avaiabe ony in the common points between Ω h,n and Ω h,n+1. On the remaining part of Ω h,n+1, i.e. ^Ω h,n+1 := Ω h,n+1 \ ( Ω h,n+1 Ω h,n ), (2.27)

28 2.2 The LDC agorithm incuding regridding 19 (a) (b) Figure 2.4: An exampe of composite grid Ω H,h,n at time t n (a) and of composite grid Ω H,h,n at time t n+1 (b). The fine grid points in (b) marked with a cross form ^Ω h,n. a soution at t n has to be computed via interpoation from u H,h,n. Therefore we introduce the operator Q n x : G( Ω H,h,n ) G( ^Ω h,n+1 ), that spatiay (hence the subscript x) interpoates u H,h,n on ^Ω h,n+1. Figure 2.4 represents an exampe in which, from t n to t n+1, the oca region has changed its ocation, its shape and its area. The fine grid points in Figure 2.4-(b) marked with a cross form ^Ω h,n+1 : in these points a fine grid approximation at time t n is not directy avaiabe and it has to be computed through interpoation from u H,h,n. From a practica point of view, we can imagine the operation performed by Q n x to be piecewise inear or piecewise quadratic interpoation. We shoud note that there is no high activity at time t n on ^Ω h,n (by definition ^Ω h,n+1 Ω h,n = ). Therefore the interpoation takes pace in a region with no high gradients. Furthermore if the time step t is sma enough, we can imagine that the area of high activity does not move much between two consecutive time steps, so that interpoation is needed on a sma region ony. Once the interpoation process has taken pace, an approximation of u is avaiabe in each point of Ω h,n+1 at time t n, with n > 1. Its expression is given by { ( ) Q n x u H,h,n, in ^Ω h,n+1, u H,h,n := u H,h,n, in Ω h,n+1 \ ^Ω h,n+1. (2.28) The approximation u H,h,n can be used to perform the next LDC time step. Note that if the region of refinement does not change during two consecutive time steps = Ω h,n ), there is no need for interpoation and u H,h,n = u H,h,n. (Ω h,n+1 A further consideration regards the choice of the oca region Ω n. So far, we have

29 20 Chapter 2 - The LDC method for paraboic partia differentia equations assumed it to be given, but in practice its position is not a priori known. At every time t n it has to be determined on the basis of the soution s features at that time eve. Many techniques have been proposed in the iterature of adaptive methods to detect where refinement is needed. In principe one can use any kind of criterion which is suitabe for one s specific appication. Throughout this thesis we wi use the method proposed in [10, 11, 65]. In [10, 11, 65] a positive weight function w ij is introduced to determine which coarse grid boxes B ij := (x i, x i+1 ) (y j, y j+1 ) require refinement. The weight function w ij is meant to be an indicator of the soution s roughness. The vaues w ij are computed on each box B ij from the gradient of the first coarse grid approximation u H,n 0. After that a smoothing fiter, an averaging and a normaization procedure are appied. At the end the mean vaue of the weight function is 1 and the boxes B ij for which w ij > ǫ are abeed for refinement. The threshod vaue ǫ is a user-specified parameter. Greater numerica vaues of ǫ resut in a tighter oca region, whie smaer vaues of ǫ make the area of Ω n arger. A suitabe vaue for ǫ shoud therefore be determined on the basis of the specific probem that has to be soved. Typica vaues for ǫ range from 1.5 to 3. Further detais on this method can be found in [10, 11, 65]. In Section 2.1 we considered the LDC time step and here we discussed a regridding strategy. We are now ready to formuate the LDC agorithm for soving paraboic partia-differentia equations. Agorithm 2.4 (LDC agorithm for paraboic probems) FOR LOOP, n = 1, 2,..., t end / t INITIALIZATION Provide initia vaues u H,h,n 1 on the coarse grid Ω H. If n = 1, set u H,h,n 1 Ω H from the initia condition. Otherwise, set u H,h,n 1 Ω = H uh,h,n 1 Ω H. Compute a goba coarse grid approximation u H,n 0 soving probem (2.3). Choose a region of refinement Ω n, introduce a fine grid Ω h,n on it and a time step δt = t/τ. Provide initia vaues ϕ H,h,n 1 on the remaining points of Ω H,h,n = Ω H,n Ω h,n. If n = 1, use the initia condition. If n > 1 and Ω H,h,n Ω H,h,n 1, set u H,h,n 1 Ω H,h,n If n > 1 and Ω H,h,n u H,h,n 1 = = u H,h,n 1 Ω. H,h,n 1 Ω H,h,n 1, set Q n x (uh,h,n 1 ), on ^Ω h,n, u H,h,n 1, on Ω H,h,n \ ^Ω h,n.

30 2.2 The LDC agorithm incuding regridding 21 Use u H,n 0 to provide a boundary condition for the oca probem. Compute a oca approximation u h,n,0 soving the oca probem (2.11). ITERATION, w = 1, 2,... Use u h,n,w 1 to compute an estimate d H,n,w 1 of the coarse grid oca discretization error as in (2.13). Compute a more accurate goba approximation u H,n w soving a modified goba probem as in (2.16). Use u H,n w to update the boundary condition for the oca probem. Compute a oca soution u h,n,w with updated boundary conditions. END ITERATION ON w END FOR LOOP ON n The soution on the composite grid at time t n is u H,h,n (remove the subscript that numbers the LDC iterations). Agorithm 2.4 shows that initia vaues u H,h,n 1 are provided in two stages. In order to compute the first coarse grid approximation u H,n 0, it is sufficient to know u H,h,n 1 on Ω H ; for this purpose, the coarse grid restriction of the composite grid soution at the previous time step can be used. Once u H,n 0 is computed and the new region of refinement Ω n is chosen, initia vaues uh,h,n 1 are provided aso at the points of Ω H,h,n that ie in the region of refinement. In this case, space interpoation is necessary whenever the new oca region does not coincide with the oca region used during the previous time step. After soving the fine grid probem too, an iterative procedure is triggered, and a goba and a oca soution at the new time eve iterativey improve each other. A suitabe stopping criterion for the LDC iteration is provided by Lemma 2.1. Note that each LDC iteration consists in the entire recomputation of the time step t. For a good performance in soving the transient probem, it is thus desirabe that ony a sma number of LDC iterations are needed at each time step. However, as it happens in stationary cases, it turns that LDC converges very fast. The convergence behavior of the LDC agorithm for paraboic probems is the topic of Chapter 3 of this thesis. When presenting the LDC method for time-dependent probems we have used the impicit Euer scheme for time discretization. We shoud notice here that this is not restrictive and that other impicit methods for time discretization (e.g. Runge- Kutta schemes) might be appied as we. Moreover we are not constrained to use the same scheme for the goba and for the oca grid. Whie it is possibe to use an expicit method on the fine grid, it is crucia for the effectiveness of LDC that an impicit time integrator is adopted gobay. In fact, if we woud use an expicit

31 22 Chapter 2 - The LDC method for paraboic partia differentia equations time integration scheme on the goba grid, the matrix M H woud be diagona and the matrices B H,Γ, BH Γ,, BH Γ,c, BH c,γ woud be a zero, see (2.7). As a consequence the effect of the defect correction on Ω H,n woud be the repacement of the coarse grid soution with the fine grid approximation. Moreover the coarse grid soution woud not change on the coarse grid points outside the area of refinement. Ceary in such a case LDC coincides with LUGR. One of the most interesting features of the LDC agorithm we are proposing is the possibiity of performing the time integration on the oca region with a time step δt < t. This is a feature, not a requirement of the method; it is very we possibe to choose the same time step on both the goba and the oca grid. In genera we expect however that a soution characterized by reativey high spatia gradients in a certain region of the domain has, in that same region, reativey high vaues of the time derivative too; an intuitive exampe is given by a front propagation. For this reason the use of a time step t on the oca grid might be not adequate to represent the fast phenomena occurring there. Moreover, if we woud perform time integration ocay with the same step as on the goba grid, the discretization error on the fine grid might be heaviy dominated by the tempora component, which woud make the effect of soving the oca probem with a grid size h < H useess. We note that it is common that adaptive grid methods for time-dependent probems provide the possibiity of using smaer time steps ocay, see for instance LUGR or [25, 57]. In the case of LDC this has a further impication: the defect correction, which is used to compute a more accurate goba approximation at t n, improves the coarse grid soution not ony in its spatia component of the error, but aso in the tempora part. In practice the parameters H, t, h and δt are to be chosen on the basis of the physica probem that has to be soved and on the discretization schemes adopted gobay and ocay. 2.3 Numerica experiments In this section we present two numerica exampes. The first one, Section 2.3.1, is a 2D numerica experiment in which the LDC technique is compared with a standard uniform grid sover. We show that LDC can achieve the same accuracy as the uniform grid sover, whie requiring the computation of a smaer number of unknowns and being thus a more efficient method. In the second exampe, Section 2.3.2, we compare LDC and LUGR. We show that LUGR may aready fai to produce reiabe resuts for a simpe 1D probem, whereas LDC eads to an accurate soution. This exampe iustrates the superior robustness of the LDC method over LUGR.

32 2.3 Numerica experiments Exampe 1: a 2D convection-diffusion probem In this section we present the resuts of a 2D numerica experiment. We choose Ω = (0, 1) 2 and Θ = (0, 2], and we sove the foowing probem u t + v u = λ 2 u, in Ω Θ, u = 0, on Ω Θ, (2.29) u = exp ( 100 ( (x 0.3) 2 + (y 0.3) 2)), in Ω, t = 0, where v = v(x, y) = (y 0.5, x + 0.5), see Figure 2.5. Note that L λ 2 v in (2.29); we choose λ = 10 4 > 0 so that L is an eiptic operator. Figure 2.6 shows the contour pots of u for different vaues of time. The soution of probem (2.29) has at each time a region of high activity that covers a imited part of Ω. Figure 2.6 aso shows (dotted ine) the ocation of the oca region Ω n in one of our LDC runs. Probem (2.29) is soved using LDC with different vaues of H, t, h and δt (see Tabe 2.1). In a the LDC runs, one and ony one LDC iteration is performed at each time step t; to reca this, in Tabe 2.1 we write LDC 1, where the subscript indicates the number of LDC iterations at each time step. In a our runs the spatia discretization is performed using finite differences; in particuar the second order centered differences scheme is appied both on the goba and on the oca grid. The time discretization is performed using the first order impicit Euer scheme both gobay and ocay. The position and the size of the oca region are determined at each time step using the aready mentioned agorithm which is described in [10, 11, 65]; we choose a threshod vaue ǫ = 3. For simpicity of impementation in a our runs the oca region Ω n has aways a rectanguar shape. Furthermore we use no safety region when computing the defect. The operator Q n x performs piecewise inear interpoation from the coarse grid vaues. Aso P h,h performs piecewise y x Figure 2.5: Pot of the veocity fied v = (y 0.5, x + 0.5).

33 Chapter 2 - The LDC method for paraboic partia differentia equations y x (a) u at t = y x (b) u at t = y x (c) u at t = 2.0 Figure 2.6: Contour pots of the soution of probem (2.29) and position (dotted ine) of Ω n at different times.

34 2.3 Numerica experiments 25 Grid and time step ǫ Tota number of unknowns H t σ = τ LDC 1 Unif. LDC 1 (coarse + fine) Unif. Unif. LDC H 0 t H 0 t H 0 t Tabe 2.1: Resuts of the 2D numerica experiment. In the tabe: H 0 = 1/20, t 0 = 0.2, and σ = H/h. inear interpoation. As a measure of the accuracy of the numerica soution found at t end = 2 using LDC, we compute the infinity norm ǫ := max (u H,N u(t end ) ) ΩH. (2.30) In (2.30) by u H,N we indicated the restriction on the coarse grid of the composite grid soution at the fina time, whie u(t end ) ΩH is the projection on Ω H of a reference soution that we computed once on a uniform grid using a very sma grid size h ref = 1/2000 and a very sma time step δt ref = These vaues for h ref and δt ref are chosen in such a way that they are a few times smaer than the smaest grid size and the smaest time step in the LDC runs. We expect the resuts of the LDC runs to have approximatey the same accuracy as the numerica soution found soving probem (2.29) on a singe goba uniform grid with grid size h and time step δt. Aso for the singe uniform grid runs we measure the infinity norm (2.30). Resuts in Tabe 2.1 show that in a the cases we considered, LDC is abe to achieve the same accuracy as the uniform grid sover. Of course LDC requires ess computationa effort than the uniform grid sover and it is a more efficient technique since the fine grid and the sma time step are used ony where it is needed. To give a rough idea of the compexity of the two methods, in Tabe 2.1 we aso report the tota number of unknowns that have to be computed to sove probem (2.29). For the uniform grid sover the tota number of unknowns is cacuated as the product

35 26 Chapter 2 - The LDC method for paraboic partia differentia equations of the number of grid points and the number of time steps. For LDC this product is given separatey for the coarse grid and for the sum of a the oca probems; the numbers in Tabe 2.1 aready take into account that in LDC 1 each time step t has to be repeated twice (computation of a first approximation at t n pus one defect correction). From the tabe we can see that in our exampe, when we use LDC, we have to compute a tota number of unknowns which is about three times ess than when the uniform grid sover is adopted. The gain increases when higher factors of grid and time refinement are used or when the high activity is more ocaized than in the exampe presented here. The LDC agorithm can be generaized for 3D probems in a straightforward manner. The gain of LDC in 3D is even higher than in 2D because the number of unknowns is proportiona to the voume rather than the area of the high activity zone. To concude, this exampe shows that LDC can achieve the same accuracy as a uniform grid sover that uses the same grid size and time step as in the LDC oca probem. Yet, LDC is a more efficient method than the uniform grid sover: in LDC the fine grid is adaptivey paced ony where it needs to be and this guarantees a saving in the tota number of unknowns that have to be computed to sove the probem Exampe 2: comparison between LDC and LUGR In this section we present the resuts of a 1D numerica experiment aimed at showing the robustness of the LDC method. We choose Ω = (0, 2) and Θ = (0, 1], and we sove the partia differentia equation with v = 1. We notice that in (2.31) u t + v u x 2 u = f, in Ω Θ, (2.31) x2 L v x + 2 x2. (2.32) Ceary L is thus a convection diffusion operator. The initia condition, the Dirichet boundary conditions and the source term f are chosen in such a way that the exact anaytica soution of the probem is ( u(x, t) = tanh ( 100(x 1/8 t) ) (1 + 1) e 2t ). (2.33) At time t > 0 the exact soution (2.33) has a region of high activity around point x a = 1/8 + t (see Figure 2.7). The probem above is soved using both LDC and LUGR. Equation (2.31) is discretized in space using finite differences; in particuar, both gobay and ocay a second order centered differences scheme is adopted. The time discretization is performed with the first order impicit Euer scheme both on the goba and on the oca

36 2.3 Numerica experiments 27 u t = 1 t = 0.8 t = 0.6 t = 0.4 t = x t = 0 Figure 2.7: Pot of the exact soution (2.33) at different vaues of time t. At each time t > 0, u has a region of high activity around point x a = 1/8 + t. grid. Like in the previous exampe, in a our runs we measure the infinity norm ǫ := max (u H,N u(t end ) ) ΩH, (2.34) where u H,N is the computed numerica soution at t end = 1, whie u(t end ) ΩH is the projection on the coarse grid of the exact soution (2.33) at the fina time. In the exampe in this section the oca region is not determined using the agorithm described in [10, 11, 65]. In this 1D probem, at each time step Ω n is chosen in such a way that it has a constant width, its eft and right bounds coincide with coarse grid points and x a ies in the midde of Ω n. We note that in [62] a precise strategy of refinement is proposed for LUGR and more than one eve of refinement is generay used. However, in order to present a fair comparison between LDC and LUGR, we imit the eves of refinement to one and we use the same strategy to determine the oca region for the two methods. In 1D there is no need to define the operator P h,h. We ony have to define Q n x : we choose it to perform piecewise inear interpoation in space. In order to speed up the convergence of LDC the defect term is computed using a safety region. In a runs of this section, Ω H def is defined as the set of coarse grid points in Ω H,n whose distance from Γ n is at east 2H. The resuts of a first numerica experiment, run #0, are presented in Tabe 2.2. In this run the width of the oca region is 0.6. In the tabe the subscript next to LDC indicates the number of LDC iterations that are performed at each time step (e.g. LDC 3 means three LDC iterations for every t n ). In this experiment we want to test the robustness of the LDC agorithm. For that reason we choose a very coarse goba grid (H = 1/25) and a rather big time step ( t = 1/5). Locay we refine in space and time and we set h = 1/125, δt = 1/25. LUGR is not abe to provide a good boundary condition for the oca probem and it fais dramaticay. LDC, on the other hand, can ead to resuts of order 10 2 accurate if enough LDC iterations (3 in our exampe) are performed at each time step. Thanks to the process of defect correction, LDC proves

37 28 Chapter 2 - The LDC method for paraboic partia differentia equations ǫ N g N LUGR = = 75 LDC = =150 LDC = =225 LDC = =300 Tabe 2.2: Resuts of run #0. N g and N indicate the sum of the dimensions of a the inear systems that have to be soved per time step t on the goba and oca grid respectivey. H t ǫ,lugr ǫ,ldc1 ǫ,lugr ǫ,ldc1 run #1 1/50 1/ run #2 1/100 1/ Tabe 2.3: Coarse grid size, time step t and resuts of run #1 and run #2. In both runs h = H/5 and δt = t/ u Exact LDC LUGR Figure 2.8: Exact soution (2.33) at t end = 1 and numerica approximations computed using LDC and LUGR in run #1. x

38 2.3 Numerica experiments 29 to be a more robust technique than LUGR. Tabe 2.2 aso shows that increasing the number of LDC iterations, we proportionay increase the number of unknowns that have to be computed per time step t: each defect correction (see Agorithm 2.4) means in fact reperforming the entire time step t again and computing new (more accurate) coarse and fine grid approximations. In particuar, we have that N LDC1 N LUGR = 2, (2.35) where N is the tota number of unknowns (goba grid + oca grid) per time step. With tota number of unknowns we mean the sum of the dimensions of a the inear systems that have to be soved per time step. In Tabe 2.3 we present the resuts of other two numerica simuations: run #1 and run #2. In both of them the width of the oca region is 0.2, the factors of grid and time refinement are both equa to 5 and we ony consider the comparison between LUGR and LDC 1. In run #1 we have a situation simiar to run #0: LUGR fais, whie the approximation computed using LDC has an accuracy of order 10 2 (see Figure 2.8). Keeping in mind that we are using a second order method in space and a first order method in time, in run #2 we take H twice as sma and t four times as sma with respect to run #1. LUGR finay gives meaningfu resuts; yet LDC is a factor 5.6 more accurate than LUGR, costing ony twice as much in terms of tota number of unknowns per time step, see (2.35).

39

40 Chapter 3 Convergence properties of LDC In Chapter 2 we introduced the LDC agorithm for paraboic probems and we expained that the method is simiar to the LUGR technique studied in [62]. Moreover, we showed by means of numerica experiments that LDC is more robust than LUGR: this is because in LDC the oca soution improves the soution gobay through a defect correction. The robustness of LDC comes at a cost though, because the defect corrections invove more computationa work. For LDC to be competitive with other techniques, it is thus desirabe that ony a sma number of iterations are necessary at every time step. This chapter is therefore focused on the convergence properties of the LDC method for paraboic probems. In particuar, we are interested in investigating the dependency of the LDC convergence rate on the time step for the coarse grid probem. The convergence properties of LDC have been previousy studied for stationary probems. In [4] a two-dimensiona Poisson s equation is considered: if the defect is computed using a safety region, it is proved that iteration errors reduce proportionay to H 2, where H is the coarse grid size. If no safety region is adopted, resuts of numerica experiments show that iteration errors reduce ineary with H. In [53] a convergence anaysis is carried out for a case where the oca domain has an annuar shape. In genera, even for rather compicated appications, it is observed by many authors (see [1, 2, 29, 33, 47]) that LDC has very good convergence properties and that one or two iterations are usuay sufficient for convergence. The conditions for one-step convergence of LDC for eiptic probems are given in [5]. The anaysis and the one-dimensiona numerica experiments of this chapter have previousy been presented in [46].

41 32 Chapter 3 - Convergence properties of LDC 3.1 The iteration matrix In this section we want to find an expression for the matrix that describes the LDC iteration at time t n. We sighty simpify the notation introduced in Chapter 2. In the previous chapter, the oca domain at time t n was denoted by Ω n, where the superscript n was introduced to take into account that the oca region might be different at different time eves. Throughout this chapter we wi ony consider the LDC iteration at one specific time step; at this specific time step we wi aways suppose the oca region to be given. The superscript n is thus not needed in this context and we wi omit it: the oca domain wi simpy be denoted by Ω. The same is done for other symbos: in this chapter the interface between coarse and fine grid at time t n wi be denoted just by Γ (not Γ n ), the coarse grid points on the interface by Γ H, the coarse grid points in the area of refinement by Ω H, etc. In order to find an expression for the matrix that describes the LDC iteration at time t n, we foow a simiar approach as done in [4] for stationary probems. In [4] the LDC iteration is expressed in terms of the iteration that takes pace on Γ H ony and it is shown that, if the iteration on Γ H converges, then the entire LDC iteration converges. Previousy, other approaches were proposed. In [34] the LDC iteration for eiptic probems is expressed in terms of grid functions that operate on Γ h, i.e. the set of fine grid points on the interface. In [28], with reference to boundary vaue probems, the LDC iteration that takes pace on the whoe set of composite grid points is considered. With u H,h,n defined as in (2.21), we introduce the iteration error of the LDC method by e H,h,n w := u H,h,n w u H,h,n. (3.1) If we subtract (2.20) and (2.22), we can write the expression for successive iteration errors M H,h e H,h,n w = S H,h e H,h,n w 1. (3.2) Note that the convergence of the LDC method does not depend on the source term, on the Dirichet boundary conditions and on the soution at the previous time step. Definitions of M H,h and S H,h enabes us to rewrite (3.2) as ( ) M h τ 0 W,Γ n 0 0 M H B H,Γ 0 0 B H Γ, M H Γ B H Γ,c 0 0 B H c,γ M H c e h,n,w e H,n,w e H,n Γ,w e H,n c,w The first equation of this system yieds 0 = X H MH RH,h e h,n 0 0,w 1 + XH BH,Γ eh,n Γ,w 1. (3.3) e h,n,w = ( (M h )τ) 1 W n,γ e H,n Γ,w. (3.4)

42 3.1 The iteration matrix 33 Repacing w with w 1 in (3.4), we can reformuate system (3.3) as M H B H,Γ 0 B H Γ, M H Γ B H Γ,c e H,n,w e H,n Γ,w e H,n c,w 0 B H c,γ M H c ) 1W X H MH ((M RH,h h )τ n,γ e H,n = 0 or, equivaenty, M H e H,n w = I 0 0 XH This eads to the foowing theorem. 0 Γ,w 1 + XH BH,Γ eh,n Γ,w 1, (3.5) (B H,Γ M H R H,h( (M h ) τ) 1 W n,γ ) e H,n Γ,w 1. (3.6) Theorem 3.1 Consider the foowing iteration that takes pace on the interface ony: e H,n Γ,w = M iter e H,n Γ,w 1, w = 1, 2,... (3.7) in which the iteration matrix M iter : G(Γ H ) G(Γ H ) is defined by M iter := ( 0 I 0 ) I (M H ) 1 0 XH (B H,Γ M H R H,h( (M h ) τ) ) 1 W n,γ. (3.8) 0 If iteration (3.7) converges, then the LDC iteration converges. Proof. It is easy to verify that (3.6) gives (3.7). Equation (3.7) describes the behavior of the component e H Γ,w of the iteration error; the other components can be expressed in terms of e H Γ,w by (3.4) and (3.6). Equations (3.4) and (3.6) show that if eh Γ,w 0 (w ), aso e H w 0 (w ). Theorem 3.1 is the time-dependent equivaent of [4, Theorem 2]. Theorem 3.1 states that, if the iteration that takes pace on the interface Γ H at time t n converges, then the entire LDC iteration at time t n converges to a fixed point. This means that for proving convergence of the LDC method for paraboic probems, it is sufficient to show that the spectra radius of the matrix M iter is ess than one. This is true if M iter < 1. Foowing the same approach as in [4], we spit the iteration matrix M iter according to M iter = M 1 M 2, (3.9)

43 34 Chapter 3 - Convergence properties of LDC where M 1 : G(Ω H ) G(ΓH ) is expressed by M 1 = ( 0 I 0 ) I (M H ) 1 0, (3.10) 0 and M 2 : G(Γ H ) G(Ω H ) by M 2 = X H (B H,Γ M H R H,h( (M h ) τ) 1 W n,γ ). (3.11) In the next section we introduce a one-dimensiona mode probem. For such a probem and for a particuar choice of the grids and the discretization schemes, the properties of M iter can be studied anayticay. 3.2 A one-dimensiona mode probem We appy the LDC method to sove the one-dimensiona heat equation u(x, t) = 2 u(x, t) t x 2 + f(x, t), in Ω = (0, 1), for t > 0, u(0, t) = ψ eft (t), for t > 0, u(1, t) = ψ right (t), for t > 0, u(x, 0) = ϕ 0 (x), in Ω = [0, 1], (3.12) where f(x, t), ψ eft (t), ψ right (t) and ϕ 0 (x) are given functions. With reference to probem (3.12), we study the convergence behavior of LDC at a generic time step t n. The LDC method is appied with the foowing settings: the goba uniform grid has grid size H = 1/N (N integer and N > 1) and grid points Ω H = {ih i = 1, 2,..., N 1}. (3.13) On Ω H we perform spatia discretization by finite differences; in particuar, we adopt the standard three-point centered differences scheme to approximate 2 / x 2. The tempora discretization is performed by the backward Euer scheme with a time step t. In this way the coarse grid operator M H is expressed by M H = I t L H = I t H (3.14)

44 3.2 A one-dimensiona mode probem 35 We et the oca region be Ω = (0, γ), with γ a mutipe of H such that 0 < γ < 1. In our anaysis we wi repace the discrete operator M h by the continuous operator M := t 2 x2. (3.15) This corresponds to etting h 0 and δt 0. This is done for anaysis purposes ony; in practice one wi aways have h > 0 and δt > 0. However, the resuts presented in [1] for stationary diffusion probems and the numerica experiments in Section 3.4 of this thesis support this approach. In [1] the LDC iteration matrix is studied both for a continuous (h = 0) and for a discrete oca probem (h > 0): the two approaches ead to the same concusions. Note that in our one-dimensiona setting the space interpoation operator P h,h reduces to the identity function. Aso note that, in 1D and with Ω = (0, γ), the set Γ H reduces to one point. As a consequence, the two operators M 1 and M 2 turn out to be a row and a coumn vector respectivey, whie M iter is their inner product, so a rea number. In this context, when writing M 1 or M 2 we wi therefore mean the standard vector infinity norm, whie M iter is the absoute vaue of the rea number M iter. For the mode probem iustrated here, we wi determine bounds for M 1 in Section and an expression for M 2 in Section Before that we emphasize the fact that, in our anaysis, we wi aways assume γ to be a given mutipe of H such that 0 < γ < 1. The specia cases γ = 0 and γ = 1 are of minor interest: if γ = 0, the oca region Ω reduces to the eft boundary point and we have no defect, whie the case γ = 1 is not interesting because Ω coincides with the goba domain Bounds for the M 1 infinity norm In this section we consider the operator M 1 as defined in (3.10), with M H given by (3.14). It is easy to verify that M 1 (M H ) 1. (3.16) Lemma 3.2 provides a first bound for the infinity norm of (M H ) 1. Lemma 3.2 With M H given by (3.14), the foowing bound for (M H ) 1 hods (M H ) 1 1. (3.17) Proof. We express M H as M H = I t L H = (1 + 2d)(I + B), (3.18)

45 36 Chapter 3 - Convergence properties of LDC where the scaar d 0 and the matrix B are given by d = t H 2, B = d d 1+2d 0 1+2d. (3.19) It is easy to verify that We write and B = 2d < 1. (3.20) 1 + 2d (I + B) 1 = I B + B 2 B 3 + (3.21) (I + B) 1 I + B + B 2 + B B = 1 + 2d. (3.22) Since we deduce (M H ) 1 = (I tl H ) 1 = d (I + B) 1, (3.23) (M H ) 1 1. (3.24) A second bound for the infinity norm of (M H ) 1 is the resut of Lemma 3.3. Lemma 3.3 With M H given by (3.14), the foowing bound for (M H ) 1 hods (M H ) 1 1 t. (3.25) Proof. Since matrix M H R N 1,N 1 is symmetric, for the symmetric diagonaization theorem, see for exampe [48, Theorem 4 at page 458], it can be written as M H = Q D Q T, (3.26) where Q R N 1,N 1 is orthogona and D R N 1,N 1 is diagona. The diagona entries of D are d j = t H 2 sin2 (jπh/2), j = 1, 2,..., N 1, (3.27) and they coincide with the eigenvaues of M H. Note that the smaest eigenvaue is d 1. The orthogona matrix Q has entries q ij given by q ij = 2H sin(i j πh), i, j = 1, 2,..., N 1. (3.28)

46 3.2 A one-dimensiona mode probem 37 From (3.28) we note that Q is aso symmetric. We can thus write It can easiy be shown that Q = max i D 1 Q = max i (M H ) 1 = Q D 1 Q. (3.29) N 1 q ij (N 1) 2H, (3.30) 1 d i j=1 N 1 j=1 q ij (N 1) 2H 1. (3.31) d 1 We note that, since we chose the integer N > 1, then H = 1/N is such that 0 H 1/2. Using the foowing inequaity ( ) sin 2 πh H, for 0 H 1/2, (3.32) 2 we can show that 1 1 = d t/h 2 sin 2 (πh/2) t/h H 4 t. (3.33) Combination of (3.29), (3.30), (3.31) and (3.33) finay yieds (M H ) 1 Q D 1 Q (N 1) 2 2H t 1 t. (3.34) Resuts of Lemmas 3.2 and 3.3 are iustrated in Figure 3.1, where (M H ) 1 is potted as a function of t for two different vaues of the grid size H. Formua (3.16) and the resuts of Lemma 3.2 and Lemma 3.3 are combined in Theorem 3.4. Theorem 3.4 The foowing bound for M 1 hods M 1 min(1, 1 ). (3.35) t Note that the bound provided by Theorem 3.4 is independent of H and thus it hods for any choice of the coarse grid size The expression of the M 2 infinity norm In this section we find an expression for the infinity norm of M 2. We et g H,n Γ R be the soution found at point x = γ by performing one time step t on the coarse grid. From the definition of M 2, see (3.11), we have that M 2 g H,n Γ = X H ( B H,Γ g H,n Γ + M H R H,h u n), (3.36)

47 38 Chapter 3 - Convergence properties of LDC H=10 1 H=10 2 min(1,1/ t) t Figure 3.1: Pot of (M H ) 1, with M H defined by (3.14), as a function of t for two vaues of the grid size H and pot of the bounds provided by Lemma 3.2 and Lemma 3.3. where u n = ( (M h )τ) 1 W n,γ g H,n Γ. (3.37) In view of (2.11) and the assumptions we made on (M h )τ, it foows that u n is the soution of the one-dimensiona heat equation u(x, t) t u(0, t) = 0, u(γ, t) = g H,n t Γ t, u(x, 0) = 0, = 2 u(x, t) x 2, for x (0, γ), t (0, t], for t (0, t], for t (0, t], for x [0, γ]. (3.38) In order to find an expression for M 2, we want to find the exact anaytica soution of probem (3.38). For that purpose we introduce the auxiiary function ( 1 v(x, t) := u(x, t) g H,n Γ 6γ t x3 γ 6 t x + xt ). (3.39) γ t Combining (3.38) and (3.39), it can be proved that v satisfies v(x, t) = 2 v(x, t) t x, for x (0, γ), t (0, t], v(0, t) = v(γ, t) = 0, for t (0, t], v(x, 0) = gh,n Γ 6γ t x3 + gh,n Γ γ x, for x [0, γ]. 6 t (3.40)

48 3.2 A one-dimensiona mode probem 39 Probem (3.40) can be soved using the technique of separation of variabes. We express its soution in the form ( ) v(x, t) = gh,n Γ v m e m2 π 2 t/γ 2 mπx sin, (3.41) t γ m=1 where the coefficients v m are to be computed from the initia condition ( ) v(x, 0) = gh,n Γ mπx v m sin = gh,n Γ t γ 6γ t x3 + gh,n Γ γ x. (3.42) 6 t m=1 We find v m = 2 t γ γ g H,n v(x, 0) dx = 2( 1)m γ 2 Γ 0 m 3 π 3. (3.43) Using (3.41), the soution of the origina probem (3.38) turns out to be u(x, t) = gh,n Γ t ( ( v m e m2 π 2 t/γ 2 mπx sin γ m=1 ) + 1 6γ x3 γ 6 x + xt γ Soution (3.44) wi now be used to express the product M 2 g H,n Γ. ). (3.44) Equation (3.36) states that M 2 g H,n Γ equas the residua of the coarse grid discretization scheme (centered differences and impicit Euer) appied to the function u(x, t) for a grid points x Ω H def. Taking aready in account that u(x, 0) = 0, we have M 2 g H,n Γ (x) = u(x, t) t (u(x + H, t) 2u(x, t) + u(x H, t)). (3.45) H2 We use the identity ( ) mπ(x + H) sin 2 sin γ m=1 ( mπx γ ( mπh = 4 sin 2γ ) + sin ) sin ( ) mπ(x H) γ ( ) (3.46) mπx to combine (3.44) and (3.45) into ( M 2 g H,n Γ (x) = gh,n ( ) ) Γ v m e m2 π 2 t/γ 2 mπx sin + 1 t γ 6γ x3 γ 6 x + 4gH,n Γ H 2 ( ) v m e m2 π 2 t/γ 2 mπh sin sin 2γ m=1 γ ( mπx γ ). (3.47) Assuming that the function u is sufficienty smooth, another expression for the product M 2 g H,n Γ (x) can be derived using Tayor expansions on the right hand side of (3.45) and the fact that u satisfies the partia differentia equation in (3.38). We obtain M 2 g H,n Γ (x) = 1 2 u (x,ϑ) 2 t 2 t u (ξ, t) 12 x 4 t H 2, (3.48)

49 40 Chapter 3 - Convergence properties of LDC with x H < ξ < x + H and 0 < ϑ < t. The time and spatia derivatives of u can be computed differentiating term by term the series in (3.44). A sufficient condition for this is that the resuting series are absoute convergent. We obtain 2 u (x,ϑ) t 2 = gh,n Γ t 4 u (ξ, t) x 4 = gh,n Γ t 2π γ 2 2π γ 2 ( ( 1) m m e m2 π 2 ϑ/γ 2 mπx sin γ m=1 m=1 ), (3.49) ( ) ( 1) m m e m2 π 2 t/γ 2 mπξ sin. (3.50) γ Ceary the series on the right hand side of (3.49) and (3.50) are absoute convergent for any positive t and ϑ. 3.3 Iteration matrix norm asymptotics With reference to the one-dimensiona heat equation (3.12) and the settings iustrated in Section 3.2, in Section we study the asymptotic expression of M iter for t 0, whie in Section we dea with M iter when t +. The imit case t + corresponds to the stationary case imit; this is of interest in practica appications when a time-dependent probem is soved to compute a stationary soution The imit for sma vaues of the time step t Combination of (3.9) with the resut of Theorem 3.4 yieds M iter M 1 M 2 M 2, for t < 1. (3.51) As a consequence, in our anaysis for t 0 we wi ony consider the infinity norm of M 2. In particuar, we wi focus on the expression for M 2 as given by (3.48). In the perspective of studying M 2 for t 1, we first sove a preiminary probem. We introduce the foowing series (cf. (3.49) and (3.50)) ( mπx ) S x (ϑ) := ( 1) m s m sin, (3.52) γ where m=1 s m := m e m2ϑ, (3.53) and we study the asymptotic behavior of S x (ϑ) for ϑ 0 and fixed 0 < x < γ. Using the fact that s m = s m for a integers m, we rewrite (3.52) as S x (ϑ) = 1 2i + m= m e m2ϑ e imπ(x/γ 1). (3.54)

50 3.3 Iteration matrix norm asymptotics 41 If a function f(y) is sufficienty smooth, the foowing reation, which is known in the iterature (cf. for instance [59]) as Poisson summation formua, hods: + m= f(m)e imω = + k= ^f(ω + 2πk). (3.55) The term ^f which appears in (3.55) is the Fourier transform of f(y), defined by For we have ^f(ω) := + f(y)e iyω dy. (3.56) f(y) = ye y2ϑ, (3.57) ^f(ω) = i π1/2 ω 2 e ω /(4ϑ). (3.58) 2ϑ3/2 Using Poisson summation formua, we can thus rewrite (3.54) as + ( ) S x (ϑ) = π3/2 x 4ϑ 3/2 γ 1 + 2k exp ( π2 (x/γ 1 + 2k) 2 ). (3.59) 4ϑ k= If we introduce the new variabes y := 1 2 ( 1 x ), α := π2 γ ϑ, (3.60) the origina probem for S x (ϑ) can be reformuated as foows: study the asymptotics of T y (α) := α3/2 2π 3/2 + k= (k y)e α(k y)2 (3.61) for α + and 0 < y < 1/2. For that, we need the resuts of the foowing emma. Lemma 3.5 The foowing identity hods: im α + k 2 e kαβ = 0, β > 0. (3.62) α + k=1 Proof. Note that by assumption β is positive. Starting from the identity it can be easiy shown that α + k=1 + k=1 e kαβ = e αβ, α > 0, (3.63) 1 e αβ k 2 e kαβ = α e αβ (1 + e αβ ) (1 e αβ ) 3, α > 0. (3.64)

51 42 Chapter 3 - Convergence properties of LDC Caim (3.62) foows immediatey since the imit for α + of the right hand side of (3.64) is 0. The resut of Lemma 3.5 is used in the proof of Theorem 3.6, which provides the asymptotic expression of T y (α) for α +. First we introduce the foowing notation: we say that f(α) is asymptoticay equivaent to g(α) for α α 0 and we write f(α) g(α), (α α 0 ), (3.65) if f(α) im = 1. (3.66) α α 0 g(α) Theorem 3.6 The foowing equivaence hods: with 0 < y < 1/2. T y (α) 1 2π 3/2 α3/2 ye αy 2, (α + ), (3.67) Proof. For proving (3.67) it is sufficient to show (cf. (3.61)) that im α ye αy2 k= (k y)e α(k y)2 = 1, with 0 < y < 1/2. (3.68) The series in (3.68) can be written in a more convenient way as foows: + k= (k y)e α(k y)2 = ye αy2 + = ye αy2 y = ye αy2 (1 + + k=1 + k=1 We proceed showing that and im α im α 1 y + k=1 ( e α(k y)2 + e α(k+y)2) + + e αk2 ( e 2αky + e 2αky) 1 y + k=1 + k=1 ((k y)e α(k y)2 (k + y)e α(k+y)2) k=1 k (e α(k y)2 e α(k+y)2) + k=1 k e αk2 ( e 2αky e 2αky)). (3.69) e αk2 ( e 2αky + e 2αky) = 0, 0 < y < 1/2, (3.70) k e αk2 ( e 2αky e 2αky) = 0, 0 < y < 1/2. (3.71)

52 3.3 Iteration matrix norm asymptotics 43 We start from (3.70). We notice that a the terms of the series in (3.70) are positive. The idea is thus to find an upper bound for the sum of the series which goes to 0 as α +. We write 0 + k=1 e αk2( e 2αky + e 2αky) 2 = 2 + k=1 + k=1 e αk2 e 2αky 2 + k=1 e αk e 2αky ( e α(1 2y) ) k 2 e α(1 2y) 1 e α(1 2y). (3.72) Ceary, if 0 < y < 1/2, the ast term in (3.72) goes to 0 for α +. This proves equation (3.70). We adopt a simiar strategy for (3.71). Aso in this case we dea with a series with positive terms. First we consider the foowing inequaity e 2αky e 2αky = +2αky 2αky e ζ dζ 4αkye 2αky, (3.73) and then we use it to find an upper bound for the sum of the series in (3.71). We write 0 1 y + k=1 + 4α k=1 k e αk2( e 2αky e 2αky) + 4α + k 2 e αk e 2αky = 4α k=1 k=1 k 2 e kα(1 2y), k 2 e αk2 e 2αky (3.74) with 0 < y < 1/2. We know from Lemma 3.5 that the ast term goes to 0 for α +. This proves (3.71). Combination of (3.69), (3.70) and (3.71) proves (3.68) and hence Caim (3.67). If we adopt again the od variabes x and ϑ, see (3.60), the resut of Theorem 3.6 can be rewritten as S x (ϑ) g(x, ϑ), (ϑ 0), (3.75) with g(x, ϑ) := π3/2 4ϑ 3/2 ( 1 x ) e π2 (1 x/γ) 2 /ϑ γ (3.76) and 0 < x < γ. In Figure 3.2, g(x, ϑ) is potted as a function of x for different vaues of ϑ. Equivaence (3.75) is used in the proof of the foowing theorem. Theorem 3.7 S x (ϑ) has the foowing properties: 1) S x (ϑ) goes exponentiay to zero as ϑ 0; 2) S x (ϑ) goes exponentiay to zero as ϑ 0 non-uniformy on 0 < x < γ;

53 44 Chapter 3 - Convergence properties of LDC ϑ=0.2 ϑ=0.1 ϑ=0.05 ϑ= x Figure 3.2: Pot of g(x, ϑ) as a function of x for different vaues of ϑ. The graph is drawn for γ = ) given a constant ǫ independent of x and ϑ and such that 0 < ǫ < γ, S x (ϑ) goes exponentiay to zero as ϑ 0 uniformy on the interva 0 < x < γ ǫ. Proof. Caim 1) can be proved using equivaence (3.75) and the fact that g goes exponentiay to zero as ϑ 0. For proving Caim 2), it is sufficient to show that sup S x (ϑ) 0<x<γ is not imited for ϑ 0. For 0 < x < γ and any fixed ϑ > 0, g is ony a function of x, it is aways positive and it has exacty one maximum (see Figure 3.2), i.e. with g max := g(x max ) = x max := γ ( 1 π 2e 1 4ϑ, (3.77) ) ϑ. (3.78) 2π Ceary g max is not imited for ϑ 0. As a consequence, because of equivaence (3.75), aso the supremum of S x (ϑ) is not imited for ϑ 0 and 0 < x < γ. This proves Caim 2). For proving Caim 3), we have to show that, for 0 < ǫ < γ and in the imit for ϑ 0, sup S x (ϑ) 0<x<γ ǫ

54 3.3 Iteration matrix norm asymptotics 45 remains imited. On the eft of x max, the function g is positive and monotonicay increasing (see again Figure 3.2). Moreover we note that im x max = γ. (3.79) ϑ 0 Hence, for 0 < ǫ < γ, the foowing identity hods in the imit for ϑ 0 sup g(x, ϑ) = g(γ ǫ, ϑ). (3.80) 0<x<γ ǫ The right hand side of (3.80) is imited for any positive vaue of ϑ and it goes to zero as ϑ 0. Because of that and of equivaence (3.75), we deduce Caim 3). At this point we have studied the properties of S x (ϑ) for ϑ 0. The resuts found for S x (ϑ) are used to state the properties of M iter for ϑ 0. Theorem 3.8 Consider the LDC method for the one-dimensiona heat probem (3.12). Consider the settings described in Section 3.2. In particuar, et Ω H be a uniform coarse grid with grid size H and, in Ω H, approximate the second space derivative by the standard three-point finite differences scheme. Perform the tempora discretization with the backward Euer scheme and time step t. Let the oca region be Ω = (0, γ), with γ a mutipe of H such that 0 < γ < 1. Locay repace the discretized operator M h with the continuous operator (3.15). Moreover, et ǫ be a constant independent of x and ϑ, and such that H < ǫ < γ. Finay, et Ω H def, the subset of ΩH in which an approximation of the oca discretization error is computed, be given by Then, the foowing resuts hod for M iter : Ω H def = (0, γ ǫ) ΩH. (3.81) 1) for any H 0, M iter goes exponentiay to zero as t 0; 2) for any H 0, M iter goes exponentiay to zero as t 0 uniformy on Ω H def. Proof. For t sma enough, combination of (3.51) with (3.48), (3.49) and (3.50) yieds M iter M 2 π t ( ) π 2 γ S ϑ x γ 2 + π H 2 ( ) π 2 6γ 2 t S t ξ γ 2, (3.82) with x Ω H def, x H < ξ < x + H and 0 < ϑ < t. Caims 1) and 2) foow immediatey using the resuts of Theorem 3.7. The fact that the norm of the iteration matrix goes to zero as t 0 is very natura. We expect this to happen in genera and not ony for the mode probem considered here. We expain this as foows: if the time step t becomes extremey sma, the soution at the new time step becomes very cose to the soution at the previous time

55 46 Chapter 3 - Convergence properties of LDC eve. In such a situation, there is itte to be corrected in the approximation at the new time step and, as a consequence, the LDC convergence rate is extremey fast. As a fina remark, we note that one basic assumption in the proof of Theorem 3.8 is the use of a safety region. As discussed ater in Sections and 3.4, this assumption is essentia to have an exponentia rate of convergence for t 0. Nevertheess, the LDC method turns out to be convergent aso if the extent of the safety region is zero. This is simiar to what happens in stationary cases. In [1, 4] the LDC iteration error for a 2D Poisson probem is proved to reduce proportionay to H 2 if ǫ > 0. However, LDC is shown to be convergent (at a ower rate) aso if ǫ = The stationary case imit In the imit case t +, we expect the LDC convergence rate for the 1D heat equation to be the same as the LDC convergence rate for the one-dimensiona Poisson s equation. When LDC is appied to a 1D Poisson probem in combination with centered differences or any other method that integrates inear functions exacty, the method reaches a fixed point in one iteration for any grid size H > 0. (If H = 0, the stationary probem is soved exacty and there is no need to use LDC). This is proved in [5] and means that, in such a case, the iteration matrix is zero. In the foowing theorem we provide a bound for M iter that hods in the imit for t + and any grid size H > 0. Ceary our resuts for the one-dimensiona heat equation fit into the theory of LDC for 1D stationary probems. Theorem 3.9 Consider the LDC method for the one-dimensiona heat probem (3.12). Consider the settings described in Section 3.2 and a grid size H > 0. Then, for any γ such that 0 < γ < 1, there exists a constant C such that M iter C t2, ( t + ). (3.83) Proof. For t > 1, combination of definition (3.9) with the resut of Theorem 3.4 yieds M iter 1 t M 2. (3.84) Hence, for proving Theorem 3.9 it is sufficient to show that, for any γ (0, 1), there exists a constant C such that We consider M 2 as expressed by (3.47) and we write M 2 C, ( t + ). (3.85) t M 2 (x) S 1 + S 2 + Spoy, (3.86)

56 3.3 Iteration matrix norm asymptotics 47 with S 1 := 1 t S 2 := 4 H 2 ( ) v m e m2 π 2 t/γ 2 mπx sin, (3.87) γ m=1 m=1 ( ) v m e m2 π 2 t/γ 2 mπh sin sin 2γ ( mπx γ ), (3.88) S poy := 1 ( 1 t 6γ x3 γ ) 6 x. (3.89) With v m given by (3.43), we prove that S 1 and S 2 are o(1/ t), ( t + ). We start from S 2. We write S 2 4 H 2 v m e m2 π 2 t/γ 2 ( ) ( ) mπh mπx sin sin 2γ γ m=1 (3.90) 8γ2 π 3 H 2 e mπ2 t/γ 2 8γ 2 e π 2 t/γ 2 = π 3 H 2. 1 e π2 t/γ 2 m=1 For any H > 0 and any γ (0, 1), the ast term in (3.90) is o(1/ t), ( t + ). A simiar procedure can be used to show that S 1 is o(1/ t), ( t + ). Having proved that two of the terms on the right hand side of (3.86) are o(1/ t), ( t + ), we dea now with the third one. Ceary, with the assumptions made, Spoy is O(1/ t), ( t + ). Therefore, there exists a constant C such that S poy C, ( t + ). (3.91) t This competes our proof Pots of the iteration matrix norm In Sections and we studied the asymptotic behavior of M iter for t 0 and t + and we found that, in both cases, the imit of M iter is zero. In other words, we know two imit situations in which the rate of convergence of LDC becomes extremey fast. This does not mean, however, that the LDC method aways converges. In this section we provide arguments in favor of the conjecture that, for our mode probem and settings, M iter is ess than one for any choice of H and t, and thus the LDC method is unconditionay convergent. For the one-dimensiona heat equation and the LDC settings discussed in Section 3.2, we can compute M iter R expicity. For that we use the origina definition (3.9) and we express M 1 by (3.10), with M H as in (3.14), and M 2 by (3.47). In Figure 3.3 we pot M iter as a function of t for different vaues of H and for γ = 0.5. In Figure 3.3-(a) M iter is computed with a safety region ǫ = 0.15, whie in Figure 3.3-(b) no safety region is adopted. We immediatey note that in both cases the

57 48 Chapter 3 - Convergence properties of LDC H=10 1 H=10 2 H=10 3 t t (a) Safety region ǫ = H=10 1 H=10 2 H=10 3 t 2 t t (b) No safety region (ǫ = 0) Figure 3.3: With reference to the mode probem and settings of Section 3.2, pot of M iter versus t for different vaues of H and for γ = 0.5; M iter is computed expicity from its definition t=10 8 t=10 7 C H H Figure 3.4: Pot of M iter ( t = 10 8 ) and M iter ( t = 10 7 ) as in Figure 3.3-(b) in terms of the grid size H.

58 3.4 One-dimensiona numerica experiments 49 maximum of M iter is aways ess than 1, which means that the LDC method is aways convergent. Moreover, for sma and big vaues of t, we observe that the asymptotic behavior is in agreement with the bounds stated in Theorems 3.8 and 3.9. For t 1 indeed LDC iteration errors reduce proportionay to t 2. When t tends to zero, M iter goes very rapidy to zero if we use a safety zone, see Figure 3.3-(a). Figure 3.3-(b) indicates that we shoud expect LDC to be convergent aso when no safety region is empoyed; in this case the iteration error reduces proportionay to t 2 when t 0. Figure 3.4 iustrates the dependency of M iter with respect to H when ǫ = 0 and t 1. For fixed time step, M iter is proportiona to H One-dimensiona numerica experiments In this section we further verify the resuts of Theorems 3.8 and 3.9 by means of some numerica experiments. One of the assumptions in the anaysis carried out in Sections 3.2 and 3.3 is that both the oca grid size h and the oca time step δt are zero. This assumption is introduced for anaysis purposes ony, namey for being abe to find the anaytica soution of the oca probem (3.38). The numerica experiments in this section, performed with positive vaues of h and δt, wi show that the resuts of Theorems 3.8 and 3.9 sti hod for a discrete oca probem. In this section we aso test the infuence of the safety region ǫ on the rate of convergence of the LDC method and we try to observe the convergence behavior of Figures 3.3 and 3.4 for a concrete exampe. We consider the appication of the LDC method to the one-dimensiona heat probem u(x, t) = λ 2 u(x, t) t x 2 + f(x, t), in Ω = (0, 1), for t > 0, u(0, t) = u(1, t) = 0, for t > 0, (3.92) u(x, 0) = exp ( 50(x 0.25) 2), in [0, 1], with λ = 0.01 and ( f(x, t) = 5 exp 50 ( x e t) ) 2. (3.93) In (3.92) the choice of initia condition, boundary conditions, diffusion coefficient and source term is such that the soution u has a region of high activity on the eft haf of the spatia domain. For this reason, we take the area of refinement as Ω = (0, γ), with γ = 0.5. In this way, as aready noted before, the set Γ H reduces to one point (x = γ) and M iter to a rea number. For the soution on the goba coarse grid we adopt the same settings as in Section 3.2: the spatia discretization is performed by centered differences, whie the Euer backward scheme is used for the tempora

59 50 Chapter 3 - Convergence properties of LDC H=1/20 H=1/60 H=1/180 t t (a) Safety region ǫ = H=1/20 H=1/60 H=1/180 t 2 t t (b) No safety region (ǫ = 0) Figure 3.5: Pot of M iter ( t) for different vaues of H as computed by soving the heat probem (3.92) M iter ( t=10 4 ), ε=0 C H H Figure 3.6: Pot of M iter ( t = 10 4 ) as in Figure 3.5-(b) in terms of the grid size H.

60 3.4 One-dimensiona numerica experiments 51 discretization. For the soution on the oca fine grid we empoy the same numerica schemes as on the goba coarse grid. We run tests aimed at measuring the convergence rate of the LDC method during one time step. We adopt the foowing strategy: starting from the initia condition, we perform one LDC time step with a chosen grid size H and a chosen time step t. The oca grid size and time step are taken as h = H/5 and δt = t/5. For measuring M iter, we act as foows: at every LDC iteration, we store the coarse grid soution on the interface Γ H. We say that the LDC iteration has converged when u H,1 Γ,w u H,1 Γ,w 1 < toerance, (3.94) for a certain w 1. In (3.94), u H,1 Γ,w denotes the soution on ΓH after one time step and w LDC iterations. In our numerica experiments, we set the vaue of the toerance to Once the converged vaue on the interface is known, the error e H Γ,w can be computed for every iteration w that has been performed. Finay, see (3.7), M iter is given by M iter = eh Γ,w e H, (3.95) Γ,w 1 for a certain w 1. In practice we take w = 1 in (3.95). Note that, if the LDC method converges in exacty one iteration, M iter = 0. We run two series of experiments on probem (3.92), the first one with a safety region, the second one with no safety region. The resuts are dispayed in Figure 3.5. In Figure 3.5-(a) M iter is potted as a function of t for different vaues of the coarse grid size H. Except for the fact that the oca probem is not soved anayticay but numericay, in this numerica experiment a the assumptions of Theorems 3.8 and 3.9 are satisfied. Note that the extent of the safety region (ǫ = 0.1) is greater than H, for every H considered. As expected, M iter goes very rapidy to zero for sma vaues of t whie, for big vaues of the time step, iteration errors reduce proportionay to t 2. For each vaue of H considered in the experiment, the maximum of M iter is aways beow 10 1 ; this means that, even in the worst case, the error e H Γ,w is reduced by a factor bigger than 10 at every LDC iteration. Figure 3.5-(b) refers to the experiment with no safety region. The behavior for t 1 is the same as before; in Theorem 3.9, in fact, no assumption is made on the extent of ǫ. For t 1 iteration errors reduce proportionay to t 2. Note that, aso in this case, the maximum of M iter is aways beow Overa the graphs in Figure 3.5 are quaitativey very simiar to the ones in Figure 3.3, where M iter was computed directy from the definition (3.9). In Figure 3.6 we finay pot, as a function of H, the vaues of M iter as they are computed by soving probem (3.92) with t = 10 4 and ǫ = 0. As aready iustrated in Figure 3.4, with fixed (and sma) t and no safety region, M iter is O(H 4 ). So far we have ony considered pure diffusion probems. Here we woud aso ike to investigate, by means of a numerica experiment, the LDC rate of convergence when the method is appied to a one-dimensiona convection-diffusion probem. We

61 52 Chapter 3 - Convergence properties of LDC H=1/20 H=1/100 H=1/500 t t (a) Safety region ǫ = H=1/20 H=1/100 H=1/500 t 2 t t (b) No safety region (ǫ = 0) Figure 3.7: Pot of M iter ( t) for different vaues of H as computed by soving the convection-diffusion equation (3.96) M ( t=10 6 ), ε=0 iter M ( t=10 6 ), ε=0.1 iter H 2 1/500 1/100 1/20 H Figure 3.8: Pot of M iter ( t = 10 6 ) as in Figure 3.7 in terms of the grid size H.

62 3.5 Two-dimensiona numerica experiments 53 consider the foowing partia differentia equation u(x, t) u(x, t) + v = λ 2 u(x, t) t x x 2 + f(x, t), in Ω = (0, 1), for t > 0, (3.96) with v = 0.1. Equation (3.96) is soved with the same initia condition, boundary conditions, diffusion coefficient and source term as probem (3.92). The convection and the diffusion term are discretized by second-order centered differences, whie the time discretization is performed by the backward Euer scheme. Moreover the same oca region is adopted as before, and the same strategy to compute M iter is empoyed. Note that in (3.96) convection is the main way of heat transport since the reative weight of convection with respect to diffusion is v λ = 0.1 = 10. (3.97) 0.01 Like before, we run two sets of numerica experiments, one with a safety region ǫ = 0.1 and another one with no safety region. The resuts are dispayed in Figure 3.7. For sma vaues of the time step t the rate of convergence M iter has the same behavior as for pure diffusion probems. Aso the maximum vaue of M iter ( t) is, for every H, about the same (ess than 10 1 ) as observed before. For big vaues of t, however, we note that M iter first decreases as O( t 2 ), but then an asymptotic vaue (greater than 0) is reached. To iustrate the dependency of the asymptotic vaue on the grid size, in Figure 3.8 we pot, as a function of H, M iter ( t = 10 6 ) as computed in the two sets of experiments on probem (3.96). From the graph we can see that, in both cases, the asymptotic vaue is proportiona to the square of the coarse grid size. 3.5 Two-dimensiona numerica experiments In this section we woud ike to study the convergence properties of LDC for twodimensiona paraboic partia differentia equations experimentay. We wi consider both a pure diffusion and a convection-diffusion probem. Like before, our goa is to measure the convergence rate of LDC during one time step. We note that in 2D M iter is indeed a matrix: its dimension, see (3.8), coincides with the number of coarse grid points in Γ H. We start considering the two-dimensiona heat probem u t = λ 2 u + f, in Ω = (0, 1) 2, for t > 0, u = 0, on Ω, for t > 0, ( ( u = exp 20 (x 0.3) 2 + (y 0.3) 2)), in Ω, (3.98) with λ = 10 2 and f = exp ( 40( (x e t ) 2 + ( y e t ) 2 )). (3.99)

63 54 Chapter 3 - Convergence properties of LDC H=1/64 H=1/128 H=1/ t (a) Safety region ǫ = H=1/64 H=1/128 H=1/256 t t (b) No safety region (ǫ = 0) Figure 3.9: Pot of ρ 2 in terms of t for different vaues of H. The vaues of ρ 2 are computed experimentay performing one time step t with LDC on probem (3.98) dt=5.5e 06 dt=1.3e 05 C H H Figure 3.10: Pot of the vaues of ρ 2 as in Figure 3.9-(b) in terms of H for two vaues of t.

64 3.5 Two-dimensiona numerica experiments 55 ǫ = 0.1 ǫ = 0.0 H ρ 2 ( t = 10 8 ) ratio ρ 2 ( t = 10 8 ) ratio 1/ / / Tabe 3.1: Vaues of ρ 2 with t = 10 8 as in Figure 3.9. In (3.98) boundary conditions, initia condition, source term and diffusion coefficient are chosen in such a way that u has a ocaized region of high activity. The area of high activity can be covered taking Ω = (0, γ) 2, with γ = 0.5. Like before, we perform one time step using LDC. Both on the goba and on the oca grid the Lapacian operator is approximated by the standard five-point centra formua, whie the time derivative is approximated by the backward Euer scheme. We test the LDC convergence with different coarse grid sizes H and different time steps t. In a tests we set h = H/2 and δt = t/2. The numerica experiments are run both with a safety region of extent ǫ > 0 and with no safety region (ǫ = 0). By safety region of extent ǫ we mean that Ω H def = { (x, y) Ω H x < γ ǫ y < γ ǫ }. (3.100) In other words, the defect term is computed ony in the coarse points Ω H whose distance from the interface Γ is arger that ǫ. In our runs we measure the differences δ w := u H,n Γ,w u H,n Γ,w 1, w = 1, 2,..., (3.101) and we say that LDC iteration has reached a fixed point when δ w < We aso compute the ratios ρ w := δ w δ w 1, w = 2, 3,... (3.102) Experimentay we observe that, for fixed H and t, the coefficients ρ w are neary constant at every iteration w. Foowing the same approach as in [1, Section 5.4], the coefficients (3.102) are thus taken as an estimate for the LDC convergence rate. In Figure 3.9 we pot the coefficients ρ 2 computed experimentay as a function of t for different vaues of H. The LDC convergence rate becomes extremey fast for t 1 if we use a safety region, see Figure 3.9-(a). If no safety region is used, see Figure 3.9-(b), the LDC convergence rate turns out to be O( t 2 ), ( t 0). This is the same behavior observed for one-dimensiona probems. If t is chosen to be much arger than 1, we are in the stationary case imit. Therefore we shoud find the same resuts as studied in [4] for the 2D Poisson s equation. In [4] the Lapacian operator is discretized by the standard five-point centered differences scheme both on the goba and on the oca grid. For such a case, the LDC convergence rate is proved to proportiona to H 2 if the safety region has an extent ǫ > 0 and independent of H. Moreover, it is shown that the LDC convergence rate is proportiona to H if ǫ = 0. In our experiments the safety region ǫ is independent of H and ρ 2 reaches an

65 56 Chapter 3 - Convergence properties of LDC H=1/64 H=1/128 H=1/ t (a) Safety region ǫ = H=1/64 H=1/128 H=1/256 t t (b) No safety region (ǫ = 0) Figure 3.11: Pot of ρ 2 in terms of t for different vaues of H. The vaues of ρ 2 are computed experimentay performing one time step t with LDC on probem (3.103). asymptotic vaue when the time step t is much arger than 1, see again Figure 3.9. In Tabe 3.1 the vaues of ρ 2 computed taking t = 10 8 are reported. Ceary our resuts are in agreement with those presented in [4]. Finay, sti with reference to probem (3.98), we woud ike to investigate the convergence rate dependency on H when t 1 and ǫ = 0. In Figure 3.10, the vaues of ρ 2 as in Figure 3.9-(b) are potted in terms of H for two different vaues of t. As observed before for the one-dimensiona case (cf. Figure 3.6), the LDC convergence rate turns out to be proportiona to H 4. We focus now on the 2D convection-diffusion probem u t + v u = λ 2 u + f 1, in Ω = (0, 1) 2, for t > 0, (3.103) with v = (0.1, 0.1) and ( (x f 1 = exp 100( e t ) 2 ( + y e t ) )) 2. (3.104)

66 3.5 Two-dimensiona numerica experiments 57 ǫ = 0.1 ǫ = 0.0 H ρ 2 ( t = 10 8 ) ratio ρ 2 ( t = 10 8 ) ratio 1/ / / Tabe 3.2: Vaues of ρ 2 with t = 10 8 as in Figure The diffusion coefficient λ, the boundary conditions and the initia condition are the same as for the pure diffusion probem (3.98). Aso in this case the soution u has a oca region of high activity that can be covered by taking Ω = (0, γ) 2, with γ = 0.5. We adopt the same discretization schemes as in the previous experiment and we discretize the convection term by second-order centered differences both on the coarse and on the fine grid. In a tests the oca probem is soved taking h = H/2 and δt = t/2. We run the same kind of experiments as before and we show the resuts in Figure Once more we observe that the rate of convergence of LDC goes extremey fast to zero for t 0 if we use a safety region, see Figure 3.11-(a). Aso in this case the LDC convergence rate is proportiona to t 2 for t 0 if ǫ = 0, see Figure 3.11-(b). In the stationary case imit ρ 2 reaches an asymptotic vaue which depends on H. The vaues of ρ 2 computed for t = 10 8 are isted in Tabe 3.2. From the tabe we can see the same kind of resuts as for the pure diffusion probem. We finay note that the pot of ρ 2 as a function of H for t 1 and ǫ = 0 woud be quaitativey simiar to Figure 3.10: aso for a 2D convection-diffusion probem the LDC convergence rate is proportiona to H 4 when t 1 and ǫ = 0. In concusion, the anaysis and the numerica experiments presented in this chapter show that the LDC method for time dependent-probems has very good convergence properties. In fact LDC turns out to be convergent for any choice of the time step t. The rate of convergence of LDC can be improved by decreasing t or by computing the defect term using a safety region ǫ > H. For sma vaues of the time step, the convergence behavior of LDC is the same for one- and two-dimensiona probems, for pure diffusion and convection-diffusion equations.

67

68 Chapter 4 A finite voume adapted LDC method In this chapter we appy the time-dependent LDC technique in combination with standard finite voume discretizations on the goba and oca grid. Unike in Chapter 2, where the oca grid is adaptivey paced at each time step where the high activity occurs, we assume that the high activity of the soution is aways ocated, at each time eve, in the same (imited) part of the goba domain. Aready for this simpe situation, the LDC method as presented in Chapter 2 is such that the discrete conservation aw, which is one of the main attractive features of the finite voume method, does not necessariy hod for the soution on the composite grid. Here, we introduce a finite voume adapted LDC method for paraboic probems for which the conservation property is preserved. In [6] a finite voume adapted LDC agorithm which is conservative on the composite grid is presented for eiptic probems. The main idea there is that the defect correction shoud be such that the integrated fuxes across the interface between the coarse and fine grid are baanced. In the time-dependent case we consider here, we extend that idea and write the defect term in such a way that the baance across the interface hods at every time eve. In doing so we dea with the compication that the time integration might be performed with different time steps on the goba and oca grid. In genera, when finite voume discretizations are combined with oca grid refinement specia care has to be taken to guarantee discrete conservation across the interface between coarse and fine regions. This is a popuar topic in the iterature. In [39], for exampe, conservative ce-centered approximations on rectanguar ocay refined grids are derived for second order convection-diffusion equations. In [21, 40] a posteriori error estimates are given for ocay refined finite voume dis-

69 60 Chapter 4 - A finite voume adapted LDC method cretizations of convection-diffusion-reaction equations. In [7] the focus is on high order approximations and a fourth-order accurate finite voume based technique with oca refinement is proposed for Poisson s equation; the method preserves discrete conservation. Paraboic partia differentia equations are treated in [51], where a conservative adaptive strategy is proposed for the computation of fuxes at the interfaces of ocay nested grids. In [17] the features of a sover for compressibe fows and fuid-structure interaction are discussed; the sover integrates dynamic adaptation, mesh generation and finite voume discretization. The method and the resuts of this chapter have been previousy presented in [45]. 4.1 Probem formuation and initiaization of LDC In order to present the LDC method for paraboic probems in combination with a finite voume discretization, we consider a two-dimensiona convection-diffusion equation for a quantity ϕ = ϕ(x, t). The equation can be expressed in integra formuation as foows t V ϕ dω + V (ϕv λ ϕ) n dγ = V s dω in Θ, for a V Ω, (4.1) where v = v(x, t) is a given veocity fied, λ = λ(x, t) > 0 is a diffusion coefficient and s = s(x, t) is a known source term. Furthermore Θ is the time interva (0, t end ], whie V is a generic voume contained in the spatia domain Ω := (0, x max ) (0, y max ). With V we indicate the boundary of V and with n the outward unit vector perpendicuar to V. We aso introduce Ω, the boundary of Ω, and we define Ω := Ω Ω. We cose probem (4.1) by prescribing the Dirichet boundary condition and the initia condition ϕ = ψ, on Ω Θ, (4.2) ϕ(x, 0) = η, in Ω. (4.3) In (4.2) and (4.3), ψ = ψ(x, t) and η = η(x) are given functions. We just consider Dirichet boundary conditions for simpicity, but the LDC method we are going to describe can easiy hande other types of boundary conditions, such as Neumann or Robin. If a the variabes in (4.1) are sufficienty smooth and the veocity fied is divergence free ( v = 0), the integra formuation (4.1) is equivaent to the differentia formuation ϕ t We introduce the fux vector f by + v ϕ (λ ϕ) = s in Ω Θ. (4.4) f = f(ϕ) = ( fx f y ) := ϕv λ ϕ, (4.5)

70 4.1 Probem formuation and initiaization of LDC 61 so that equation (4.1) can be rewritten as ϕ dω + f n dγ = t V V V s dω in Θ, for a V Ω. (4.6) We assume that ϕ, the continuous soution of (4.6) that satisfies (4.2) and (4.3), at each time in Θ presents a region of high activity that covers a (sma) part of Ω Composite grid definition Probem (4.6) is first discretized in space on a goba uniform coarse grid using the finite voume method. We consider, in particuar, a standard vertex-centered approach. This is just for notationa convenience and the usage of other approaches, e.g. ce-centered discretization, woud guarantee the same properties in the method we are describing. We introduce the grid size parameters H x = x max /N x and H y = y max /N y, where N x and N y are positive integers, and the grid points (x i, y j ) := (ih x, jh y ), i = 0, 1,..., N x, j = 0, 1,..., N y, (4.7) (x i+1/2, y j ) := ((i + 1/2)H x, jh y ), i = 0, 1,..., N x 1, j = 0, 1,..., N y, (4.8) (x i, y j+1/2 ) := (ih x, (j + 1/2)H y ), i = 0, 1,..., N x, j = 0, 1,..., N y 1. (4.9) We define Ω H := {(x i, y j )} Ω, Ω H := Ω H Ω, Ω H := Ω H \ Ω H. (4.10) We want to compute a discrete approximation of ϕ at the points of the computationa grid Ω H. Each point of Ω H is the center of a contro voume V i,j := (x i 1/2, x i+1/2 ) (y j 1/2, y j+1/2 ). (4.11) The midpoints of the interfaces of voumes V i,j form a dua grid ( ) V H := {(x i+1/2, y j )} {(x i, y j+1/2 )} Ω, (4.12) on which we wi define discrete fuxes. Figure 4.1 represents the goba coarse grid we have introduced in Ω. The figure is drawn for N x = 5 and N y = 7; grid points Ω H are marked with a circe, whie rhombi and sma squares identify points of Ω H and V H respectivey. We denote by G(Ω H ), G( Ω H ) and G(V H ), the space of grid functions that operate on Ω H, Ω H and V H respectivey. We introduce the foowing notation: for a certain T H G(Ω H ), we write T H = {(T H i,j ), i = 1, 2,..., N x 1, j = 1, 2,..., N y 1}, with T H i,j := TH (x i, y j ). Simiary it is done for eements in G( Ω H ) and G(V H ). Given a certain F H G(V H ), we introduce the centra difference operator H Σ : G(VH ) G(Ω H ) by ( H Σ F H ) i,j := F H i,j+1/2 FH i,j 1/2 + FH i+1/2,j FH i 1/2,j. (4.13)

71 62 Chapter 4 - A finite voume adapted LDC method Figure 4.1: Goba coarse grid. We aso define T(ϕ) G(Ω H ), S G(Ω H ) and F(ϕ) G(V H ) as foows: T i,j (ϕ) := ϕ dω, (4.14) V i,j S i,j := s dω, (4.15) V i,j F i+1/2,j (ϕ) := F i,j+1/2 (ϕ) := yj+1/2 y j 1/2 f x (x i+1/2, η, t)dη, (4.16) xi+1/2 x i 1/2 f y (ξ, y j+1/2, t)dξ. (4.17) The term F is caed integrated fux and in its expression the fux vector that we defined in (4.5) occurs. Finay we use the operators and definitions that we have introduced so far to write the conservation aw in (4.6) for V = V i,j : t T i,j(ϕ) + ( H ΣF(ϕ) ) i,j = S i,j, in Θ. (4.18) In a finite voume approach the continuous fuxes that appear in (4.16) and (4.17) are approximated by finite differences; furthermore a quadrature rue has to be used in order to approximate a the integras (4.14)-(4.17). Here we wi not be specific on the particuar schemes to be empoyed. Suitabe choices are the standard secondorder centra formua for the fuxes in (4.16) and (4.17), and the midpoint rue for the integras in (4.14)-(4.17). This is for exampe what is done in the numerica experiments presented in Section 4.4. We denote the spatia discretization of T, F and S by T H G(Ω H ), F H G(V H ) and S H G(Ω H ) respectivey and we ca ϕ H = ϕ H (t) G( Ω H ) the spatia approximation of ϕ = ϕ(x, t). The finite voume

72 4.1 Probem formuation and initiaization of LDC 63 discretization appied to (4.18) eads to a set of (N x 1)(N y 1) ordinary differentia equations that have to be satisfied by ϕ H d dt TH (ϕ H ) + H Σ FH (ϕ H ) = S H, in Θ. (4.19) System (4.19) has sti to be discretized in time in order to be soved numericay. For this we divide the time interva Θ into N t 1 subintervas such that t end /N t =: t. We aso introduce t n := n t, with n = 0, 1, 2,..., N t and by ϕ H,n we denote an approximation of ϕ(x, t n ) on Ω H. Because of the high activity of the soution, a coarse grid approximation computed at time t n using a time step t might be not accurate enough to adequatey represent ϕ n := ϕ(t n ). Therefore we want to find a more accurate (both in space and time) oca approximation of ϕ n and eventuay use it to correct and improve the goba coarse grid soution. For this purpose we choose Ω, an open subset of Ω such that the oca high activity of ϕ is entirey contained in Ω, for a t Θ. Since we do not consider regridding in this chapter, we use no superscript n in the definition of the oca domain (cf. definition of the oca domain in Section 2.1). Adopting the same notation as for the goba domain, Ω indicates the boundary of Ω and Ω := Ω Ω. For convenience, the oca region Ω is chosen in such a way that hods with (x i, y j ) Ω H Ω W i,j Ω (4.20) W i,j := (x i 1, x i+1 ) (y j 1, y j+1 ). (4.21) This condition means that if a coarse grid point (x i, y j ) ies in Ω, then its eft, right, top and bottom neighbors ie in Ω. Note that V i,j W i,j, so that W i,j Ω impies V i,j Ω. Aso, Ω is not a union of contro voumes V i,j. In Ω, ike in Ω, we appy a vertex-centered finite voume method. Foowing the same procedure adopted for the goba domain, in Ω we introduce a oca fine grid (sizes h x < H x and h y < H y ), which we denote by Ω h. We aso define, cf. (4.10), Ω h := Ω h Ω, Ω h := Ω h \ Ω h. (4.22) It is of practica convenience that points of Ω H that ie in the area of refinement beong to Ω h too, and that boundaries of contro voumes in the oca fine grid coincide with boundaries of contro voumes in the goba coarse grid. For that reason, we assume the factors of grid refinement σ x := H x h x, σ y := H y h y, (4.23) to be odd integers. The union of coarse and fine grid defines the composite grid Ω H,h := Ω H Ω h. Figure 4.2 shows an exampe of a composite grid. In the figure the grid points in Ω H have been marked with an empty circe, whie the empty rhombi denote points beonging to Ω H. The region with gray background is Ω ; in that region the sma circes are used to mark the fine grid points Ω h, whie the sma rhombi indicate Ω h. From the figure one can see that coarse grid points ying

73 64 Chapter 4 - A finite voume adapted LDC method Figure 4.2: Exampe of a composite grid. in the area of refinement Ω beong to the fine grid too. In this exampe the factors of grid refinement are σ x = σ y = 3, but in genera one can have σ x σ y. As for the LDC method described in Chapter 2, the time integration on the fine grid is performed with a smaer time step than the one used gobay. Therefore, together with the factors of grid refinement we aso introduce the factor of time refinement τ. We et τ be an integer 1 and we divide the time interva (t n 1, t n ), with n 1, into τ subintervas. The time step for the fine grid probem is thus defined as δt := t/τ. We aso introduce t n 1+k/τ := t n 1 +kδt, with k = 0, 1, 2,..., τ, and by ϕ h,n 1+k/τ we denote an approximation of ϕ on the oca fine grid Ω h at time t n 1+k/τ. In practice the factors of grid and time refinement are to be chosen on the basis of severa factors, such as the physica probem that has to be soved, the discretization parameters (H x, H y and t) adopted for the coarse grid probem, and the discretization schemes used gobay and ocay Computation of a composite grid soution In this section we wi first find an expression for ϕ H,n and then we wi define the oca probem that eads to determining ϕ h,n, a oca better approximation of ϕ n. We assume that the LDC technique has been appied in the time interva [t n 2, t n 1 ], with n > 1. This means that a discrete probem has been soved on the goba and on the oca grid, and that the foowing approximation of ϕ n 1 is avaiabe on the

74 4.1 Probem formuation and initiaization of LDC 65 composite grid Ω H,h : ϕ H,h,n 1 := { ϕ h,n 1, in Ω h, ϕ H,n 1, in Ω H,h \ Ω h. (4.24) Since regridding is not considered in this chapter, expression (4.24) can directy be used to provide initia vaues for the computation of the next time step. In practice, there is need to use the subscript in the definition of ϕ H,h,n 1. This subscript was in fact introduced in Chapter 2 to take into account that the soution at time t n 1 might be computed on a different composite grid than the one used to perform the next time step. The composite grid soution ϕ H,h,n 1 aso incudes vaues on the boundary of the goba and oca grid. For n = 1, we simpy have the initia condition (4.3) ϕ H,h,0 = η Ω H,h. (4.25) We indicate by ϕ H,h,n 1 Ω H the restriction of ϕ H,h,n 1 on Ω H. Now, we can appy an impicit time discretization scheme to (4.19). We choose the impicit Euer scheme. However, as noted in Chapter 2, this is not restrictive and other impicit methods ike the Runge-Kutta schemes might be used as we. In Chapter 2 it is aso expained why time integration by an expicit method on the coarse grid is of minor interest in LDC. A coarse grid approximation ϕ H,n can finay be computed soving T H (ϕ H,n ) T H (ϕ H,h,n 1 Ω H) + H Σ FH (ϕ H,n ) t = S H,n t, (4.26) with ϕ H,n = ψ(t n ), on Ω H. If we now want to sove a discrete anaogue of (4.6) on Ω h {t n 1+k/τ, k = 1,..., τ}, we have to provide conditions on the boundary of the oca fine grid for each time t n 1+k/τ, with k = 1,..., τ. For Ω Ω, i.e. the part of the oca area s boundary in common with the goba boundary, we can appropriatey use the origina boundary condition (4.2). As for the rest of the oca area s boundary, namey Γ := Ω \ ( Ω Ω ), we introduce the interpoation operator in space P h,h. With Γ H := Γ Ω H and Γ h := Γ Ω h, the operator Ph,H G(Γ H ) G(Γ h ) spatiay interpoates ϕ H,n Γ H on Γ h. In LDC, typica choices for P h,h are piecewise inear and piecewise quadratic interpoation (cf. [30]). With P h,h we are abe to prescribe artificia Dirichet boundary conditions on Γ h at t n. Since we need boundary conditions not ony at t n, but for a the t n 1+k/τ, with k = 1, 2,..., τ, we define another interpoation operator R k t G(Γh ) G(Γ h ) G(Γ h ). The operator R k t performs inear time interpoation between the time eves t n 1 and t n ; in particuar, R k t interpoates between the restriction of ϕ H,h,n 1 on Γ h, see (4.24), and P h,h (ϕ H,n ). In this way, R k t enabes us to specify artificia Dirichet boundary conditions on Γh for every t n 1+k/τ, with k = 1, 2,..., τ. We can syntheticay write the boundary condition for the oca probem as ϕ h,n 1+k/τ = ψ h,n 1+k/τ, on Ω h, for k = 1, 2,..., τ, (4.27)

75 66 Chapter 4 - A finite voume adapted LDC method where ψ h,n 1+k/τ := ψ(t n 1+k/τ ), on Ω h \ Γh, for k = 1,..., τ, ( ) Γ h, P h,h (ϕ H,n ), on Γ h, for k = 1,..., τ. R k t ϕ h,n 1 (4.28) From (4.28) we can see that on Γ h the boundary condition for the oca probem depends on ϕ H,n, the approximation of ϕ n that we computed on the goba coarse grid soving (4.26). Note that on Γ H the fine and coarse grid approximation coincide at t n. This can easiy be verified, since ϕ h,n Γ H = ψ h,n Γ H = P h,h (ϕ H,n ) Γ H = ϕ H,n Γ H. (4.29) It is aso important to emphasize that on Ω h \ Γh the boundary conditions for the oca probem come from a proper discretization of the boundary conditions for the origina continuous probem. For simpicity we ony considered Dirichet boundary conditions (see (4.2)) in our description, but ceary other types, such as Neumann or Robin boundary conditions, coud be impemented straightforwardy. Note however that we aways impose Dirichet boundary conditions on Γ h. If we now introduce a oca discretization of (4.6), we are abe to formuate a oca probem from which we can compute ϕ h,n. The oca approximation ϕ h,n is regarded to be more accurate than ϕ H,n since it is computed using a finer grid (h < H) and a smaer time step (δt t). If again the Impicit Euer scheme is used, the oca probem that enabes us to determine ϕ h,n is T h (ϕh,n 1+k/τ ) T h (ϕn 1+(k 1)/τ,h ) + h Σ Fh (ϕh,n 1+k/τ )δt = S h,n 1+k/τ δt, for k = 1, 2,..., τ, (4.30) ϕ h,n 1+k/τ = ψ h,n 1+k/τ, on Ω h, for k = 1, 2,..., τ. The procedure (4.30) is initiaized using (4.24) if n > 1, or by a proper discretization of the origina initia condition (4.3) if n = 1. The coarse and fine grid approximation computed at t n define the composite grid soution { ϕ h,n ϕ H,h,n, on Ω h = ϕ H,n, on Ω H,h \ Ω h. (4.31) At this point we have competed the initiaization of the LDC agorithm for paraboic probems in combination with a finite voume discretization. In the next section we expain how, through a defect correction, we can obtain a more accurate composite grid soution at time t n. 4.2 The finite voume adapted defect term The crucia part of the LDC agorithm is how the oca soution is used to improve the goba approximation ϕ H,n through an approximation of the oca discretization

76 4.2 The finite voume adapted defect term 67 error or defect. In particuar, ike it is done in [6] for eiptic probems, we are interested in expressing the defect in such a way that the resuting composite grid discretization satisfies a discrete conservation aw. The defect d H,n G(Ω H ) is defined as d H,n := T H (ϕ n Ω H) T H (ϕ n 1 Ω H) + H ΣF H (ϕ n Ω H) t SH,n t. (4.32) In (4.32) we have pugged the exact anaytica soution ϕ into the coarse grid discretization scheme (4.26). We consider now the continuous equation (4.6), which is vaid for a voumes V Ω and thus for any contro voume V i,j Ω too. If we appy time integration between t n 1 and t n, we obtain or ϕ n dω ϕ n 1 dω + V i,j V i,j = tn t n 1 tn t n 1 V i,j f n dγ dt V i,j s dω dt, for i = 1, 2,..., N x 1, j = 1, 2,..., N y 1, (4.33) tn tn T(ϕ n ) T(ϕ n 1 ) + H ΣF(ϕ) dt = t n 1 S dt. t n 1 (4.34) Combination of (4.34) and (4.32) yieds ( ) ( ) d H,n = T H (ϕ n Ω H) T(ϕ n ) T H (ϕ n 1 Ω H) T(ϕ n 1 ) + H Σ ( F H (ϕ n Ω H) t tn t n 1 F(ϕ) dt ) ( S H,n t tn t n 1 S dt ), (4.35) which is the definition of defect that we wi use in practice. If we woud know the vaues of d H,n, we coud use them to compute a more accurate (both in space and time) approximation of ϕ n on the goba grid. This coud be done by adding d H,n on the right hand side of (4.26). However, since we do not know the exact soution of our partia differentia equation, we cannot compute the vaues of d H,n. What we can do, though, is to use the oca soution ϕ h,n to get an estimate of d H,n. We introduce so that, cf. (4.35), d H,n T := T H (ϕ n Ω H) T(ϕ n ), (4.36) d H,n F d H,n S := F H (ϕ n Ω H) t tn t n 1 F(ϕ) dt, (4.37) tn := S H,n t S dt, (4.38) t n 1 d H,n = d H,n T d H,n 1 T + H Σ dh,n F d H,n S. (4.39) In the foowing section we wi approximate each of the terms that appear on the right hand side of (4.39).

77 68 Chapter 4 - A finite voume adapted LDC method Approximation of the defect We start by considering d H,n T. After soving the goba and the oca probem, the foowing approximations are avaiabe for an arbitrary T i,j (ϕ n ): 1) the coarse grid approximation T H i,j (ϕh,n ); 2) a coarse grid approximation that based on the composite grid soution (4.31), i.e. T H i,j (ϕh,h,n Ω H); 3) a sum of fine grid approximations T sum,i,j(ϕ h,n ) := (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 T h,i+p/σ x,j+q/σ y (ϕ h,n ). (4.40) In T,i,j sum the subscript reminds us that this third approximation is oca and it exists ony for (x i, y j ) Ω, i.e. for the coarse grid points ying in the area of refinement. The approximations above are considered to be isted in order of increasing accuracy. The second approximation is more accurate than the first one because in the area of refinement it expoits the vaues computed on the fine grid with a time step δt t. The approximation number three can be considered more accurate than the second one because it computes spatia integras using the fine grid discretization. We introduce Ω H := Ω H Ω h and we use the best avaiabe information to define T best (ϕ H,h,n ) G(Ω H ) as { T sum T best (ϕ H,h,n (ϕ h,n ), on Ω H, ) := (4.41) T H (ϕ H,h,n ), on Ω H \ Ω H. Definition (4.41) is used to introduce the foowing approximation for d H,n T : d H,n T = T H (ϕ n Ω H) T(ϕ n ) T H (ϕ H,h,n Ω H) T best (ϕ H,h,n ) =: d H,n T. (4.42) Combining (4.41) and (4.42) it is easy to show that d H,n T = 0, on Ω H \ Ω H. (4.43) We consider now the fux discretization error d H,n F, see (4.37), and in particuar F n i+1/2,j := tn t n 1 F i+1/2,j (ϕ) dt, (4.44) where F i+1/2,j is an arbitrary horizonta fux. In the expression of F n i+1/2,j both space (cf. (4.16)) and time integras appear; these can be approximated in different ways, using the coarse or the fine grid size, the t or the δt time discretization. Beow we ist the possibe approximations for F n i+1/2,j :

78 4.2 The finite voume adapted defect term 69 1) an H- t approximation, i.e. an approximation based on the H space discretization and on the t time discretization: F n i+1/2,j tn t n 1 F H i+1/2,j ( ϕ H (t) ) dt F H i+1/2,j (ϕh,n ) t; (4.45) 2) an (H, h)- t approximation based on the composite grid soution (4.31): F n i+1/2,j tn t n 1 F H i+1/2,j ( ϕ H,h ) (t) Ω dt F H H i+1/2,j (ϕ H,h,n Ω H) t; (4.46) 3) a oca h- t approximation based on the sum of fine grid fuxes. We introduce and we write F sum,i+1/2,j (ϕh (t)) := F n i+1/2,j tn t n 1 F sum,i+1/2,j (σ y 1)/2 q= (σ y 1)/2 F h,i+1/2,j+q/σ y (ϕ h (t)), (4.47) ( ϕ h (t) ) dt F sum,i+1/2,j (ϕh,n ) t; (4.48) 4) a oca h-δt approximation based on the δt time discretization: F n i+1/2,j tn = t n 1 F sum,i+1/2,j (ϕh τ k=1 τ tn 1+k/τ (t)) dt t n 1+(k 1)/τ F sum,i+1/2,j (ϕh F sum,i+1/2,j (ϕh,n 1+k/τ k=1 )δt. (t)) dt (4.49) As before the various approximations are isted in order of increasing accuracy. The third and the fourth approximation can ony be expressed for points (x i+1/2, y j ) Ω. Anaogous approximations are avaiabe for the other fux terms, i.e F i 1/2,j, F i,j+1/2, F i,j 1/2. We can therefore define F best G(V H ) and F best G(V H ) as τ F sum F best (ϕ H,h,n ) := F best (ϕ H,h,n (ϕ h,n 1+k/τ )δt, on V H Ω, ) t := k=1 F H (ϕ H,h,n Ω H) t, esewhere. We can now approximate d H,n F as d H,n F = F H (ϕ n Ω H) t tn t n 1 F(ϕ) dt F H (ϕ H,h,n Ω H) t Fbest (ϕ H,h,n ) t =: d H,n F. (4.50)

79 70 Chapter 4 - A finite voume adapted LDC method Combination of (4.50) with the definition of F best (ϕ H,h,n ) yieds d H,n F = 0, on V H \ (V H Ω ). (4.51) Simiar considerations hod for an arbitrary source term Si,j n := which can be approximated through: tn t n 1 S i,j (t) dt, (4.52) 1) an H- t approximation S n i,j tn t n 1 S H i,j (t) dt SH,n i,j t. (4.53) This approximation is goba and hods for a 1 i N x 1 and 1 j N y 1; 2) a oca h- t approximation based on a sum of fine grid approximations and on a t time integration. We introduce and we write S sum,i,j(t) := (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 S n i,j tn t n 1 S sum,i,j The approximation (4.55) hods for (x i, y j ) Ω H ; 3) a oca h-δt approximation S h,i+p/σ x,j+q/σ y (t), (4.54) (t) dt Ssum,n,i,j t. (4.55) S n i,j = tn S sum,i,j(t) dt t n 1 τ k=1 tn 1+k/τ t n 1+(k 1)/τ S sum,i,j (t) dt τ k=1 S sum,n 1+k/τ,i,j δt, (4.56) defined for points (x i, y j ) Ω H. Gathering the best avaiabe information, we define S best,n G(Ω H ) and S best,n G(Ω H ) as τ S sum,n 1+k/τ S best,n := S best,n,i,j δt, on Ω H, t := k=1 (4.57) S H,n t, on Ω H \ Ω H.

80 4.2 The finite voume adapted defect term 71 It is now possibe to provide an approximation for d H,n S : d H,n S tn := S H,n t S dt S H,n t S best,n t =: d H,n S. (4.58) t n 1 We note that combination of (4.57) and (4.58) yieds d H,n S = 0, on Ω H \ Ω H. (4.59) The finite voume adapted LDC agorithm In the previous section we have found an approximation for a the terms that appear on the right hand side of (4.39). where d H,n G(Ω H ) is defined by (cf. (4.42), (4.50) and (4.58)) d H,n := d H,n T d H,n F d T H,n 1 + H Σ d H,n S ( ) = T H (ϕ H,h,n Ω H) T best (ϕ H,h,n ) + t H Σ ( ) T H (ϕ H,h,n 1 Ω H) T best (ϕ H,h,n 1 ) ( ) ( F H (ϕ H,h,n Ω H) Fbest (ϕ H,h,n ) t S H,n S best,n). (4.60) The approximation d H,n ceary depends on the soution computed on the fine grid in the time interva [t n 1, t n ]. We can finay compute a more accurate goba approximation of ϕ n soving the modified coarse grid probem T H (ϕ H,n 1 ) T H (ϕ H,h,n 1 Ω H) + H Σ FH (ϕ H,n 1 ) t = S H,n t + d H,n, (4.61) with ϕ H,n 1 = ψ(t n ), on Ω H. In (4.61) we caed the new approximation ϕ H,n 1, where the new subscript is used to distinguish the new approximation from the previous one ϕ H,n, from now on referred as ϕ H,n 0. Once ϕ H,n 1 is computed, we are abe to define new boundary conditions, see (4.27), for a new oca probem on Ω h and this triggers an iterative procedure which is formaized in Agorithm 4.1. As for the goba grid soution, in Agorithm 4.1 an extra subscript is added to number the different approximations computed ocay; the same is done for the defect term. Agorithm 4.1 (Finite voume adapted LDC agorithm) FOR LOOP, n = 1, 2,..., N t INITIALIZATION Compute a goba approximation ϕ H,n 0 G(Ω H ) soving probem (4.26), with ϕ H,n 0 = ψ(t n ) on Ω H. Compute a oca approximation ϕ h,n,0 G(Ωh ) soving the oca probem (4.30). Define ϕ H,h,n 0 G(Ω H,h ) as in (4.31).

81 72 Chapter 4 - A finite voume adapted LDC method ITERATION, w = 1, 2,... Compute d H,n w 1 G(ΩH ), an approximation of the oca discretization error d n H G(ΩH ), through ( ) d H,n w 1 = T H (ϕ H,h,n w 1 Ω H) T best (ϕ n H,h,w 1 ) ( ) T H (ϕ H,h,n 1 Ω H) T best (ϕ H,h,n 1 ) ( ) + t H Σ F H (ϕ H,h,n w 1 Ω H) Fbest (ϕ n H,h,w 1 ) ( t S H,n S best,n). (4.62) Compute a more accurate goba coarse grid approximation ϕ H,n w G(ΩH ) soving the modified probem T H (ϕ H,n w ) T H (ϕ H,h,n 1 Ω H) + H Σ FH (ϕ H,n w ) t = S H,n t + d H,n w 1, (4.63) ϕ H,n w = ψ(t n ), on Ω H. Use ϕ H,n w to update the boundary condition ψ h,w on Ωh. Sove the foowing oca probem with updated boundary conditions T h(ϕh,n 1+k/τ,w ) T h(ϕh,n 1+(k 1)/τ,w ) + h Σ Fh (ϕh,n 1+k/τ,w ϕ h,n 1+k/τ,w ϕ H,h,n w = )δt = S h,n 1+k/τ δt, = ψ h,n 1+k/τ,w, on Ω h, for k = 1, 2,..., τ. Define the composite grid approximation { ϕ h,n,w, on Ωh END ITERATION ON w (4.64) ϕ H,n w, on Ω H \ Ω H. (4.65) Ca ϕ h,n and ϕ H,n the atest soutions found on the oca and goba grid respectivey (remove the ast subscript); the soution on the composite grid at time t n is: { ϕ h,n ϕ H,h,n, := in Ω h, (4.66) ϕ H,n, in Ω H,h \ Ω h. END FOR

82 4.2 The finite voume adapted defect term 73 This is the LDC agorithm for time-dependent probems as presented in Chapter 2, but now adapted to a setting with a finite voume discretization. Like done in [6] for stationary cases, the expression of the defect term d H,n w 1 is such that the resuting composite grid discretization is sti conservative. This property of the finite voume adapted LDC method is discussed in Section 4.3. Before that, Section 4.2.3, we give a practica way to compute the defect term (4.62) Practica considerations on the defect term In practice the defect term d H,n w 1 is not computed through a straightforward appication of (4.62). That woud in fact be quite onerous, since it woud require to sum fine grid vaues at a the points Ω h and for a the times t n 1+k/τ, with k = 1, 2,..., τ. In the perspective of simpifying the computation of d H,n w 1, in Lemma 4.2 we state a oca conservation aw that hods just for the points (x i, y j ) Ω H. This is obtained summing the fine grid baance of each fine grid contro voume at each time t n 1+k/τ, with k = 1, 2,..., τ. Lemma 4.2 The fine grid approximation is such that T sum,i,j(ϕ h,n,w) T sum,i,j(ϕ H,h,n 1 Ω h ) + = ( τ H Σ k=1 τ k=1 F sum (ϕ h,n 1+k/τ,w ) S sum,n 1+k/τ,i,j δt for (x i, y j ) Ω H. (4.67) ) i,j δt Proof. We consider the fine grid baance in (4.64) and we sum it for a the σ x σ y fine grid contro voumes that partition a coarse grid contro voume V i,j Ω. We obtain (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 = + (σ x 1)/2 (σ x 1)/2 ( p= (σ x 1)/2 q= (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 T,i+p,j+q h (ϕh,n 1+k/τ,w ) T,i+p,j+q h (ϕh,n 1+k/τ,w ) (σ y 1)/2 ( ( S h,n 1+k/τ,i+p,j+q δt h ΣF h (ϕ h,n 1+k/τ,w )δt ) ) i+p,j+q = ) for (x i, y j ) Ω H, k = 1, 2,..., τ. (4.68)

83 74 Chapter 4 - A finite voume adapted LDC method Figure 4.3: The coarse grid points Ω H can be divided into three compementary subsets: Ω H (trianges), ΓH (squares) and Ω H c (circes). Using definitions (4.40), (4.47), (4.54), and the fact that interna fuxes cance, we can rewrite (4.68) as T,i,j sum (ϕh,n 1+k/τ,w If we now sum over k, we get ( ) T,i,j sum (ϕh,n 1+(k 1)/τ,w ) + T,i,j sum (ϕh,n,w ) T,i,j sum (ϕh,h,n 1 Ω h ) + H Σ Fsum ) (ϕ h,n 1+k/τ,w )δt i,j = S sum,n 1+k/τ,i,j δt for (x i, y j ) Ω H, k = 1, 2,..., τ. (4.69) = τ ( k=1 τ ( k=1 H Σ Fsum S sum,n 1+k/τ,i,j ) (ϕ h,n 1+k/τ,w )δt i,j ) δt, for (x i, y j ) Ω H, (4.70) which is equivaent to (4.67). The resuts in Lemma 4.2 are used in the proof of Theorem 4.3, which gives us a practica way to compute d H,n w 1. First we define the new set ΩH c := Ω H \ (Ω H Γ H ). In this way the coarse grid points Ω H are divided into three distinct groups, namey Ω H, ΓH and Ω H c. From the definitions of the three subsets, it is easy to verify that Ω H = Ω H ΓH Ω H c. In Figure 4.3 the coarse grid points ΩH are marked with trianges, whie squares and circes denote points Γ H and Ω H c respectivey. The simpification stated in Theorem 4.3 hods for Ω H and ΩH c.

84 4.2 The finite voume adapted defect term 75 Theorem 4.3 The defect term d H,n w 1 G(ΩH ) can be written as ( d H,n w 1) i,j = T H i,j (ϕh,h,n + w 1 Ω H) TH i,j (ϕh,h,n 1 Ω H) ( H ΣF H (ϕ H,h,n w 1 Ω )i,j H) t S H,n i,j t, for (x i, y j ) Ω H, 0 for (x i, y j ) Ω H c. (4.71) Proof. Consider a point (x i, y j ) Ω H. We have ( ) ( d H,n (4.62) w 1) i,j = Ti,j(ϕ H H,h,n w 1 Ω H) T,i,j(ϕ sum h,n,w 1) ( + ( ( T H i,j(ϕ H,h,n 1 w 1 H Σ S H,n i,j ( F H (ϕ H,h,n t Ω H) T sum,i,j(ϕ H,h,n 1 w 1 Ω h ) w 1 Ω H) t τ τ k=1 k=1 ) S sum,n 1+k/τ,i,j δt F sum ) ) ) (ϕ H,h,n 1+k/τ w 1 )δt i,j (4.72) (4.67) = T H i,j(ϕ H,h,n + w 1 Ω H) Ti,j(ϕ H H,h,n 1 Ω H) ( ) H Σ FH (ϕ H,h,n w 1 Ω H) i,j t SH,n i,j t. This proves the first part of (4.71). The second part can be proved combining (4.62) for a point (x i, y j ) Ω H c with (4.43), (4.51) and (4.59). Theorem 4.3 gives us formuas to compute the finite voume adapted defect term on Ω H ΩH c. Therefore we need the origina definition (4.62) for points ΓH ony. We note however that even on Γ H things can be simpified significanty. Combination of (4.62) for a point (x i, y j ) Γ H with formuas (4.43) and (4.59) yieds in fact ( ) d H,n w 1 = t H Σ F H (ϕ H,h,n w 1 Ω H) Fbest (ϕ n H,h,w 1), on Γ H. (4.73) The ony term on the right hand side of (4.62) that we expicity have to compute for a point (x i, y j ) Γ H is thus the sum of the fine grid fuxes. Note that Theorem 4.3 not ony gives us a practica way to compute the finite voume adapted defect term (4.62), but it aso estabishes the connection between the LDC method with the finite voume adapted defect term introduced here and the LDC method with the standard defect term presented in Chapter 2. On Ω H the standard defect term is computed pugging the fine grid soution at time t n into the coarse grid discretization scheme. This is what happens in the first part of (4.71) too. Esewhere

85 76 Chapter 4 - A finite voume adapted LDC method the standard defect is zero. This is not true for the finite voume adapted defect term, which is zero on Ω H c ony. On ΓH it is not zero and the baance of integrated fuxes guaranteed by (4.73) is such that the finite voume adapted LDC method produces a soution that satisfies a discrete conservation aw on the composite grid. This is expained in Section Conservation properties of LDC We discuss a few properties of the LDC agorithm described in Sections 4.1 and 4.2. The time-dependent LDC technique is an iterative procedure that impicity gives a discretization of the convection-diffusion probem on a composite grid at the discrete time eves t n = n t, with n = 1, 2,..., N t. Throughout this section, we wi assume that at each time t n the LDC iteration converges. A sufficient condition for the iterative procedure to be convergent is that for every n = 1, 2,..., N t the vector norm ϕ H,n w Condition (4.74) impies that ψ h,n 1+(k 1)/τ,w ϕ H,n w 1 0, (w ). (4.74) ψ h,n 1+(k 1)/τ,w 1 0, (w ), k = 1, 2,..., τ, (4.75) and therefore ϕ h,n,w ϕh,n,w 1 0, (w ). (4.76) When the LDC iteration at t n has converged, the subscript w is removed, and the converged goba and oca soution are caed ϕ H,n G( Ω H ) and ϕ h,n G( Ω h ) respectivey. They define a composite grid approximation ϕ H,h,n G( Ω H,h ) as in (4.66). In Lemma 4.4 we show that the converged coarse grid soution ϕ H,n and the converged fine grid soution ϕ h,n coincide in Ω H for n > 1. Note that this property is automaticay verified for n = 0 (initia condition). Lemma 4.4 Assume that the oca coarse grid stationary homogeneous system ) Ti,j ( H (ν) + H Σ FH (ν) = 0, i,j for (x i, y j ) Ω H, ν = 0, for (x i, y j ) Γ H, (4.77) has ony the zero soution in G(Ω H ΓH ). Then the imit soution (ϕ H,n, ϕ h,n ) of the LDC iteration at time t n (n > 1) satisfies ϕ H,n = ϕ h,n, on Ω H. (4.78)

86 4.3 Conservation properties of LDC 77 Proof. From (4.63) and (4.71), we obtain ( ) Ti,j H (ϕh,n ) + H Σ FH (ϕ H,n ) i,j ) = Ti,j H (ϕh,h,n Ω H) + ( H Σ FH (ϕ H,h,n Ω H) i,j for (x i, y j ) Ω H. (4.79) or ) Ti,j(ϕ H H,n ϕ H,h,n Ω H)+ ( H ΣF H (ϕ H,n ϕ H,h,n Ω H) = 0 for (x i, y j ) Ω H. (4.80) i,j Moreover, from (4.29) and (4.66), we can deduce that ϕ H,n Γ H ϕ H,h,n Γ H = 0. Hence, ν = ϕ H,n ϕ H,h,n Ω H G(ΩH ΓH ) satisfies system (4.77). From the assumption, ν is nu on Ω H ΓH, which is equivaent to (4.78) because ϕ H,h,n ϕ h,n on Ω H (see (4.66)). The resuts of Lemma 4.4 wi be used in the proof of Theorem 4.5, which gives the discrete conservation aw that is satisfied by the converged composite grid soution ϕ H,h,n. Before that, we consider for a whie what happens when probem (4.6) is discretized on a goba grid Ω H by the finite voume method. Again, for simpicity, we consider the impicit Euer scheme for time discretization. The foowing conservation aw Ti,j H (ϕh,n ) Ti,j H (ϕh,n 1 ) + ( H Σ FH (ϕ H,n ) ) t = SH,n i,j i,j t (4.81) hods for any contro voume V i,j Ω H. Formua (4.81) is a discrete equivaent of the continuous conservation aw (4.6). Summation of the discrete conservation aws that hod for individua contro voumes V i,j eads to a conservation aw on the union of these contro voumes. This is because fuxes over interna faces cance: in a time step t, the infux into V i,j Ω H out a neighboring contro voume V,m Ω H is baanced by the outfux from V i,j to V,m in the same time step. Theorem 4.5 states that this property is aso verified for the imit of the LDC iteration ϕ H,h,n, which is computed on a composite grid using two different time integration steps t and δt. This is the generaization to time-dependent probems of the conservation property presented in [6] for the LDC method in stationary cases. Theorem 4.5 Under the assumption of Lemma 4.4, the composite grid soution satisfies the foowing system of discrete conservation aws: T best (ϕ H,h,n ) T best (ϕ H,h,n 1 ) + H Σ Fbest (ϕ H,h,n ) t = S best,n t. (4.82) Proof. Combination of (4.63) and (4.62) yieds T H (ϕ H,n ) + H Σ FH (ϕ H,n ) t = T H (ϕ H,h,n Ω H) T best (ϕ H,h,n ) + T best (ϕ H,h,n 1 ) ( ) + H Σ F H (ϕ H,h,n ) Ω H Fbest (ϕ H,h,n ) t + S best,n t. (4.83)

87 78 Chapter 4 - A finite voume adapted LDC method Using Lemma 4.4, we have T H (ϕ H,n ) = T H (ϕ H,h,n Ω H), F H (ϕ H,n ) = F H (ϕ H,h,n Ω H). (4.84) Substitution of (4.84) into (4.83) gives (4.82). We note that, for (x i, y j ) Ω H, the conservation aws (4.82) reduce to T,i,j(ϕ sum h,n ) T sum,i,j(ϕ h,n 1 ) + ( τ H Σ F sum (ϕ h,n 1+k/τ k=1 )δt ) i,j = τ k=1 S sum,n 1+k/τ,i,j δt. (4.85) This is the same reation we deduced in Lemma 4.2 summing the conservation aws that hod for each of the σ x σ y fine grid contro voume that partition V i,j Ω for a the times t n 1+k/τ, with k = 1, 2,..., τ. For (x i, y j ) Ω H c, baance (4.82) becomes ( ) Ti,j(ϕ H H,n ) Ti,j(ϕ H H,n 1 ) + H ΣF H (ϕ H,n ) t = SH,n i,j t, (4.86) i,j which is ceary the set of conservation aws that correspond to the coarse grid discretization with time step t. For points (x i, y j ) Γ H, Theorem 4.5 guarantees that, in a time step t, the discrete infux into V i,j out of a neighboring voume V,m, with (x, y m ) Ω, matches the tota discrete outfux from V,m to V i,j in the same time step t. The atter is the sum of a the fine grid fuxes at a the intermediate times t n 1+k/τ, with k = 1, 2,..., τ. In practice, when we sum discrete conservation aws that hod on individua contro voumes V i,j, the finite voume adapted LDC method is such that interna fuxes cance ike it happens for the finite voume discretization on a singe grid. We note that (4.82) hods for the fuy converged composite grid soution ϕ H,h,n. However, as it happens for the standard LDC method for paraboic probems, the convergence of the finite voume adapted LDC agorithm is generay very fast and ony very few iterations suffice. When presenting the finite voume adapted LDC method, we considered a vertexcentered discretization for both coarse and fine grid, but we noted that other approaches, ike a ce-centered discretization, coud be used as we. Figure 4.4 shows an exampe of composite grid where the goba coarse grid is treated with a cecentered approach, i.e. the domain boundary coincides with contro voume boundaries. As a consequence of this choice, the fine grid is treated ike in the ce-centered method aong Ω Ω, i.e. the part of the oca region s boundary which is in common with the physica boundary, and ike in the vertex-centered method aong the interface Γ. This approach is foowed in the numerica experiments we present in Section 4.4, Chapter 5 and Chapter 6.

88 4.4 Numerica experiments 79 Figure 4.4: Exampe of a composite grid where a ce-centered approach is used on the goba coarse grid. 4.4 Numerica experiments We present two numerica experiments that iustrate the accuracy and the efficiency of the LDC method with a finite voume adapted defect term. In Section 4.4.1, we compare the method to a singe uniform grid sover. We show that LDC can achieve the same accuracy as the uniform grid sover, whie computing a numerica approximation of the soution in a significanty smaer number of grid points and thus being a more efficient method. In Section 4.4.2, we compare the LDC agorithm with the standard choice for the defect term and the LDC agorithm with the finite voume adapted defect term. Ony for the atter a discrete conservation property hods on the composite grid Exampe 1: comparison with a uniform grid sover In this section we consider a two-dimensiona convection-diffusion equation. We choose Ω = (0, 1) (0, 1) and Θ = (0, t end ], with t end = 1.5, and we sove the foowing probem ϕ t + v ϕ = 2 ϕ + s in Ω Θ, ϕ = ψ, on Ω Θ, (4.87) ϕ = η, in Ω, t = 0,

89 80 Chapter 4 - A finite voume adapted LDC method t (a) ϕ ex (x, y, t = 1.5) (b) ϕ ex (x = 0, y = 0, t) Figure 4.5: The exact soution ϕ ex. where v = (1, 1). The source term s(x, t), the initia condition η(x) and the Dirichet boundary conditions are chosen in such a way that the exact anaytica soution of the probem is ϕ ex (x, y, t) = ( 1 tanh ( 100((x 0.1) + (y 0.1)) )) e t sin 2 (3πt). (4.88) At each time eve in Θ the exact soution (4.88) has a region of high activity in the bottom eft corner of Ω. Figure 4.5 shows the pot of the exact soution (4.88) at the fina time t end and the tempora evoution of ϕ ex at the boundary point (0, 0). We sove probem (4.87) by means of the LDC method with a finite voume adapted defect term and the oca region Ω = (0, 0.275) (0, 0.275). For this probem the appication of LDC with the standard choice for the defect term woud give simiar resuts. In fact, the choice of Ω is such that the interface Γ is away from the high activity; hence, the fuxes across Γ are reativey sma at every time t Θ. As a consequence, the baance of fuxes across the interface is not crucia to obtain an accurate numerica approximation of the soution. On the coarse grid Ω H we use a ce-centered finite voume approach. The fuxes (4.16) and (4.17) are approximated by centered differences; in particuar, the standard second-order centra formuas are used. Integras (4.14)-(4.17) are computed with the midpoint rue and the operator P h,h performs piecewise inear interpoation in space. The time integration is performed using the impicit Euer method both gobay and ocay. We perform severa runs with different vaues of grid sizes and time steps. Because of the symmetry of the soution (4.88) about the ine y = x, we aways set H x = H y and h x = h y. In a the tests ony one LDC iteration is performed at each time step t. As a measure of the accuracy of the various numerica soutions, for each run we measure the

90 4.4 Numerica experiments 81 Number of discretized equations Grid & time step ǫ 2 soved per time step t Unif. H x t σ x LDC Unif. LDC Unif. LDC H 0 t H 0 t Tabe 4.1: Resuts of the 2D numerica experiment. A the tests are run with H y = H x and τ = σ y = σ x. In the tabe: H 0 = 0.05, t 0 = 0.1. scaed Eucidean norm ǫ 2 = ϕh,h,nt Ω H ϕ ex (x, y, t end ) Ω H 2 Nx N y. (4.89) In (4.89), the term ϕ H,h,Nt Ω H represents the restriction on the coarse grid of the composite grid soution at the fina time t end, whie in the denominator we have the square root of the tota number of coarse grid points N x N y. The resuts of each of the LDC runs are compared to the numerica soution found soving probem (4.87) on a singe goba uniform grid. The grid sizes and the time step in each uniform grid run are the same as the fine grid sizes and the oca time step of the corresponding LDC test. Aso for the singe uniform grid runs we measure the scaed Eucidean norm (4.89). Resuts in Tabe 4.1 show that in a the cases we considered, LDC can achieve the same order of accuracy as the uniform grid sover. Of course LDC is a more efficient method than the uniform grid sover, since a fine grid size and a sma time step are adopted ony where the arge variations occur. To give an estimate of the compexity of the two methods, in Tabe 4.1 we aso report the tota number of discretized equations that are soved at every time step t. For LDC this vaue is the sum of two terms: the number of coarse grid points and the factor of time refinement τ mutipied by the number of fine grid points. For the uniform grid sover it is the product of the number of grid points in the uniform grid and the factor of time refinement τ of the corresponding LDC run. The ast coumn of Tabe 4.1 shows that, in our exampe, LDC computes the soution at a number of grid points one order of magnitude smaer than the uniform grid sover. We note however that in LDC the soution is first computed and then corrected. This means that, if we aways perform exacty one LDC iteration per time step t, both the coarse and fine grid discretized equations are soved twice at every time eve t n. As a consequence, the actua speed-up of LDC with respect to the uniform grid sover in terms of computationa time can be estimated as haf of the numbers in the ast coumn of Tabe 4.1. Finay, note that one is

91 82 Chapter 4 - A finite voume adapted LDC method not necessariy forced to set the factor of time refinement τ equa to the factors of grid refinement σ x and σ y. For rea appications one woud rather first choose the grid sizes and the time steps on the two grids so that both the goba and the oca scaes are propery resoved, and then set the factors of refinement consequenty. In genera we expect τ to be neither too big nor too sma with respect to the factors of grid refinement. In the first case the spatia components of error might be dominant in the oca probem, whie in the second case the oca time scaes might not be propery resoved Exampe 2: comparison with the standard LDC method In this section we present a two-dimensiona probem for which conservation of fuxes across the interface between coarse and fine grid pays a crucia rue. For such a probem, we expect the finite voume adapted LDC agorithm to be more accurate than the LDC method presented in [44], where no specia precautions are taken in order to guarantee conservation on the composite grid. We choose Ω = (0, 2) (0, 1) and Θ = (0, t end ], with t end = 20, and we sove ϕ t + (vϕ ϕ ) = 0, in Ω Θ, ( ) vϕ ϕ n = 0, on Ω Θ, ϕ = 1, in Ω, t = 0. (4.90) The veocity fied is v = (v x, v y ) = (1 + 5 sin(πt/2), sin(πt/2)). The choice of the boundary conditions is such that integration of the partia differentia equation over Ω yieds the foowing goba conservation aw where M(t) is defined by M(t) := d M(t) = 0, (4.91) dt Ω ϕ(x, y, t) dx dy. (4.92) The soution ϕ is periodic in time with period T = 4 and varies most in the regions of Ω next to the corners P 0 = (0, 0) and P 1 = (2, 1). The veocity fied is such that the ampitude of the osciations in P 1 is arger than in P 0. Moreover the veocity fied is such that, unike the exampe in Section 4.4.1, the soution is not symmetric about the ine y = x. Figure 4.6 shows the soution ϕ at two different time eves. Because of the two oca regions of high activity, probem (4.90) is soved by means of LDC. On the goba coarse grid we appy the finite voume method with a cecentered approach. As in the previous exampe, we approximate integras by the midpoint rue, and fuxes by second-order centered differences. Two oca fine grids

92 4.4 Numerica experiments y 0 0 x y 0 0 x 1 2 (a) ϕ(x, y, t = 1.0) (b) ϕ(x, y, t = 3.0) Figure 4.6: Soution of probem (4.90) at two different time eves t Figure 4.7: M(t) for the LDC agorithm with the standard choice of the defect term (dashed ine) and with the finite voume adapted defect term (soid ine).

93 84 Chapter 4 - A finite voume adapted LDC method t t (a) ϕ(x = 0, y = 0, t) (b) ϕ(x = 2, y = 1, t) Figure 4.8: Time evoution of ϕ in P 0 (a) and in P 1 (b) for the LDC agorithm with the standard choice of the defect term (dashed ine) and with the finite voume adapted defect term (soid ine). are paced in the corners next to P 0 and P 1. The time integration is performed using the impicit Euer method both gobay and ocay. Aso in this case the operator P h,h performs piecewise inear interpoation in space. Figures 4.7 and 4.8 report the numerica resuts for both the LDC method with the standard choice of the defect term and the finite voume adapted defect term. The resuts are obtained using the coarse grid sizes H x = H y = 0.05 and the time step t = 0.1. The oca regions are Ω,0 = (0, 0.275) (0, 0.125) and Ω,1 = (1.575, 2) (0.775, 1). The area of the two oca regions refects the fact that, as noted before, arger osciations occur in P 1 than in P 0. On both oca regions the grid sizes are h x = h y = 0.01, whie the time step is δt = Ony one LDC iteration is performed at each time step t. Athough one iteration might not be sufficient to have fu convergence of LDC at a time steps, Figure 4.7 shows that the pot of M(t) as computed by the LDC method with the finite voume adapted correction term is neary constant. The standard choice for the defect term, on the other hand, yieds an error in M which is of order 150% after ony five periods T. As a consequence, the ampitude of the osciations at the two corners P 0 and P 1 increases unreaisticay, see Figure 4.8. We note that in [6] a one-dimensiona version of the probem iustrated in this section is presented. There is, however, a crucia difference between the method presented here and in [6]: in our approach a sma time step δt is adopted ony on the oca grid(s) to resove the reativey fast phenomena occurring there. On the goba coarse grid a arger time step t is sufficient to catch the reativey sow variations of the soution. In [6] the same time step is used on the different grids; in this way the time step on the coarse grid might be forced to be unnecessariy sma, making the method ess efficient than the one presented in this thesis.

94 Chapter 5 Generaizations of the LDC method In Chapter 4 we introduced a finite voume adapted LDC agorithm such that the computed composite grid soution satisfies a system of discrete conservation aws, see Theorem 4.5. However, we restricted ourseves to cases where the high activity of the soution is aways ocated in the same imited part of the goba domain Ω and we did not incude a regridding strategy in Agorithm 4.1. In this chapter we want to overcome this imitation and we introduce a conservative regridding procedure to be incorporated in the finite voume adapted LDC agorithm. The agorithm is then extended to incude mutipe eves of refinement. 5.1 A conservative regridding strategy In LDC, a regridding strategy is needed when the high activity of the soution is ocated in different regions of the goba domain at different time eves. In order to propery resove the oca sma scaes of the soution and to avoid unnecessary refinement, the oca fine grid has to be adapted to the soution s behavior at each time step. In Chapter 2 we expained that this can be done measuring certain features of the first goba approximation computed at the new time eve. Once the new oca region has been chosen, initia vaues ϕ H,h,n are to be provided in a the points of the new composite grid (see Agorithm 2.4) in order to perform the next LDC time step. Consider now the finite voume adapted LDC agorithm and assume that the soution s high activity is ocated in different regions of the goba domain Ω at differ-

95 86 Chapter 5 - Generaizations of the LDC method Ω h,n Ω h,n+1 Figure 5.1: Exampe of composite grid for the finite voume adapted LDC method. In this exampe the region of refinement changes during two consecutive time steps. Ony three coarse grid points beong to both Ω h,n and Ω h,n+1. ent time eves. Aso assume that the region of refinement needs to be updated at time t n, i.e. assume that Ω n+1 Ω n. With (x i, y j ) defined as in (4.7) and with the same notation as in Chapters 2 and 4, we introduce the foowing sets of coarse grid points. First we define Υ H,n 1, the set of coarse grid points that ie in the region of refinement in the time step [t n 1, t n ], but not anymore during the time step [t n, t n+1 ]: Υ H,n 1 := { (x i, y j ) Ω H (x i, y j ) Ω h,n \ ( Ω h,n+1 Ω h,n )}. (5.1) Then, we introduce Υ H,n 2, the set of coarse grid points that beong to a region that does not require refinement during the time step [t n 1, t n ], but it does during the time step [t n, t n+1 ]: Υ H,n 2 := { (x i, y j ) Ω H (x i, y j ) Ω h,n+1 \ ( Ω h,n+1 Ω h,n )}. (5.2) Figure 5.1 represents an exampe of composite grid where Ω n+1 does not coincide with Ω n ; in the figure, the coarse grid points beonging to ΥH,n 1 are marked with a circe, whie points in Υ H,n 2 are marked with a square. In order to perform the next LDC time step, one has to provide initia vaues ϕ H,h,n in a the points of the new composite grid Ω H,h,n+1 = Ω H Ω h,n+1. In Section 2.2 we described a procedure to do so. Briefy, the procedure consisted in taking the soution computed at t n in a the common points between Ω H,h,n and Ω H,h,n+1, and performing spatia interpoation for the remaining part of Ω H,h,n+1, see Formua (2.28). This strategy works fine when LDC is appied in combination with the finite differences method, cf. numerica exampes in Section 2.3. However, one woud ike to obtain a soution that satisfies a system of discrete conservation aws when the finite voume adapted LDC

96 5.1 A conservative regridding strategy 87 method is used. Now, if we woud provide initia vaues ϕ H,h,n described in Section 2.2, in genera we woud get using the method Ti,j best (ϕ H,h,n ) T best i,j (ϕ H,h,n ), for (x i, y j ) Υ H,n 1 Υ H,n 2, (5.3) where Ti,j best is the best approximation avaiabe for the integra of ϕ over a coarse grid contro voume V i,j, see definition (4.41). This choice of initia vaues is not desirabe because it impies that conservation aws that hod for two consecutive time intervas, say [t n 1, t n ] and [t n, t n+1 ], cannot be summed up to give a conservation aw that hods on [t n 1, t n+1 ]. Therefore we woud ike to modify the way in which initia vaues ϕ H,h,n are provided on the two sets Υ H,n 1 and Υ H,n 2 in order to guarantee that Ti,j best (ϕh,h,n If condition (5.4) is satisfied, then (cf. (4.82)) ) = Ti,j best (ϕh,h,n ), (x i, y j ) Ω H. (5.4) T best (ϕ H,h,n ) T best (ϕ H,h,n 1 ) + H Σ Fbest (ϕ H,h,n ) t = S best,n t, (5.5) T best (ϕ H,h,n+1 ) T best (ϕ H,h,n ) + H Σ Fbest (ϕ H,h,n+1 ) t = S best,n+1 t, (5.6) can be added together to give T best (ϕ H,h,n+1 ) T best (ϕ H,h,n 1 ) + 1 H Σ Fbest (ϕ H,h,n+j ) t = j=0 1 S best,n+j t. (5.7) Equation (5.7) represents the set of conservation aws satisfied by the composite grid soution during the interva [t n 1, t n+1 ]. In the foowing section we outine an aternative way to compute the initia vaues ϕ H,h,n which ensures discrete conservation. j= Providing initia vaues on the new composite grid We first consider the coarse grid points in Υ H,n 1. We reca that these points were ocated in the area of refinement during the time step [t n 1, t n ], but they are not anymore during [t n, t n+1 ]. Hence we have to describe how we shoud restrict the computed oca fine grid soution at time t n to coarse grid points ony. Reca that ϕ H,h,n denotes the fuy converged LDC soution at time t n on the composite grid Ω H,h,n. So, according to definition (4.41), we have T best i,j (ϕ H,h,n ) = T sum,i,j(ϕ H,h,n ) for (x i, y j ) Υ H,n 1. (5.8) Since ϕ H,h,n is the initia soution for the new time step [t n, t n+1 ] and because the coarse grid points Υ H,n 1 are not in the area of refinement anymore during [t n, t n+1 ], it foows that Ti,j best (ϕ H,h,n ) = T H i,j(ϕ H,h,n ) for (x i, y j ) Υ H,n 1. (5.9)

97 88 Chapter 5 - Generaizations of the LDC method Therefore, in order to guarantee (5.4), we propose to set ϕ H,h,n in such a way that Ti,j H (ϕh,h,n ) = T,i,j sum (ϕh,h,n ) for (x i, y j ) Υ H,n 1. (5.10) If the midpoint rue is used to compute quadratures, formua (5.10) can be written as ϕ H,h,n (xi,y j ) = 1 H x H y (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 ( ϕ h,n,i+p,j+q h x h y ), (5.11) for (x i, y j ) Υ H,n 1. In practice equation (5.10) impies changing the soution computed during the previous time step at a certain number of coarse grid points. However, if the time step t is sma enough, it is reasonabe to assume that the area of refinement does not change much between two consecutive time steps, so that the set Υ H,n 1 contains a sma number of points ony. Note that, according to (5.10) and (5.11), the new regridding strategy is such that initia vaues ϕ H,h,n in a coarse grid point (x i, y j ) Υ H,n 1 depend on the soution computed at t n at a the fine grid points incuded in the contro voume whose center is (x i, y j ) Υ H,n 1 ; in Section 2.2 we simpy set ϕ H,h,n (xi,y j ) = ϕ H,h,n (xi,y j ). We now consider the set Υ H,n 2. We reca that the area covered by the coarse grid voumes whose centers form Υ H,n 2 needs to be refined in the new time step (from t n to t n+1 ), but not during the od one (from t n 1 to t n ). We now outine how to choose the proongation of ϕ H,h,n to ensure discrete conservation. According to definitions (4.41) and (5.2), we have in this case T best i,j (ϕh,h,n ) = T H i,j (ϕh,h,n ) Ti,j best (ϕ H,h,n ) = T sum,i,j(ϕ H,h,n for (x i, y j ) Υ H,n 2, (5.12) ) for (x i, y j ) Υ H,n 2. (5.13) In order to ensure that (5.4) hods, we have thus to guarantee that T sum,i,j(ϕ H,h,n ) = T H i,j(ϕ H,h,n ) for (x i, y j ) Υ H,n 2. (5.14) We note that (5.14) does not give us a way to compute the initia vaues ϕ H,h,n, but it is just a set of constraints that have to be satisfied by ϕ H,h,n. We propose the foowing strategy to determine ϕ H,h,n : first, compute the vaues of ϕ H,h,n at the fine grid points ^Ω h,n+1, see (2.27), using the interpoation operator Q n x ike in Agorithm 2.4. Then scae the computed vaues in such a way that (5.14) is satisfied. For the specia case of the midpoint rue for the integras, the scaing procedure is as foows: for each coarse grid contro voume whose center (x i, y j ) beongs to Υ H,n 2, compute i,j := 1 (σ x 1)/2 (σ y 1)/2 ϕ H,n ( ) i,j H x H y ϕ H,h,n (xi+p,y h x h j+q ) h x h y. y p= (σ x 1)/2 q= (σ y 1)/2 (5.15)

98 5.1 A conservative regridding strategy 89 Note that i,j is nothing ese than the expression for the midpoint rue of the difference Ti,j H (ϕh,h,n ) T,i,j sum(ϕh,h,n ), for (x i, y j ) Υ H,n 2. After that introduce the coefficients ϕ H,h,n (xi+p,y α i+p,j+q := j+q ), (5.16) (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 ϕ H,h,n (xi+p,y j+q ) with (σ x 1)/2 p (σ x 1)/2 and (σ y 1)/2 q (σ y 1)/2. By construction the coefficients α i+p,j+q are such that (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 α i+p,j+q = 1. (5.17) Finay, for each coarse grid contro voume whose center (x i, y j ) beongs to the set Υ H,n 2, do ϕ H,h,n (xi+p,y j+q ) ϕ H,h,n (xi+p,y j+q ) + α i+p,j+q i,j, (5.18) for (σ x 1)/2 p (σ x 1)/2 and (σ y 1)/2 q (σ y 1)/2. In (5.18) the symbo means repacement (cf. [19]). It is easy to verify that the new vaues of ϕ H,h,n computed by (5.18) are such that (σ x 1)/2 (σ y 1)/2 p= (σ x 1)/2 q= (σ y 1)/2 ( ) ϕ H,h,n (xi+p,y j+q ) h x h y = ϕ H,n i,j H x H y, (5.19) for (x i, y j ) Υ H,n 2. Equation (5.19) is the particuar expression for the midpoint rue of (5.14). If quadratures other than the midpoint rue are used, the scaing procedure can be done ikewise. At this point we can incorporate the conservative regridding strategy described above into the finite voume adapted LDC technique. We obtain the foowing agorithm. Agorithm 5.1 (Finite voume adapted LDC agorithm with regridding) FOR LOOP, n = 1, 2,..., t end / t INITIALIZATION Provide initia vaues ϕ H,h,n 1 on Ω H. If n = 1, then set H from the initia condition. Otherwise, set ϕ H,h,n 1 Ω ϕ H,h,n 1 Ω H = ϕh,h,n 1 Ω H. Compute a goba approximation ϕ H,n 0. Use ϕ H,n 0 to determine the oca region Ω n.

99 90 Chapter 5 - Generaizations of the LDC method Provide the initia vaues ϕ H,h,n 1 at the remaining points of Ω H,h,n. If n = 1, use the initia condition. If n > 1 and Ω H,h,n Ω H,h,n 1, set ϕ H,h,n 1 Ω H,h,n If n > 1 and Ω H,h,n ϕ H,h,n 1 = = ϕ H,h,n 1 Ω H,h,n 1. Ω H,h,n 1, first et Q n x (ϕh,h,n 1 ), on ^Ω h,n, ϕ H,h,n 1, on Ω H,h,n \ ^Ω h,n, with ^Ω h,n defined as in (2.27). Then, modify the vaues of ϕ H,h,n 1 in a the composite grid points incuded in the coarse grid contro voumes whose centers form Υ H,n 1 Υ H,n in such a way that (5.10) and (5.14) are satisfied. Use ϕ H,n 0 to provide artificia boundary conditions for the oca probem. Compute a oca approximation ϕ h,n,0. ITERATION, w = 1, 2,... Estimate the defect as expained in Chapter 4. Correct the goba soution. Update the boundary conditions for the oca probem. Compute an updated oca soution. END ITERATION ON w END FOR LOOP ON n The soution on the composite grid Ω H,h,n at time t n is ϕ H,h,n (remove the subscript that numbers the LDC iterations). We note that in Agorithm 5.1 initia vaues ϕ H,h,n 1 are provided in two stages. On the coarse grid they are immediatey avaiabe from the soution at the previous time step. Locay, on the other hand, they are provided ony after a first coarse grid soution ϕ H,n 0 has been computed and the new region of refinement has been chosen. This is simiar to what happens in Agorithm 2.4. However, appication of the conservative regridding strategy described above impies modification of ϕ H,h,n 1 at the composite grid points incuded in the coarse grid contro voumes whose centers form Υ1 H,n 1 Υ2 H,n 1. In Agorithm 5.1 the coarse grid approximations ϕ H,n w, with w 1 and n > 1, are thus not computed with the same initia vaues used to compute ϕ H,n 0. As a consequence, it might be possibe that one extra LDC iteration is necessary for convergence in those time steps in which the conservative regridding procedure takes pace.

100 5.1 A conservative regridding strategy A numerica exampe In this section we test the conservative regridding strategy introduced above in a concrete exampe and we compare it to the procedure described in Section 2.2. We choose Ω = (0, 1) 2 and Θ = (0, t end ], with t end = 3, and we consider the foowing two-dimensiona convection-diffusion probem u t + v u = λ 2 u, in Ω Θ, u n = 0, on Ω Θ, (5.20) u = 100 exp ( 50 ( (x 0.3) 2 + (y 0.8) 2)), in Ω, t = 0. In (5.20), by n we denoted the outward unit vector perpendicuar to Ω. The diffusion coefficient and the veocity fied are chosen as λ = 10 3 and v = ( 2x (x 1) (y 0.5), 2y(y 1)(x 0.5) ), (5.21) respectivey. It is easy to verify the v = 0. In this way probem (5.20) can be written in integra formuation and soved using the finite voume adapted LDC method. On the goba grid we use a ce-centered approach. Both gobay and ocay we approximate integras by the midpoint rue, and fuxes by second-order centra formuas. Time discretization is performed by the backward Euer scheme on both grids. In this exampe we et the operator P h,h perform piecewise quadratic interpoation. The high activity of u is not confined to the same imited part of Ω for a times t Θ. Therefore we incorporate a regridding strategy in the finite voume adapted LDC agorithm. At every time step, the oca region Ω n is determined using the method described in [10, 11, 65]. For that we use a threshod vaue ǫ = 3, see Section 2.2. We run two kinds of experiments: in the first one initia vaues ϕ H,h,n on Ω H,h,n+1 are prescribed using the regridding strategy discussed in Section 2.2; in the second one we appy the conservative regridding method described in this chapter. In both cases we et the operator Q n x perform piecewise quadratic interpoation from the coarse grid soution. We note that v n = 0 on Ω, and that homogeneous Neumann boundary conditions are prescribed on the whoe boundary. This yieds to the fact that the foowing identity hods d M(t) = 0, (5.22) dt with M(t) := Ω u dx dy. (5.23) In our experiments we want to test how we the two regridding strategies can reproduce the conservation aw (5.22). The resuts are potted in Figure 5.2. They are

101 92 Chapter 5 - Generaizations of the LDC method t Figure 5.2: Pot of M(t) as computed by soving probem (5.20) with the finite voume adapted LDC agorithm. The soid ine refers to the non-conservative regridding strategy, the dashed ine to the conservative regridding strategy. computed using H x = H y = , t = , h x = h y = H/3 and δt = t/3. In both cases two LDC iterations are performed at every time step. When using the conservative regridding strategy we aso perform one extra LDC iteration (i.e. three in tota) at the time steps when the oca domain changes. Figure 5.2 shows that the graph of M(t) computed with the non-conservative regridding strategy (soid ine) has jumps. These occur every time the area of refinement has to be updated to foow the soution s behavior. The jumps are due to the fact that the non-conservative regridding strategy does not necessariy guarantee (5.4). Between two jumps M(t) is neary constant. Figure 5.2 aso shows that the conservative regridding strategy introduced in this chapter yieds no jumps in M(t), which is neary constant ti t end = 3. We note, however, that Agorithm 5.1 impies that (5.22) is reproduced exacty ony if the LDC iteration is fuy converged at each time step (see Theorem 4.5). In our experiment we fix a-priori the number of LDC iterations to be performed at every time step; therefore it coud happen that we have no fu convergence of LDC at certain time eves. The reative variation µ(t) := M(t) M(0) M(0) (5.24) computed by appying Agorithm 5.1 is potted in Figure 5.3. From the graph we can see that µ(t) is very sma, but non zero, when two LDC iterations per time step are performed. However, the conservation properties of the method can be further improved by performing more LDC iterations per time step. In our case the maximum of µ is reduced by roughy a factor of 10 2 for every extra LDC iteration per time step. Finay note that in the experiment with the non-conservative regridding

102 5.2 LDC agorithm with mutipe eves of refinements LDC iterations / time step LDC iterations / time step LDC iterations / time step t Figure 5.3: Time evoution of µ, see (5.24), in the LDC experiment with the conservative regridding strategy. strategy, see again Figure 5.2, we have max (µ(t)) , (5.25) whie the maximum of µ(t) is just (see Figure 5.3) if the conservative regridding strategy with two LDC iterations per time step is appied. 5.2 LDC agorithm with mutipe eves of refinements In this section we extend the LDC agorithm for time-dependent probems to incude mutipe eves of refinement. The basic idea is the foowing: we start performing the LDC iteration between a goba coarse grid soution (eve 0) and a oca fine grid soution (eve 1 of refinement). When this iteration has converged, the artificia boundary conditions for the probem at eve 1 have come to a fixed point. The LDC iteration can now be performed between the soution at eve 1 and a soution at eve 2. When aso this iteration has converged, we can further refine and iterate at higher eves unti the required accuracy is reached. We note that every new oca region shoud be chosen according to the criteria mentioned in the previous chapters for the LDC with a singe eve of refinement. In particuar, grid points at eve that ie in the area of refinement shoud beong to the grid at eve + 1 too. Moreover, if one uses the finite voume adapted LDC method, condition (4.20) shoud be satisfied for a eves. At each eve the area of refinement can be determined measuring

103 94 Chapter 5 - Generaizations of the LDC method certain features of the soution at that eve; for instance, the method described in [10, 11, 65] can be empoyed. We observe that at each subeve not ony a finer grid spacing, but aso a smaer time step is adopted. The LDC time marching has to take this into account. In Figure 5.4 the procedure to perform one LDC time step with two eves of refinement is iustrated. The figure refers to a case in which one and ony one LDC iteration is performed at each eve, and the factor of time refinement is aways τ = 2. Like in the LDC method with a singe eve of refinement, the first step of the agorithm is the computation of a first goba soution (eve 0) at t n (a). This provides artificia boundary conditions to a oca probem at eve 1, which is then integrated from t n 1 ti t n (b). The soution at eve 0 is now improved by a defect correction (c). By assumption the boundary conditions for the probem at eve 1 have aready reached a fixed point. If this shoud not be the case, we coud perform more iterations between eve 0 and eve 1. The next step is anyhow appying the LDC method between eve 1 and eve 2. For that we integrate the eve 1 probem from t n 1 ti t n 1/2 (d) and we provide artificia boundary conditions for a probem at eve 2. This is then soved (e) and, after that, the eve 1 soution at t n 1/2 is improved by a defect correction (f). We move on by soving the eve 2 probem with updated boundary conditions and performing the second haf time step at eve 1 (e). The LDC iteration is now performed between t n 1/2 and t n. A eve 2 probem is soved (h), the soution at eve 1 is corrected (i) and finay the soution at the finest eve is recomputed with updated boundary conditions (j). This procedure defines a composite grid soution at time t n. We woud ike to remind here that one is not forced to use the same numerica schemes at each eve. Aso the factors of grid and time refinement can be different at different eves. The ony requirement is that no expicit schemes are used at a eves except at the finest. As noted before (Section 2.2), this is for the effectiveness of the defect correction. We aso want to highight the foowing: if the finite voume adapted LDC method is appied at each eve in combination with the conservative regridding strategy described in Section 5.1, the resuting soution turns out to be conservative in a discrete sense on the mutieve composite grid, union of the goba and a the ocay nested grids. This is not verified in a practica exampe here, but it wi become cear from the numerica experiments discussed in the next chapter. Finay, note that in [1, 3] LDC is appied to sove a stationary combustion probem and mutipe eves of refinement are adopted. The approach there consists in computing first an approximation of the soution on a grids, from the coarsest to the finest eve. After that, the soution on the finer grids is used to update the soution on the coarser grids through a defect correction; this is done in an opposite order, i.e. from the finest ti the coarsest eve. The entire cyce is then repeated again unti a fixed point in the iteration is reached. This approach is efficient (see numerica resuts in [1, 3]) for stationary probems. However, it does not have a straightforward generaization to time-dependent partia differentia equations that are soved ocay not ony with a finer grid size, but aso with a smaer time step. For this reason we proposed an approach that consists in iterating ti convergence at a cer-

104 5.2 LDC agorithm with mutipe eves of refinements 95 t n 1 t n t t n 1 t n (a) Compute soution at eve 0 (b) Compute soution at eve 1 t t n 1 t n t t n 1 t n t (c) Correct soution at eve 0 (d) Update soution at eve 1 ti t n 1/2 t n 1 t n t t n 1 t n t (e) Compute soution at eve 2 ti t n 1/2 (f) Correct soution at eve 1 ti t n 1/2 t n 1 t n t t n 1 t n t (g) Update soution at eve 2 ti t n 1/2, then at eve 1 ti t n (h) Compute soution at eve 2 ti t n t n 1 t n t t n 1 t n t (i) Correct soution at eve 1 ti t n (j) Update soution at eve 2 ti t n Figure 5.4: An LDC time step with two eves of refinement.

105 96 Chapter 5 - Generaizations of the LDC method tain eve before moving on to a finer one. With the new features introduced in this chapter, namey mutipe eves of refinement and a conservative regridding strategy, the LDC method is now compete and can be used to sove reaistic physica probems. The foowing chapter is concerned with the soution of transport probems in transient fow fieds by means of LDC.

106 Chapter 6 Soving transport probems using LDC In this chapter, we wi consider the transport of passive tracers in a given veocity fied. A passive tracer is a diffusive contaminant in a fuid fow that is present in such ow concentration that it does not infuence the dynamics of the fow. Understanding the infuence of the main fow on the tracer materia is crucia in many engineering appications, such as mixing in chemica reactors or transport of fy ashes in burners. Knowedge of the passive tracer behavior is aso of fundamenta importance in enviromenta sciences for studying dispersion of poutants (e.g. chemica species, radioactive components) in the atmosphere or in the oceans. Transport of passive tracers is modeed by an advection-diffusion equation, and it has been studied from a phenomenoogica and experimenta point of view by many authors, see review artices [26, 52, 67] and references therein. In turbuent fows, one of the main difficuties of the probem from a computationa point of view is the scae separation between main fow eddies and transport processes: the atter occur at a scae which can be a few times smaer than the smaest ength scaes of the turbuent fow [26]. Considering the fact that running Direct Numerica Simuations (DNS) of the main fow is aready chaenging in simpe geometries, but sti feasibe using for exampe spectra methods, we see that it is much more difficut to propery simuate the transport processes. In many practica appications, however, the tracer materia is mainy confined in a very imited part of the computationa domain. The LDC method for time-dependent PDEs we deveoped in the previous chapters seems thus to be an appeaing technique for soving the transport probem. Using LDC, in fact, a fine grid spacing and a sma time step can be adopted ony where the highest concentration of the tracer materia occurs. Moreover, when the LDC iteration is fuy converged, the finite voume adapted LDC method in combination with a conservative regridding strategy guarantees conservation of the tracer

107 98 Chapter 6 - Soving transport probems using LDC materia. In this chapter we study a test probem in which the main fow equations are sti soved by spectra methods, whie the soution of the transport probem is computed on composite grids by means of LDC. 6.1 Mathematica mode We consider a dipoe-wa coision probem in the two-dimensiona spatia domain Ω = (0, 2) ( 1, 1). In this test case a veocity dipoe is initiay ocated at the center of the domain Ω and it starts traveing towards one of the was. After a certain time the dipoe hits the wa and secondary vortices are formed. We pace some tracer materia around the point where the dipoe hits the wa, and we run simuations to see how the tracer detaches from the wa and is transported by the veocity fied. We indicate by ϕ(x, y, t) the continuous distribution of the passive tracer, and by v(x, y, t) the given veocity fied. The transport of the passive tracer is modeed by the foowing dimensioness advection-diffusion equation ϕ t + v ϕ = 1 Pe 2 ϕ, in Ω. (6.1) In (6.1), the Pécet number is defined as Pe := V D/k, where V is a characteristic veocity of the fow, D the haf-width of the rectanguar domain Ω (i.e. D = 1 in our case) and k the tracer s moecuar diffusivity. Approach (6.1) is caed Euerian in contrast with the Lagrangian approach, where trajectories of individua tracer partices are computed, see [56]. Partia differentia equation (6.1) is soved with the no-fux boundary condition ϕ n = 0, on Ω, (6.2) where n is the outward unit vector perpendicuar to the boundary. The initia condition for ϕ is the Gaussian distribution ( ) ϕ(x, y, t = 0) = 1 2π σ 2 exp (x 1.0)2 + (y + 1.0) 2 0 2σ 2, (6.3) 0 with σ 0 = 0.1/ 2. The Gaussian distribution (6.3) is centered in the point (1.0, 1.0) and about 95% of the tracer materia is confined within a distance of 2σ from the center of the distribution. For soving advection-diffusion equation (6.1), the veocity fied v must be given. In practice, v(x, y, t) is computed soving the two-dimensiona Navier-Stokes equations in the veocity-vorticity formuation. This system incudes the foowing dimensioness equation for the vorticity ω(x, y, t) ω t + v ω = 1 Re 2 ω, in Ω. (6.4)

108 6.1 Mathematica mode 99 1 Ω top Ω eft Ω right Ω bottom Figure 6.1: Partition of Ω in four subsets. In (6.4) the Reynods number of the fow is defined as Re := V D/ν, where V and D are the same characteristic veocity and ength that appear in the definition of the Pécet number, whie ν is the kinematic viscosity of the fuid. The other equations of the system are the continuity equation and the definition of vorticity (see [22] for the detais). In [24] it is shown that this system is equivaent to equation (6.4) couped with 2 v = k ω, in Ω, (6.5) k v = ω, on Ω, (6.6) where k is the unit vector perpendicuar to the fow domain. In order to prescribe boundary conditions for the veocity fied, the boundary Ω is divided into four subsets Ω eft, Ω right, Ω top, Ω bottom, see Figure 6.1. Each of the four subsets coincides with one of the four sides of the square Ω. The boundary conditions are chosen as foows v Ωeft = v Ωright, (periodic boundary conditions), (6.7) v Ωtop = v Ωbottom = 0, (no-sip boundary conditions). (6.8) The initia condition for the vorticity is the dipoe given by ω(x, y, t = 0) = ω e (1 (r 1 (x, y)/r 0 ) 2) exp ( (r 1 (x, y)/r 0 ) 2) ω e (1 (r 2 (x, y)/r 0 ) 2) ( exp (r 2 (x, y)/r 0 ) 2). (6.9) The dipoe is the sum of two monopoes of equa ampitude ω e and the same radius r 0, but opposite sign. In (6.9), the terms r 1 (x, y) and r 2 (x, y) represent the

109 100 Chapter 6 - Soving transport probems using LDC distance from the centers C 1 := (x 1, y 1 ) and C 2 := (x 2, y 2 ) of the two monopoes, viz. r 1 (x, y) := (x x 1 ) 2 + (y y 1 ) 2, r 2 (x, y) := (x x 2 ) 2 + (y y 2 ) 2. (6.10) In our simuation, we take r 0 = 0.1, C 1 = (0.9, 0) and C 2 = (1.1, 0). Once the vorticity at t = 0 is given, the veocity fied is computed consequenty integrating equation (6.5) and imposing v = 0 on Ω. The ampitude ω e of the two monopoes is chosen in such a way that the tota kinetic energy E of the resuting fow, namey E(t) := 1 v(x, y, t) 2 dx dy, (6.11) 2 Ω is such that E(0) = 2, see [38]. With our settings, it turns out that ω e 300. Note that the same initia condition is chosen in [23] for a case with nonperiodic boundary conditions in both x- and y-direction. Interaction between dipoes and boundaries is aso studied in [49]. The settings chosen for the main fow probem are such that the soution v is symmetric with respect to the ine x = 1. From equations (6.1)-(6.3) it is possibe to see that, if the veocity fied is symmetric with respect to x = 1, aso the distribution of the passive tracer turns out to be symmetric with respect to the same ine. For this reason, we sove the transport probem ti t end = 1 ony in the haf domain Ω haf := (0, 1) ( 1, 1), with homogeneous Neumann boundary conditions aong x = 1. Moreover, the settings of our probem are such that where M(t) := d M(t) = 0, (6.12) dt Ω ϕ(x, y, t) dx dy. (6.13) This means that the tota amount of passive tracer materia is conserved. 6.2 Numerica method We sove the transport probem (6.1)-(6.3) using the finite voume adapted LDC method described in Chapter 4. The finite voume adapted LDC agorithm is appied with mutipe eves of refinement (see Section 5.2) and in combination, at each eve, with the conservative regridding introduced in Section 5.1. Note that in the test case we are considering, the time variation of ϕ is not uniform. Before that the veocity dipoe hits the wa, in fact, the distribution of the passive tracer varies sowy in time; this is because the advection term is neary zero and ϕ is ony affected by the diffusion process. After that the dipoe hits the wa, on the other hand, the advection term becomes dominant and ϕ changes very rapidy in

110 6.2 Numerica method 101 time. For this reason, we do not use a constant time step t to integrate the coarse grid probem; rather, t is chosen at every time eve on the basis of the variations in the coarse grid soutions occurred during the previous time step. Before computing the first goba coarse approximation ϕ H,n 0, the foowing agorithm is appied to determine the proper time step t to be used. Agorithm 6.1 (Determine t for the goba coarse grid probem) Compute δ := t ϕ H,h,n 1 Ω H ϕ H,h,n 2 Ω H. If δ > to 1, then reduce t, i.e. et t = max { t/k, t min }. If δ < to 2, then increase t, i.e. et t = min{ t K, t max }. If to 1 δ to 2, then t remains unchanged. The terms t min, t max, K > 1, to 1 and to 2 are user-specified parameters that can be determined from the resuts of previous numerica experiments. Note that δ is computed from absoute and not reative differences because in our appication ϕ is neary zero in the greatest part of Ω haf. The parameters t min and t max contro the range of variation of t and they aways guarantee that t min t t max. Agorithm 6.1 is not appied during the first time step (n = 1); at the very beginning t is specified directy by the user. Aso the system of Navier-Stokes equations in the veocity-vorticity formuation is discretized in time and space in order to be soved numericay. The numerica method we consider is based on equations (6.4)-(6.6) and it is described in great detai in [38]. Here we just mention that the tempora discretization of (6.4) is performed approximating the advection term by a second-order expicit Adams- Bashforth scheme, and the diffusion term by the second-order impicit Crank-Nicoson scheme (see [20] for the definitions). This combination is aso known as ABCN scheme. This choice is convenient because ω n, i.e. the vorticity at time t n, can be computed expicity from the vaues of the veocity fied at the two previous time eves, namey t n 1 and t n 2. Once ω n is known, v n is computed from Poisson s equation (6.5) straightforwardy. This procedure has to be initiaized at the first time step. For this a second-order Runge-Kutta method is used, so that the time discretization is overa second-order accurate. We woud aso ike to mention that the spatia discretization is performed by a spectra method. For that, both ω and v are expanded in a doube truncated series. In x-direction (periodic direction) a discrete Fourier series is used, whie Chebyshev poynomias are empoyed in y-direction (nonperiodic direction). When the series expressions for ω and v are pugged into (6.4) and (6.5), one obtains a system of equations for the spectra coefficients of ω and v. The physica vaues of vorticity and veocity are then computed performing an inverse transform. This is done efficienty using fast Fourier transform (FFT) methods. Specia care has to be taken to impose the boundary conditions (6.7)-(6.8). For that the Lanczos tau method is appied, see [20]. The method consists of dropping the two highest modes in the Fourier-Chebyshev expansion in each direction, and evauating the coefficients of these modes expicity in terms of (6.7)-(6.8). Since a boundary condition for ω is

111 102 Chapter 6 - Soving transport probems using LDC Figure 6.2: A Fourier-Chebyshev grid not a priori given, the infuence matrix technique described in [22, 24] is adopted. Appication of the infuence matrix method ensures that the computed vaues of v and ω satisfy (6.6). Moreover the computed veocity fied is divergence free. We finay note that, when the fast Fourier transform agorithm is appied to convert the spectra coefficient of veocity and vorticity into the physica space, the physica vaues v and ω are computed at the grid points (x i, y j ) defined by x i := 2i, i = 0, 1,..., N, (6.14) N ( ) j π y j := cos, j = 0, 1,..., M. (6.15) M In the definitions above, N and M are the number of Fourier and Chebyshev modes considered in the series expansion, whie points y j are caed Gauss-Lobatto points. Figure 6.2 represents an exampe of a Fourier-Chebyshev grid on the domain Ω. The figure is drawn for N = M = 12. The Fourier-Chebyshev spectra method described in [38] is based on the method previousy proposed in [22]. In [22] non-periodic boundary conditions for v are considered in both x- and y-direction, the variabes are expanded in a doube truncated series of Chebyshev poynomias and the computationa grid consists of Gauss-Lobatto points in both directions. A 3D veocityvorticity sover for cyindrica geometries is described in [55]. The sover is based on a Fourier expansion in the azimutha direction, whie Chebyshev poynomias are used in the nonperiodic radia and axia directions.

112 6.3 Numerica resuts Impementation In order to sove the transport equation (6.1) we deveoped a C++ code. The code impements the finite voume adapted LDC method with the conservative regridding strategy introduced in Section 5.1, and it can hande mutipe eves of refinement. The Fourier-Chebyshev method described in [38] for the soution of the Navier-Stokes equations is impemented in a FORTRAN code that we run as externa users. We couped that code with the C++ code we deveoped to sove (6.1) by means of LDC. In practice, we use the output of the spectra code (i.e. the veocity fied) as an input to sove the transport probem. As noted before the FORTRAN code, after performing the inverse Fourier-Chebyshev transform, returns the physica vaues of the veocity fied on a Fourier-Chebyshev grid ike the one in Figure 6.2. In the finite voume LDC, however, we need to know v on the boundary of the coarse and fine grid contro voumes that form the composite grid, see Figure 4.4; this is for the evauation of fuxes (4.5). In the C++ code the physica veocity fied is thus interpoated in space on the composite grid used by LDC; for that we use piecewise inear interpoation in space. Furthermore, the goba and oca LDC time steps might not coincide with the time step chosen to integrate the Navier-Stokes equations. Therefore the veocity fied must be interpoated in time too. Aso in this case we appy piecewise inear interpoation. Note that both (6.1) and (6.4) are advection-diffusion equations. Since the soution of (6.4) is aready impemented in the Fourier-Chebyshev code, we can use the spectra method to sove the transport probem (6.1) too. This provides a numerica soution that can be compared to the one computed by LDC. In genera, spectra methods are widey used in fow simuations for their exponentia convergence behavior, and their reduced numerica damping and dispersion properties [20]. However, spectra methods cannot easiy cope with compicated geometries. Moreover, spectra methods are certainy not optima for soutions with extremey ocaized properties: this is because the computationa grid is fixed in time and grid refinement wi impy a fine grid spacing on the whoe domain, not ony where the high activity in the soution occurs. LDC, on the other hand, is combined with traditiona techniques ike finite differences or finite voumes and it can be easiy impemented in domains that are for exampe unions of rectanges. Furthermore, LDC guarantees not ony adaptive grid refinement but aso adaptive time integration, so that sma time steps are adopted ony to resove the fast variations in the soution. 6.3 Numerica resuts The soution of the dipoe-wa coision probem incuding the transport equation is computed running the couped FORTRAN C++ code. The Navier-Stokes equations in veocity-vorticity formuation are integrated in the domain Ω with Re = 250 ti the fina time t end = 1. The foowing discretization parameters are used: N = 128

113 104 Chapter 6 - Soving transport probems using LDC Fourier modes, M = 128 Chebyshev modes, and the time step t fow = The computed veocity fied is symmetric with respect to the ine x = 1. The transport equation (6.1) is soved on Ω haf with Pe = 500 ti t end = 1. For soving the transport probem, the finite voume adapted LDC method is appied with two eves of refinement. On the goba coarse grid we use a ce-centered approach. Both at eve 1 and 2 the operator P h,h performs piecewise quadratic interpoation and exacty one LDC iteration is done at each (sub)time step. The oca regions are chosen using the agorithm described in [10, 11, 65]; at eve 1 the threshod vaue is ǫ = 1.5, at eve 2 we choose ǫ = 2. At both eves we et the operator Q n x perform piecewise quadratic interpoation. Athough LDC is not restricted to rectanguar subdomain, we ony empoy rectanguar oca regions for simpicity of impementation. The computed resuts are presented in Figures 6.3 and 6.4. In the figures, the distribution ϕ of the passive tracer in Ω haf, the position of the oca grids and the veocity fied are shown for some vaues of t [0, t end ]. These vaues are not chosen uniformy in the interva [0, t end ]. Before t 0.4, in fact, the tracer distribution does not vary a ot; this is because, in this phase, the dipoe traves towards the wa y = 1 and v is neary zero in the area where the passive tracer mainy is, see Figures 6.3 (a)-(b). On the other hand, ϕ changes very rapidy in time immediatey after the dipoe hits the wa, see Figures 6.3 (c)-(d) and Figures 6.4 (a)-(b)-(c). At the fina time, Figure 6.4 (d), the passive tracer is more uniformy distributed on a reativey arger area. The distribution of the passive tracer shown in Figures 6.3 and 6.4 is computed approximating the fuxes by the second-order centered differences scheme, and the integras by the midpoint rue at a eves. The goba grid spacing is such that H x = H y = 1/20, whie the factors of grid refinement are σ x = σ y = 5 at every eve. The time discretization is performed by the backward Euer scheme at eves 0 and 1, and by the so-caed ϑ-method at eve 2. The ϑ-method coincides with the first-order backward Euer scheme if ϑ = 1, with the first-order forward Euer scheme if ϑ = 0, and with the second-order Crank-Nicoson scheme if ϑ = 0.5. In our simuations we set ϑ = 0.51, so that, in practice, the method is sti second-order accurate whie having better damping properties than the Crank-Nicoson scheme. At each time step, we determine the time step t empoyed to integrate the coarse grid probem using Agorithm 6.1. In our simuation we take t min = , t max = , K = 1.05, to 1 = and to 2 = The first time step is performed with t = At both eves, the factor of time refinement is τ = 5 at every time. The time step t actuay used in our simuation is potted as a function of t in Figure 6.5. The figure shows that the minimum time step is adopted from t 0.4 to t 0.5. Indeed the most rapid variations in the soution occur during this time interva (see again Figures 6.3 and 6.4). We note that if we woud adopt a constant time step t = t min = the tota number of LDC time steps necessary to integrate the transport probem ti t end = 1 woud be N t := t end / t min = 1/ = Appication of Agorithm 6.1, on the other hand, requires the computation of N t = 2920 time steps ony.

114 6.3 Numerica resuts (a) t = 0.00 y x (b) t = 0.30 y x (c) t = 0.40 y x (d) t = 0.45 y x Figure 6.3: Dipoe-wa coision probem with Re = 250 and Pe = 500: distribution of the passive tracer and ocation of the oca grids (eft), veocity fied (right). Part 1.

115 106 Chapter 6 - Soving transport probems using LDC (a) t = 0.50 y x (b) t = 0.60 y x (c) t = 0.70 y x (d) t = 1.00 y x Figure 6.4: Dipoe-wa coision probem with Re = 250 and Pe = 500: distribution of the passive tracer and ocation of the oca grids (eft), veocity fied (right). Part 2.

116 6.3 Numerica resuts x t max = t 0 min = t Figure 6.5: Time step t used to sove the transport probem t Figure 6.6: Pot of µ(t) = M(t) M(0) /M(0), with M(t) defined by (6.13). The C++ code computes the tota amount of passive tracer M(t), see (6.13), at each time eve. In order to see how we the LDC method reproduces the conservation aw (6.12), the reative variation µ(t) = M(t) M(0) /M(0) is potted as a function of time in Figure 6.6. The figure shows that the maximum of µ is not higher than As in the exampe discussed in Section 5.1.2, µ(t) is not zero because the a-priori chosen number of LDC iterations might not be sufficient to guarantee fu convergence of the LDC iteration at every (sub)time step Comparison between LDC and the spectra method The transport probem with Pe = 500 is aso soved by the spectra method impemented in the FORTRAN code. The structure of the code is such that, as externa users, it is not straightforward to expoit the symmetry of the probem with respect

117 108 Chapter 6 - Soving transport probems using LDC 0.6 Spectra LDC 0.6 Spectra LDC y 0.8 y x x (a) t = 0.5 (b) t = 0.6 Figure 6.7: Contour pots of ϕ as computed by LDC and by the spectra method at two time eves. to x = 1; therefore the transport equation is soved on the entire domain Ω. The time step adopted is the same as the one used to integrate the Navier-Stokes equations, i.e. t fow = In this way the veocity fied does not have to be interpoated in time to be avaiabe for soving the transport equation. Spatia discretization is performed using 768 Fourier modes and 768 Chebyshev modes. With these settings, grid sizes and time steps are very simiar in the spectra and in the LDC method. With reference to the x-direction, the grid size in the spectra method is equa to 2/768 = , whie it is equa to H x /σ 2 x = (1/20)/5 2 = at the finest grid (eve 2) of LDC. Moreover, the minimum time step used for the finest LDC grid is t min /τ 2 = /5 2 = , which is of the same order of magnitude as t fow. We compare the distribution of the passive tracer computed by LDC and by the spectra method in Figure 6.7. In the figure contour pots of the two soutions are drawn for two different time eves, namey t = 0.5 and t = 0.6; the eves are chosen in the interva where ϕ varies most rapidy in time. In both cases we can observe ony margina differences between the two soutions. A more quantitative comparison between the two methods can be carried out considering the integra quantities x m (t) := 1 x ϕ(x, y, t) dx dy, Σ x (t) := 1 (x x m ) 2 ϕ(x, y, t) dx dy, M(t) M(t) y m (t) := 1 M(t) Ω Ω yϕ(x, y, t) dx dy, Σ y (t) := 1 M(t) Ω Ω (y y m ) 2 ϕ(x, y, t) dx dy, with M(t) defined by (6.13). The point with coordinates (x m, y m ) is the center of

118 6.3 Numerica resuts t t t (a) (y m ) LDC (b) (Σ x ) LDC (c) (Σ y ) LDC Figure 6.8: Pots of the integra quantities y m, Σ x and Σ y as computed by the LDC code t t t (a) r(y m ) (b) r(σ x ) (c) r(σ y ) Figure 6.9: Pots of the reative differences between the integra quantities y m, Σ x and Σ y computed by LDC and by the spectra method. mass of the passive tracer distribution, whie (Σ x, Σ y ) represents its variance. We note that the behavior of x m (t) is not of specia interest since the symmetry of the transport probem with respect to x = 1 impies that x m (t) = const = 1. We wi therefore focus on the other three integra quantities (i.e. y m, Σ x, Σ y ) and we wi adopt the foowing notation: the symbos (y m ) LDC and (y m ) spectra are used to indicate the approximations of y m computed by LDC and by the spectra method respectivey, and their reative difference is denoted by r(y m ) := (y m ) LDC (y m ) spectra (y m ). (6.16) spectra Simiar notation is used for Σ x and Σ y. The approximations (y m ) LDC, (Σ x ) LDC and (Σ y ) LDC are potted as a function of time in Figure 6.8, whie the reative differences r(y m ), r(σ x ) and r(σ y ) are shown in Figure 6.9. From Figure 6.9 we can see

119 110 Chapter 6 - Soving transport probems using LDC Method Number of discretized equations soved Ratio Spectra eve LDC eve eve Tabe 6.1: Tota number of discretized equations soved by both spectra and LDC code in the transport probem with Pe = 500. that the reative differences between the integra quantities computed by LDC and by the spectra method are at most of the order of Finay, we woud ike to give an estimate of the compexity of the two methods for soving the transport probem with Pe = 500. For this reason we cacuate the tota number of discretized equations soved by both codes from t = 0 ti t = t end. The resuts are given in Tabe 6.1. For the spectra method the tota number of discretized equations to be soved is simpy given by the product of the Fourier- Chebyshev modes adopted in the doube series expansion ( ) and the number of time steps performed (t end / t fow = ). For the LDC method it is the sum of the number of discretized equations soved at each eve. The numbers reported in Tabe 6.1 aready take into account that the mutieve LDC time marching strategy requires discretized equations to be soved more than once at each eve, see Figure 5.4. In order to have a fair comparison between the two techniques, a the numbers cacuated for LDC are sti to be mutipied by 2; this is because the spectra method soves the transport probem on the entire Ω, whie LDC on Ω haf ony. The ast coumn of Tabe 6.1 shows that LDC soves significanty ess discretized equations than the spectra method: the ratio between the tota number of discretized equations soved by the spectra and by the LDC code is in fact more than 5. This resut coud be further improved by modifying the C++ code we deveoped in such a way that the oca subdomains do not necessariy have to be rectanguar. This woud certainy be beneficia in those time steps where the oca high activity is not aigned with the main axes, see for exampe Figure 6.3-(d) or Figure 6.4-(a). Note that our compexity estimate does not consider factors ike the overhead in the C++ code due to the space-time interpoation to provide veocity vaues where needed by LDC. This effect might pay a non-margina roe. In fact, when the actua computationa times of the two codes are compared, we see that they are of the same order of magnitude. Interpoation costs coud become even higher when higher resoutions are needed, for exampe in probems with higher Reynods and Pécet numbers than the ones considered here. A natura way to overcome this woud be to extend the LDC method to systems of time-dependent PDEs and appy it to the soution of the Navier-Stokes equations too; Figures 6.3 and 6.4 indicate in fact that aso the soution of the fow probem has a oca region of high activity. In this way, one coud tacke probems with high Re and Pe without having to dea with uniform

120 6.3 Numerica resuts 111 fine meshes and time steps ike required by spectra methods; using LDC, one can adopt reativey sma grid sizes and time steps to resove the oca high activity ony. In concusion, the exampe presented in this section suggests that the LDC method for paraboic partia differentia equations we deveoped and anayzed in this thesis is a vaid aternative to spectra methods for the soution of transport probems with highy ocaized properties.

121

122 Chapter 7 Concusions and recommendations In this thesis we present, anayze and impement a new static-regridding technique for paraboic partia differentia equations: Loca Defect Correction. The method generaizes the LDC technique initiay introduced for the efficient soution of eiptic probems with highy ocaized properties. In particuar, the origina idea of oca defect correction is extended in the sense that the defect term is used to correct the error of the goba coarse grid approximation not ony due to the spatia dicretization, but aso to the time discretization. This can be achieved by soving the oca probem not ony with a finer grid size, but aso with a smaer time step than the one adopted gobay. One of the main advantages of LDC is that one may work with uniform structured grids and uniform grid sovers ony. In comparison with dynamic-regridding methods, LDC requires very itte tuning of user-specified parameters; basicay, just one parameter has to be set in LDC: this is for the adaptive choice of the oca domain at each time eve. We discuss some properties of the LDC agorithm for paraboic probems. In particuar, we show that coarse and fine grid soution coincide at the common points between the two grids, and we derive the system of discretized equations that the composite grid soution satisfies. The LDC method is tested on some concrete exampes, that iustrate its accuracy, efficiency and robustness. A direction for future research is to provide error bounds for the LDC composite grid soution at a generic time eve. The error estimates shoud take into account a the approximations introduced by the numerica method: space and time discretization on the goba and on the oca grid, space and time interpoation to provide artificia boundary for the oca probem, space interpoation to provide initia vaues in a the points of the new composite grid during regridding.

123 114 Chapter 7 - Concusions and recommendations The LDC method produces a composite grid soution at the new time eve combining a goba and a oca approximation iterativey. The convergence behavior of the LDC agorithm is studied in detai in this thesis: a genera expression for the iteration matrix of the method is derived, and the properties of the iteration matrix are studied both anayticay and by means of numerica experiments. The resuts of the anaysis iustrate how the coarse grid discretization parameters (grid size and time step) and the choice of the so-caed safety region infuence the convergence rate of the LDC agorithm. In genera, we observe that LDC converges for any choice of the discretization parameters and that iteration errors are reduced by severa orders of magnitude at every iteration. In this thesis we aso propose a finite voume adapted LDC agorithm for timedependent probems that yieds discrete conservation on the composite grid. This resut can be achieved by writing the defect term in such a way that a baance of fine and coarse grid fuxes is guaranteed across the interface between the two grids at each time eve. The method can hande the fact that time integration on the oca grid is performed with a smaer time step than the one adopted gobay. The finite voume adapted LDC agorithm is successfuy tested in some numerica experiments, and then extended to incorporate a conservative regridding strategy and mutipe eves of refinement. Further improvements of the method might be the impementation of genera Runge-Kutta schemes for time integration, or the design of a taiored strategy for detecting the oca high activity at each (sub)time eve. The mutieve conservative LDC agorithm is finay appied to sove transport probems in transient fow fieds. With reference to a dipoe-wa coision probem, the resuts computed by LDC are compared to the soution computed by a Chebyshev-Fourier spectra method. The two techniques give very simiar soutions when simiar grid spacing and time steps are adopted; yet, LDC requires the soution of a smaer number of discretized equations. As mentioned before, the efficiency of LDC coud be further improved by aowing the oca subdomains to have a more genera shape than rectanges, for exampe union of rectanges. The resuts presented and, in genera, the attractive properties of LDC make future research on oca defect correction methods worthwhie. Interesting topics woud certainy be the soution of systems of time-dependent partia differentia equations and the appication of LDC to more compex transport and fow probems.

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125 116 Bibiography [10] B. A. V. Bennett and M. D. Smooke. Loca rectanguar refinement with appication to axisymmetric aminar fames. Combustion Theory and Modeing, 2: , [11] B. A. V. Bennett and M. D. Smooke. Loca rectanguar refinement with appication to nonreacting and reacting fuid fow probems. Journa of Computationa Physics, 151: , [12] M. J. Berger and P. Coea. Loca adaptive mesh refinement for shock hydrodynamics. Journa of Computationa Physics, 82:64 84, [13] M. J. Berger and R. J. Leveque. Adaptive mesh refinement using wavepropagation agorithms for hyperboic systems. SIAM Journa on Numerica Anaysis, 35: , [14] M. J. Berger and J. Oiger. Adaptive mesh refinement for hyperboic partia differentia equations. Journa of Computationa Physics, 53: , [15] L. K. Bieniasz. Use of dynamicay adaptive grid techniques for the soution of eectrochemica kinetic equations: Part 5. A finite-difference, adaptive space/time grid strategy based on a patch-type oca uniform spatia grid refinement, for kinetic modes in one-dimensiona space geometry. Journa of Eectroanaytica Chemistry, 481: , [16] J. G. Bom and J. G. Verwer. VLUGR3: a vectorizabe adaptive grid sover for PDEs in 3D, part I: agorithmic aspects and appications. Appied Numerica Mathematics, 16: , [17] F. Bramkamp, P. Lamby, and S. Müer. An adaptive mutiscae finite voume sover for unsteady and steady state fow computations. Journa of Computationa Physics, 197: , [18] A. Brandt. Muti-eve adaptive soutions to boundary vaue probems. Mathematics of Computation, 31: , [19] W. L. Briggs, V. E. Henson, and S. F. McCormick. A mutigrid tutoria. SIAM, Phiadephia, 2nd edition, [20] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectra Methods in Fuid Dynamics. Springer-Verag, [21] C. Carstensen, R. D. Lazarov, and S. Tomov. Expicit and averaging a posteriori error estimates for adaptive finite voume methods. SIAM Journa on Numerica Anaysis, 42: , [22] H. J. H. Cercx. A spectra sover for the Navier-Stokes equations in the veocity-vorticity formuation for fows with two nonperiodic directions. Journa of Computationa Physics, 137: , [23] H. J. H. Cercx and C. H. Bruneau. The norma and obique coision of a dipoe with a no-sip boundary. Computers and Fuids, 35: , 2006.

126 Bibiography 117 [24] O. Daube. Resoution of the 2D Navier-Stokes equations in veocity-vorticity form by means of an infuence matrix technique. Journa of Computationa Physics, 103: , [25] C. Dawson and R. Kirby. High resoution schemes for conservation aws with ocay varying time steps. SIAM Journa on Scientific Computing, 22: , [26] P. E. Dimotakis. Turbuent mixing. Annua Review of Fuid Mechanics, 37: , [27] P. J. J. Ferket. Couping of a goba coarse discretization and oca fine discretizations. In W. Hackbusch and G. Wittum, editors, Numerica Treatment of Couped Systems, voume 51 of Notes on Numerica Fuid Mechanics, pages 47 58, Braunschweig, Vieweg. [28] P. J. J. Ferket. Soving boundary vaue probems on composite grids with an appication to combustion. PhD thesis, Eindhoven University of Technoogy, Eindhoven, [29] P. J. J. Ferket and A. A. Reusken. Further anaysis of the oca defect correction method. Computing, 56: , [30] P. J. J. Ferket and A. A. Reusken. A finite difference discretization method on composite grids. Computing, 56: , [31] J. E. Faherty, R. M. Loy, M. S. Shephard, B. K. Skymanski, J. D. Teresco, and L. H. Ziantz. Adaptive oca refinement with octree oad baancing for the parae soution of three-dimensiona conservation aws. Journa of Parae and Distributed Computing, 47: , [32] M. Graziadei. Using oca defect correction for aminar fame simuation. PhD thesis, Eindhoven University of Technoogy, Eindhoven, [33] M. Graziadei, R. M. M. Mattheij, and J. H. M. ten Thije Boonkkamp. Loca defect correction with santing grids. Numerica Methods for Partia Differentia Equations, 20:1 17, [34] W. Hackbusch. Loca defect correction and domain decomposition techniques. In K. Böhmer and H. J. Stetter, editors, Defect Correction Methods. Theory and Appications, Computing, Supp. 5, pages , Wien, New York, Springer. [35] W. Huang. Practica aspects of formuation and soution of moving mesh partia differentia equations. Journa of Computationa Physics, 171: , [36] W. Huang, Y. Ren, and R. D. Russe. Moving mesh partia differentia equations (MMPDES) based on the equidistribution principe. SIAM Journa on Numerica Anaysis, 31: , 1994.

127 118 Bibiography [37] W. Huang and R. D. Russe. Moving mesh strategy based on a gradient fow equation for two-dimensiona probems. SIAM Journa on Scientific Computing, 20: , [38] W. Kramer. Dispersion of passive tracers in two-dimensiona bounded fows. PhD thesis, Eindhoven University of Technoogy, Eindhoven, to appear in [39] R. D. Lazarov, I. D. Mishev, and P. S. Vassievski. Finite voume methods with oca refinement for convection-diffusion probems. Computing, 53:33 57, [40] R. D. Lazarov and S. Tomov. A posteriori error estimates for finite voume eement approximations of convection-diffusion-reaction equations. Computationa Geosciences, 6: , [41] S. McCormick. Fast adaptive composite grid (FAC) methods: Theory for the variationa case. In K. Böhmer and H. J. Stetter, editors, Defect Correction Methods. Theory and Appications, Computing, Supp. 5, pages , Wien, New York, Springer. [42] S. F. McCormick and J. Thomas. The fast adaptive composite grid (FAC) method for eiptic equations. Mathematics of Computation, 46: , [43] R. Minero, M. J. H. Anthonissen, and R. M. M. Mattheij. Loca defect correction for time-dependent partia differentia equations. In Proceedings of the 16th Internationa Conference on Domain Decomposition Methods, Avaiabe onine at [44] R. Minero, M. J. H. Anthonissen, and R. M. M. Mattheij. A oca defect correction technique for time-dependent probems. Numerica Methods for Partia Differentia Equations, 22: , [45] R. Minero, M. J. H. Anthonissen, and R. M. M. Mattheij. Soving paraboic probems using oca defect correction in combination with the finite voume method. Numerica Methods for Partia Differentia Equations, In press. [46] R. Minero, H. G. ter Morsche, and M. J. H. Anthonissen. Convergence properties of the oca defect correction method for paraboic probems. Technica Report CASA 05-40, Eindhoven University of Technoogy, Eindhoven, November Submitted for pubication. [47] V. Nefedov and R. M. M. Mattheij. Loca defect correction with different grid types. Numerica Methods for Partia Differentia Equations, 18: , [48] D. Norman. Introduction to Linear Agebra for Science and Engineering. Addison-Wesey Pubishers Limited, [49] P. Orandi. Vortex dipoe rebound from a wa. Physics of Fuids A, 2: , [50] T. Pewa and E. Müer. AMRA: an adaptive mesh refinement hydrodynamic code for astrophysics. Computer Physics Communications, 138: , 2001.

128 Bibiography 119 [51] O. Rousse, K. Schneider, A. Tsiguin, and H. Bockhorn. A conservative fuy adaptive mutiresoution agorithm for paraboic PDEs. Journa of Computationa Physics, 188: , [52] B. Sawford. Turbuent reative dispersion. Annua Review of Fuid Mechanics, 33: , [53] I. E. M. Severens, M. J. H. Anthonissen, R. M. M. Mattheij, and J. M. L. Maubach. Appication of the oca defect correction method in discrete eement method simuations. Technica Report CASA 04-17, Eindhoven University of Technoogy, Eindhoven, August [54] M. Sizov, M. J. H. Anthonissen, and R. M. M. Mattheij. Anaysis of oca defect correction and high-order compact finite differences. Numerica Methods for Partia Differentia Equations, 22: , [55] M. F. M. Speetjens and H. J. H. Cercx. A spectra sover in the veocity-vorticity formuation for ow Reynods number fows in 3D cyindrica cavities. Internationa Journa of Computationa Fuid Dynamics, 19: , [56] M. F. M. Speetjens, H. J. H. Cercx, and G. J. F. van Heijst. A numerica and experimenta study on advection in three-dimensiona Stokes fows. Journa of Fuid Mechanics, 514:77 105, [57] Z. Tan, Z. Zhang, Y. Huang, and T. Tang. Moving mesh methods with ocay varying time steps. Journa of Computationa Physics, 200: , [58] H. Tang and T. Tang. Adaptive mesh methods for one- and two-dimensiona hyperboic conservation aws. SIAM Journa on Numerica Anaysis, 41: , [59] C. R. Traas, H. G. ter Morsche, and R. M. J. van Damme. Spines en Waveets. Epsion Uitgaven, Utrecht, In Dutch. [60] R. A. Trompert. Loca uniform grid refinement and systems of couped partia differentia equations. Appied Numerica Mathematics, 13: , [61] R. A. Trompert. Loca-uniform-grid refinement and transport in heterogeneous porous media. Advances in Water Resources, 16: , [62] R. A. Trompert. Loca Uniform Grid refinement for Time-Dependent Partia Differentia Equations. PhD thesis, University of Amsterdam, Amsterdam, [63] R. A. Trompert and J. G. Verwer. Anaysis of the impicit Euer oca uniform grid refinement method. SIAM Journa on Scientific Computing, 14: , [64] R. A. Trompert and J. G. Verwer. Runge-Kutta methods and oca uniform grid refinement. Mathematics of Computation, 60: , 1993.

129 120 Bibiography [65] B. A. Vadati (Bennett). Soution-adaptive gridding methods with appication to combustion probems. PhD thesis, Yae University, New Haven, CT, [66] J. U. Wapper. Die okae Defektkorrekturmethode zur adaptiven Diskretisierung eiptischer Differentiageichungen mit finiten Eementen. PhD thesis, Christian-Abrechts-Universität, Kie, In German. [67] Z. Warhaft. Passive scaars in turbuent fows. Annua Review of Fuid Mechanics, 32: , [68] P. A. Zegeing and H. P. Kok. Adaptive moving mesh computations for reaction-diffusion systems. Journa of Computationa and Appied Mathematics, 168: , 2004.

130 Index ABCN scheme, 101 Adams-Bashforth scheme, 101 AMR, 2, 3 asymptotic equivaence, 42 backward Euer, 11, 34, 65 baance of fuxes, across the interface, 6, 59 boundary conditions artificia, 4, 12 13, 65 different types, 10, 60, 66 fow probem, 99 in spectra method, 101 C++ code, 103, 110 ce-centered approach, 78 Chebyshev poynomias, 101, 102 compexity, 25, 81, 110 conservation in LDC during regridding, numerica exampes, 79 84, system of discrete cons. aws, 77 contro voume, 61 convergence of LDC 1D Poisson s equation, 46 1D convection-diffusion probem, 51 1D diffusion probem, 47, 49 2D Poisson s equation, 6, 31, 55 2D convection-diffusion probem, 56 2D diffusion probem, 53 in one iteration, 46 Crank-Nicoson scheme, 101, 104 defect, 14, 67 approximation, differentia formuation, 60 diffusion coefficient, 60 dipoe, 99 Direct Numerica Simuations (DNS), 1, 97 divergence free, 60, 102 dynamic-regridding methods, 2 expicit time integration in LDC, 21 FAC, 4 factor of time refinement, 64 factors of grid refinement, 63 fast Fourier transform (FFT), 101 fux, 60 FORTRAN code, 103, 107 Fourier transform, 41 Fourier-Chebyshev grid, inverse transform, 103 Gauss-Lobatto points, 102 grid composite, 4, 9, 63 goba coarse, 10, 34, 61 oca fine, 10, 63 grid function, 11, 61 heat equation one-dimensiona, 34, 38, 49 two-dimensiona, 53 homogeneous system, 76 impicit Euer, see backward Euer infuence matrix, 102

131 122 Index initia vaues for LDC time step, 21, 65, integra formuation, 60, 91 integra quantities, integrated fux, 62 interface, 12 interpoation compute veocity fied, 103 oca boundary conditions, 4, 13, 65 piecewise inear, 23, 27, 80, 84 piecewise quadratic, 104 regridding, 19, 88 iteration error, 32 iteration matrix, 33 kinetic energy of a fow, 100 Lanczos tau method, 101 LDC agorithm eiptic probems, 4 finite voume adapted, 71 finite voume adapted with regridding, 89 mutipe eves of refinement, 93 paraboic probems, 20 LDC convergence, see convergence of LDC LDC iteration, 4, 14 fixed point, 17 stopping criterion, 21 oca discretization error, see defect oca region choice of, 19, 63 circuar shape, 6 santwise direction, 6 LUGR, 3, 9 vs. LDC, midpoint rue, 62 in numerica experiments, 80, 82, 91, 104 regridding, MLAT, 4, 5 MMPDE, 2 moving-grid methods, 2 3 non-uniform convergence, partition of coarse grid points, 12 Pécet number, 98 Poisson summation formua, 41 Poisson s equation, 5, 6, 31, 46 for the veocity fied, 101 regridding strategy, 18 19, repacement of coarse grid soution ocay, 22 symbo, 89 residua, 39 restriction operator, 14 Reynods number, 99 Runge-Kutta schemes in LDC, 21, 65, 114 in spectra method, 101 safety region, 14 15, separation of variabes, 39 source term, 10, 60 spectra method, 101 vs. LDC, 103, spectra radius, 33 static-regridding methods, 2 symmetric diagonaization theorem, 36 Tayor expansions, 39 ϑ-method, 104 time marching, 8, 94 uniform convergence, uniform grid, 3, 113 coarse, see grid, goba coarse fine, see grid, oca fine uniform grid sover, 3, 18, 113 vs. LDC, 22 26, veocity-vorticity formuation, see Navier- Stokes equations vertex-centered approach, 61 vorticity, 98 Navier-Stokes equations, 98 99

132 Summary The soutions of partia differentia equations (PDEs) describing physica phenomena are often characterized by oca regions of high activity, i.e., regions where spatia gradients are quite arge compared to those in the rest of the domain, where the soution presents a reativey smooth behavior. Exampes are encountered in many appication areas; one of them is the transport of passive tracers in turbuent fow fieds. A passive tracer is a diffusive contaminant in a fuid fow that is present in such ow concentration that it does not infuence the dynamics of the fow. A few exampes of passive tracer transport from everyday ife are the exhaust gases from chimneys, smoke from a cigarette, dust partices spread by the wind, etc. Understanding the infuence of the main fow on the tracer materia is crucia in many engineering appications, such as mixing in chemica reactors or transport of fy ashes in burners. Knowedge of the passive tracer behavior is aso of fundamenta importance in environmenta sciences for studying dispersion of poutants (e.g. chemica species, radioactive components) in the atmosphere or in the oceans. Since fiaments of tracer materia are often concentrated ony in a very imited part of the computationa domain, an efficient numerica soution of this type of probems requires the usage of adaptive grid techniques. In adaptive grid methods, a fine grid spacing and a reativey sma time step are adopted ony where the reativey arge variations occur, so that the computationa effort and the memory requirements are minimized. This thesis focuses on a new adaptive grid method for efficienty soving paraboic PDEs characterized by highy ocaized properties: Loca Defect Correction (LDC). In LDC the PDE is integrated on a goba uniform coarse grid and on a oca uniform fine grid; the atter is adaptivey paced at each time eve where the high activity in the soution occurs. At each time step, goba and oca soution are iterativey combined to utimatey produce a soution on the composite grid, union of goba and oca grid. In particuar, the goba approximation provides artificia boundary conditions for the oca fine grid probem, whie the oca approximation is used to estimate the coarse grid oca discretization error (or defect) and then to improve the soution gobay by means of a defect correction. In the agorithm we propose, the oca probem is soved not ony with a smaer grid size, but aso with a smaer time step than the one used for the goba probem. In this way the oca soution

133 124 Summary corrects the error in the goba approximation due not ony to spatia discretization, but aso to tempora discretization. When the LDC iteration has come to a fixed point, it can be proved that the goba and the oca soution at the new time eve coincide at the common points between the two grids. The method described in this thesis extends the LDC technique that was initiay introduced in the iterature for soving eiptic PDEs. The new LDC agorithm is tested in some concrete exampes that iustrate its accuracy, efficiency and robustness. LDC is an iterative process that can be used for practica appications ony if it converges sufficienty fast. In this thesis the convergence properties of the LDC method for time-dependent probems are studied in detai, both anayticay and by means of numerica experiments. For both one- and two-dimensiona probems we investigate the dependency of the LDC convergence rate on the discretization parameters (grid size and time step) for the coarse grid probem. In genera, it is observed that LDC converges for any choice of the discretization parameters and that iteration errors are reduced by severa orders of magnitude at every iteration. When the coarse and the fine grid probem are discretized appying the finite voume method, specia care is needed to guarantee that a discrete conservation property hods for the LDC soution on the composite grid. In fact, if no specia precautions are taken, the standard LDC method for time-dependent probems is such that fuxes across the interface between goba and oca grid are not necessariy in baance. In this thesis we propose a finite voume adapted LDC agorithm for paraboic PDEs. In this agorithm the defect term is adapted in such a way that, at each time step, fuxes across the interface between goba and oca grid are in baance at convergence of the LDC iteration. The finite voume adapted LDC agorithm is then extended to incude a conservative regridding strategy. The strategy guarantees that the composite grid soution satisfies a discrete conservation aw aso when the oca region is moved in time to foow the behavior of the soution. The LDC technique is not restricted to one eve of refinement. In this thesis a mutieve LDC method for time-dependent probems is introduced. The time marching strategy is such that time integration at the finer eves can be performed with smaer time steps. Finay, the new, fast converging, conservative and mutieve LDC agorithm is appied to sove a transport probem with highy ocaized properties. In particuar, we test the LDC method on a dipoe-wa coision probem. The probem is soved both by LDC and by a Chebyshev-Fourier spectra method. When the two numerica soutions are compared, we see that the two methods yied very simiar resuts. LDC, however, is specificay meant for soving probems whose soutions exhibit oca regions of high activity, and for this reason it turns out to be a ess compex agorithm than the Chebyshev-Fourier spectra method.

134 Samenvatting Opossingen van partiëe differentiaavergeijkingen (PDV en) die fysische verschijnseen beschrijven worden vaak gekenmerkt door okae gebieden met hoge activiteit. Dit zijn gebieden waar de paatsafgeeiden groot zijn in verhouding tot de gradiënten in de rest van het domein waar de opossing reatief gad is. Er zijn ta van voorbeeden te vinden in verschiende toepassingsgebieden zoas bijvoorbeed het transport van merkstof in turbuente stromingen. Een merkstof is een verontreiniging die zich in een voeistofstroming verspreidt en in dusdanig age concentraties aanwezig is dat de dynamica van de stroming niet wordt beïnvoed. Aedaagse voorbeeden van merkstoffen zijn de gassen uit een schoorsteen of uitaat, rook van een sigaret en door de wind getransporteerde stofdeetjes. Het begrijpen van de invoed van de hoofdstroming op de merkstofdeetjes is beangrijk voor ta van technische toepassingen zoas het mixen in een chemische reactor of het transport van viegas in een brander. Kennis van het gedrag van een merkstof is ook van fundamentee beang in miieustudies naar de verspreiding van vervuiing (zoas chemische stoffen of radioactieve componenten) in de atmosfeer of oceaan. Omdat fiamenten van merkstofmateriaa vaak geconcentreerd zijn in een kein dee van het rekendomein zijn adaptieve roostertechnieken nodig om dit soort probemen efficiënt op te ossen. In adaptieve roostermethoden wordt aeen een keine maaswijdte en tijdstap genomen in die gebieden waar de reatief grote veranderingen paatsvinden. Hierdoor wordt de hoeveeheid rekenwerk en geheugengebruik geminimaiseerd. Dit proefschrift richt zich op een nieuwe adaptieve roostermethode voor het efficiënt opossen van paraboische PDV en met sterk okae eigenschappen: Lokae Defect Correctie (LDC). In LDC wordt de PDV geïntegreerd op een gobaa uniform grof rooster en een okaa uniform fijn rooster. Laatstgenoemde wordt in eke tijdstap daar gepaatst waar zich de hoge activiteit in de opossing bevindt. In eke tijdstap worden de gobae en okae opossing op een iteratieve manier gecombineerd om uiteindeijk tot een opossing te komen op het samengestede rooster, de vereniging van het gobae en okae rooster. In het bijzonder evert de gobae opossing artificiëe randvoorwaarden voor het okae fijne roosterprobeem. Daarentegen wordt de okae opossing gebruikt om de okae discretisatiefout (of defect) van het grove rooster te schatten en vervogens de gobae opossing te verbeteren met een defectcorrectie. In het voorgestede agoritme wordt het okae probeem niet aeen opge-

135 126 Samenvatting (Summary in Dutch) ost met een keinere maaswijdte maar ook met een keinere tijdstap dan die van het gobae probeem. Op deze manier corrigeert de okae opossing zowe de fout ten gevoge van de paats- as tijdsdiscretisatie in de gobae benadering. Wanneer de LDC-iteratie een vast punt heeft bereikt kan aangetoond worden dat de gobae en okae opossing op het nieuwe tijdsniveau hetzefde zijn in de punten waar beide roosters overappen. De methode die wordt beschreven in dit proefschrift breidt dus de bestaande LDC-techniek voor eiptische PDV en uit. De nieuwe LDC-techniek wordt getest aan de hand van een aanta concrete voorbeeden die de nauwkeurigheid, efficiëntie en robuustheid iustreren. LDC is een iteratief proces dat voor ta van praktische toepassingen gebruikt kan worden mits de convergentie sne genoeg is. In dit proefschrift worden de convergentie-eigenschappen van LDC voor tijdsafhankeijke probemen tot in detai bestudeerd, zowe anaytisch as met behup van numerieke experimenten. Voor zowe één- as tweedimensionae probemen wordt de convergentiesneheid van LDC onderzocht as functie van de discretisatieparameters (maaswijdte en tijdstap) van het grove roosterprobeem. In het agemeen convergeert LDC voor eke wiekeurige keuze van discretisatieparameters en bovendien wordt de iteratiefout enkee ordes keiner per stap. Wanneer het grove en fijne roosterprobeem gediscretiseerd worden met de eindige voume methode moet aandacht besteed worden aan het intact houden van discrete behoudseigenschappen van de LDC-opossing op het samengestede rooster. As er geen maatregeen worden getroffen za de standaard LDC-methode voor tijdsafhankeijke probemen de fux over de grens tussen het gobae en okae rooster niet meer in baans houden. In dit proefschrift wordt een LDC-techniek voor paraboische PDV en gepresenteerd die is aangepast voor eindige voume discretisaties. In dit agoritme wordt de defectterm aangepast zodat in eke tijdstap de fux over de grens tussen het gobae en okae rooster in evenwicht is nadat de LDC-iteratie is geconvergeerd. Het aangepaste LDC-agoritme is uitgebreid met een conservatieve regridding techniek. Deze strategie garandeert dat de opossing op het samengestede rooster vodoet aan een discrete behoudswet zefs wanneer het okae gebied verschuift in de tijd om het gedrag van de opossing te vogen. De LDC-techniek is niet beperkt tot één niveau van verfijning. In dit proefschrift wordt een LDC-methode met meer niveaus voor tijdsafhankeijke probemen geïntroduceerd. De strategie om tijdstappen te maken zorgt ervoor dat de tijdsintegratie op de fijnere niveaus met keinere stapgroottes uitgevoerd kan worden. Tensotte wordt het nieuwe, sne convergerende, conservatieve LDC-agoritme met meer niveaus toegepast op een transportprobeem met zeer okae eigenschappen. Concreet wordt de LDC-methode getest voor een dipoo-muur-botsingsprobeem. Dit probeem wordt zowe opgeost met LDC as met een Chebyshev-Fourier spectraamethode. Wanneer beide numerieke opossingen met ekaar worden vergeeken, bijkt dat ze sterk overeenstemmen. Echter, LDC is speciaa bedoed voor het opossen van probemen waar de opossing okae gebieden heeft met hoge activiteit. Om die reden is LDC een minder compex agoritme dan de Chebyshev-Fourier spectraamethode.

136 Acknowedgements At the end of this thesis I woud ike to give thanks to a of the peope that, in many different ways, have heped me during the time I spent at the Eindhoven University of Technoogy. First of a, I woud ike to thank my promotor prof.dr. Bob Mattheij for giving me the opportunity to work on my PhD in the CASA group. With my copromotor dr.ir. Martijn Anthonissen I had many usefu discussions on LDC, and his comments and remarks on a first draft of this thesis have ed to significant improvements in its stye and content. I woud aso ike to mention dr.ir. Pauine Vosbeek who heped me at the beginning of my PhD, and a the peope invoved in the SMARTER project: prof.dr. Herman Cercx, ir. Werner Kramer, dr. Hans Kuerten, ir. Joost de Hoogh, dr.ir. Ion Barosan, prof.dr.ir. Jack van Wijk. I thank a of them, especiay ir. Werner Kramer who deveoped the spectra code I used for the resuts of Chapter 6. I woud aso ike to express my specia gratitude to dr. Hennie ter Morsche: his hep and usefu suggestions have been fundamenta for the anaysis of the convergence properties of LDC (Chapter 3 of this thesis). During my four years as a PhD student I have reay enjoyed being part of the CASA group. Therefore I woud ike to thank a (but reay a) the present and the past members of this group. Some of them, though, deserve a specia word of thanks. This is certainy the case for my friends Dragan Bežanović and Migue Patricio: thanks a ot for the time we spent together, not ony at the university. Specia thanks aso go to Mark van Kraaij, whose concrete contribution to this thesis is the Ducth transation of the summary (aso thanks again to Martijn Anthonissen). During these years it has been a peasure for me to share the office with Sandra Bruin, Pieter Heres, Evgueni Shcherbakov, and more recenty with Godwin Kakuba and Zoran Iievski. Thanks a ot to a of you, and aso to Bas van der Linden, Pau de Haas and Enna van Dijk who have answered an enormous amount of sma questions prompty and efficienty for me during my PhD. A peasant moment during my working days was the unch break at Kennispoort. Here, I woud ike to thank a those peope that reguary shared this moment; some of them have aready been mentioned before, the others are: Caro De Faco, Yves van Gennip, Kamyar Maakpoor, Jos and Peter in t Panhuis, Jurgen Tas, and Erwin Vondenhoff.

137 128 Acknowedgements Fortunatey, a PhD student does not spend a of his time at the university. During these years I have had penty of opportunities to appreciate the cooking abiities of my friend Giovanni Russeo, and I woud ike to thank him for the numerous times he invited me to have dinner at his pace. Remaining in the food arena, I have reay enjoyed the time I spent at Ristorante Pisa in Eindhoven; therefore I woud ike to thank Leontine & Franco Goracci, Wadie Hanna and a the other staff members of the restaurant. During my stay in the Netherands I have ived in three different houses. I woud ike to thank a of the peope I shared with, especiay Massimo Ciacci, Stefano Atomare & Jitka Outratova, Stefano Angeocoa. In my spare time I have reay had a great time paying indoor footba with the teams Od Soccers and Ape d Huez. I woud ike to thank a the payers of these two teams and wish them good uck for future competitions. Athough iving abroad, I have not ost contact with my friends in Itay. I woud ike to take this opportunity to send my greetings to a of them, especiay to those who personay came to visit me in Eindhoven: Edoardo Depiano, Samuee Lanza, Abino Mettifogo, Eena Negri & Caudio Bongianni, Eena Prodi, and Norma Taotta. Of course, I aso traveed many times to Itay. It was possibe to do this quite often thanks to the ow fares offered by some no-fris fight companies; as a reguar customer I woud ike to ask them to keep on offering this great service. Finay, I woud ike to thank my mum Mariea and my brother Paoo: thanks a ot for your ove and support and, Paoo, good uck with your PhD! I am aso thankfu to a of my reatives; in particuar, I woud ike to express my affection for my grandmothers nonna Emma (who is not physicay here anymore) and nonna Lea. Last, but by no means east, I woud ike to thank my girfriend Lidia: this thesis is dedicated to her for a the ove and support she gave to me during these years, her patience when things were not so rosy, and her genera open-mindness. Remo Minero Eindhoven, Apri 2006

138 Curricuum vitae Remo Minero was born in Borgosesia (Itay) on August 28, He competed pre-university education at Liceo Scientifico Avogadro in Biea (Itay) in After that he enroed at Poitecnico di Torino where he gratuated cum aude in Nucear Engineering in In his Master s thesis he worked on the therma-hydrauic modeing of high temperature superconducting cabes for power transmission; the project was founded by Pirei Cabes and Systems. In 2002 he moved to the Netherands where he joined the Scientific Computing Group of the Eindhoven University of Technoogy; the group ater became the Centre for Anaysis, Scientific Computing and Appications (CASA). His research work in this group has ed to this thesis.