High Order ADER FV/DG Numerical Methods for Hyperbolic Equations

Transcription

1 Monographs of the School of Doctoral Studies in Environmental Engineering 10 High Order ADER FV/DG Numerical Methods for Hyperbolic Equations Cristóbal E. Castro 2007

2 Based on the Doctoral Thesis in Environmental Engineering (XIX cycle) defended in February 2007 at the Faculty of Engineering of the University of Trento Supervisor: Professor E. F. Toro OBE On the cover: Two dimensional shock wave interaction over a solid triangle. Numerical solution of the Euler equations on triangular mesh. Schlieren image for density. c Cristóbal E. Castro (text and images when not differently specified) Direttore della collana: Marco Tubino Segreteria di redazione: Laura Martuscelli e Fabio Bernardi Università degli Studi di Trento, Italia May 2007 ISBN:

3 To my wife Marta and my family

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5 Acknowledgements The author wishes to thank professor Toro for his guide, for shearing his knowledge and for his valuable advises, and his colleagues Vladimir, Martin and Michael for their contribution to this work and the good time spent at the Golden Rose. The author also wishes to thank all persons working in the Faculty of Engineering, specially Laura, Elena, Eleonora and his colleagues of the PhD program Dawa, Marco, Oscar, Silvia and Stefano. Special gratitude to the good friends made during these three years: Alessandro, Matteo, Sara, Daniela, Alessandra, Luca, Matteo, Raquel, Javier, Veronica, Ainhoa and the partners of football. The author also wishes to thank Ing. Valerio Caleffi and the group of University of Ferrara. v

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7 Contents 1 Introduction 1 2 Hyperbolic systems of balance laws The balance laws Characteristic curves The Riemann problem Hyperbolicity Characteristic variables The linear Riemann problem Characteristic fields for non-linear systems Rarefaction waves Shock waves Contact discontinuity wave Numerical methods for hyperbolic equations Review of numerical methods for hyperbolic equations Finite volume schemes on triangular meshes Discontinuous Galerkin schemes Reconstruction procedure Finite Volume schemes in one space dimension Classical Riemann solvers Introduction Euler equations Introduction Equations and eigenstructure Non-linear Riemann solver Contact waves Rarefaction waves Shock waves vii

8 Contents Iterative procedure to solve the star values Linear Riemann solvers for conservative variables Solution for the stiffened-gas EOS Solution for the van der Waals EOS The Baer-Nunziato equations Introduction Equations and eigenstructure Non linear Riemann solver Fluid-2 contact to the left of fluid-1 contact Fluid-2 contact to the right of fluid-1 contact EVILIN Riemann solver Stratified Riemann solver Linear Riemann solver Shallow water equations Introduction Equations and eigenstructure Non linear Riemann solver Linear Riemann solver The Derivative Riemann Problem Introduction The Derivative Riemann Problem The Mathematical Problem A Known Method of Solution Other Methods of Solution Interaction of Power-Series Expansions Interaction of Time Derivatives Assessment of the Derivative Riemann Problem solvers Reference solution The Derivative Riemann Problem for the Euler equations Test 1: smooth initial conditions Test 2: Initial data with discontinuous derivatives Tests with discontinuous initial conditions The Derivative Riemann Problem for the Baer-Nunziato equations Test 1: Advection of the void fraction Test 2: Mild jump in fluid-1 pressure viii

9 Contents Test 3: Large jumps in pressures and densities Discussion of results ADER schemes on triangular meshes Introduction Evaluation of the volume integral Evaluation of the surface integral DRP over triangular meshes ADER TT-DRP scheme over triangular meshes ADER CT-DRP scheme over triangular meshes ADER HEOC-DRP scheme over triangular meshes Numerical results Euler equations Two dimensional convergence test: Vortex evolution Shock wave reflection problem CPU cost of Riemann solvers Baer-Nunziato Test 1: Shock tube problem for liquid and gas Test 2: Numerical convergence Shallow water Convergence test Idealised circular dam Dam with channel with 45 o bend Malpasset Summary, conclusions and future work 135 A Equations of state for gas dynamics 137 A.1 Van der Waals equation of state A.2 Stiffened-gas equation of state Bibliography 141 ix

10 Contents x

11 List of Figures 2.1 Family of curves for the linear advection equation The classical Riemann problem for the linear advection equation Wave pattern generated from the Riemann problem for a linear system Elementary wave configuration of the i-th characteristic field Wave structure for a 3 3 hyperbolic system Discrete domain for a Finite Difference numerical method The triangular element T m is defined in the physical domain Wave structure for the 2 dimensional Euler equations Shock wave travelling to the right Wave structure of the Baer-Nunziato system Uncoupled representation of the wave structure of the BN system Possible structure of the Riemann problem. In this case λ 3 < λ Possible structure of the Riemann problem. In this case λ 5 < λ EVILIN Riemann solver for the non-conservative formulation Stratified approach Reference coordinate system for shallow water system Wave structure for the 2 dimensional shallow water equations The classical Riemann problem for a typical 3 3 non-linear system The Derivative Riemann Problem for a typical 3 3 non-linear system Illustration of the HEOC Derivative Riemann Problem solver Euler equation: Shock-tube test problem Euler equation: Test 1: DRP solution Euler equation: Fifth order DRP solutions for Test Euler equation: DRP solution for Test 2 and p = Euler equation: DRP solution for Test 2 and p = Euler equation: DRP solution for Test 2 and p = xi

12 List of Figures 6.7 Euler equation: DRP solution for Test 2 and p = Baer-Nunziato equation: DRP solution for Test 1. Void fraction Baer-Nunziato equation: DRP solution for Test 1. Error in the void fraction Baer-Nunziato equation: DRP solution for Test 2. Fluid-1 pressure Baer-Nunziato equation: DRP solution for Test 2. Error in the fluid-1 press Baer-Nunziato equation: DRP solution for Test 3. Fluid-2 velocity Baer-Nunziato equation: DRP solution for Test 3. Error in the fluid-2 vel Triangular element T m and integration point x h Surface-time integration over the j-th edge of the normal flux Shock wave problem. Schlieren image for density at time t = Shock wave problem. Schlieren image for density at time t = Shock wave problem. Schlieren image for density at time t = Shock wave problem. Schlieren image for density at time t = Shock wave problem. Three dimensional image for the density Wave pattern for Baer-Nunziato Test Numerical results for Baer-Nunziato Test 1 and 100 cell Numerical results for Baer-Nunziato Test 1 and 800 cell Circular dam-break problem. Radial cut of the free surface elevation Circular dam-break problem. Radial cut of the velocity Circular dam-break problem. Half plane of the solution at different times Dam with channel. Unstructured grid with 9228 elements Dam with channel. Discontinuity in the free surface Dam with channel. Reconstruction of the bottom elevation Dam with channel. Free surface contour plot for ADER FV Dam with channel. Free surface contour plot for ADER DG Malpasset dam: Computational domain formed by triangles Malpasset dam: Numerical solution xii

13 List of Tables 4.1 Stratified solver. Assignment for intercell vectors Q (l) i DRP Euler solution: Errors in L 2 norm for the vector Q for Test Toro-Titarev solver. Errors for Test HEOC solver: Errors for Test Castro-Toro solver. Errors for Test Euler Convergence rates test: second order method Euler Convergence rates test: third order method Euler Convergence rates test: fourth order method Euler Convergence rates test: fifth order method CPU computational cost for different Riemann solvers Euler: Initial conditions for shock tube problem for liquid and gas Baer-Nunziato: CPU Computational cost for different numerical methods L 1, L 2 and L inf norm for EVILIN method with MUSCL_Hancock L 1, L 2 and L inf norm for StraEv method with MUSCL_Hancock L 1, L 2 and L inf norm for StraEx method with MUSCL_Hancock Shallow water convergence rates test: FV TT-DRP second order method Shallow water convergence rates test: FV TT-DRP third order method Shallow water convergence rates test: FV TT-DRP fourth order method Shallow water convergence rates test: FV TT-DRP fifth order method Shallow water convergence rates test: FV CT-DRP second order method Shallow water convergence rates test: FV CT-DRP third order method Shallow water convergence rates test: FV CT-DRP fourth order method Shallow water convergence rates test: FV CT-DRP fifth order method Shallow water convergence rates test: FV HEOC-DRP second order method Shallow water convergence rates test: FV HEOC-DRP third order method Shallow water convergence rates test: FV HEOC-DRP fourth order method Shallow water convergence rates test: FV HEOC-DRP fifth order method Shallow water convergence rates test: DG CT-DRP second order method xiii

14 List of Tables 8.24 Shallow water convergence rates test: DG CT-DRP third order method Shallow water convergence rates test: DG CT-DRP fourth order method Shallow water convergence rates test: DG CT-DRP fifth order method xiv

15 List of Tables xv

16 List of Tables xvi

17 1 Introduction Numerical methods are a part of applied mathematics and enter the scene when mathematical knowledge is applied to other disciplines. Numerical methods are used to solve partial differential equations (PDEs) when analytical solution are not possible. Numerical methods approximate the theoretical solution of the PDEs making use of approximation theories and computational capabilities. In most of the cases PDEs represent physical processes of practical interest, like transport phenomena, wave propagation or interaction between fluids and solids. PDEs also have an intrinsically mathematical meaning. Researchers try to solve PDEs in order to forecast for example the transport of a pollutant, the arriving time of an ocean wave or a seismic wave or the behaviour of a wing under the effect of the air. All these physical situations are defined inside a physical domain which can be a river, the ocean and the coast of a country, the exterior of an airplane, and a particular instant or period of time. For many years the only possibility of studying these physical phenomena has been using an experimental approach, but in many cases this is very difficult or even impossible because of the extension of the physical area of study or the impossibility to reproduce the real conditions in the laboratory. In other cases the financial costs are too high, for example the construction and use of wind tunnels is very expensive. In these situations numerical methods represent a real alternative to predict the evolution of physical phenomena. The numerical approach applied to industry improves the efficiency of the production, reducing the time of designing and costs. Also, they are often used by public service institutions to create emergency plans in cases of inundations due to rivers, tsunamis or dam-break situations. PDEs and the physical domain are continuous elements in the sense that contain infinite data. For example the velocity in all three spatial dimensions of all points that belong to the volume of a river, for all the time period during which one intends to solve the equations, give rise to a infinite number of items of information. Because of this tremendous amount of information, the spatial-temporal domain is subdivided into a finite set of elements or volumes, generating the computational domain. Inside of each element of volume the physical properties or variables are averaged. 1

18 1. Introduction Over the computational domain the continuous solution of the PDEs is approximated in a piece-wise manner in each single volume. The final goal is to reduce the difference between the numerical solution and the theoretical one. Four aspects are considered for engineering applications: a practical physical problem to solve, a mathematical model that represents the problem with a set of partial differential equations, a numerical method to solve the PDEs and a computational capability. Different classes of errors are introduced in these four aspects. For example, the process of translating the physical problem into mathematical model of PDEs, simplifications and assumptions are made. Simplifications like considering a one dimensional problem or axial symmetry, or assumptions like constant density or inviscid flows. Because of this, the solution of the mathematical model is different from the solution of the real problem. This type of error is associated to the process of defining the mathematical model that describes the physical problem of interest and is called the modeling error. Other type of errors are introduced when the numerical method is applied to the computational domain and used to approximate the PDEs. This is the numerical error. A related source of error arise from the subdivision process of the physical domain where, for example, a particular detail of the riverbed topography is averaged. This error is called discretization error. Another source of error is associated to the use of the computer. No matter which computer is used for implementing the numerical method and approximate the solution of the PDEs, the machine-precision must be considered, for example when the zero is expressed as In this thesis we concentrate on numerical methods and we intent to minimize the numerical error associated with the approximation of the solution of the PDEs. The PDEs that we intent to approximate are first order hyperbolic partial differential equations in one and two spatial dimensions. We construct high-order numerical methods in space and time, where the order of accuracy of the approximation is theoretically arbitrary. Because of the nature of hyperbolic PDEs, the goal is to obtain a numerical method that combines high-order accuracy in smooth regions with a sharp non-oscillatory solution near shocks. These methods are known as high-resolution schemes and was not until 1970s when scientists succeeded in combining these two properties. These modern numerical methods started with the famous work of Godunov [40] in 1959, when he proposed the construction of numerical methods by solving the local Riemann problem at the interface between two volumes. These methods are called Godunov-type methods. At the same time, Godunov presented a very important theorem about monotonicity of the solution: if the scheme is linear and monotone then is at most first-order accurate. Monotonicity is a very desired property that prevents the generation of spurious 2

19 1. Introduction oscillations in the vicinity of strong gradients. Contemporary with Godunov s work, Lax and Wendroff [55] proposed a second-order Finite Difference (FD) scheme, producing sharp resolution of shocks but highly oscillatory. Additional dissipative terms need to be introduced in order use this method. Improvements of this scheme was presented in [109] and [72]. The one proposed by MacCormack [109] was very efficient and extensively used in industrial applications. In hydraulic engineering it is well known and widely used together with TVD (Total Variation Diminished) limits, see for example the article of Garcia-Navarro and Alcrudo [32]. The strong first-order accurate limitation presented by Godunov s theorem applies only to linear schemes. The way to circumvent this is by constructing non-linear schemes. One way of constructing non-linear schemes is by using piece-wise polynomial reconstruction of the data subject to some constraints. Nevertheless, the reconstruction must be bounded in order to reduce the oscillatory nature of the high order spatial representation. One of the first works on reducing spurious oscillations is due to Kolgan [66] who proposed to apply the principle of minimal values of derivatives, producing a non-oscillatory secondorder Godunov-type scheme. Another well-known approach was proposed by van Leer in his series of articles [102; 103; 105; 106; 107] starting in The idea is to develop high-order schemes using high-order reconstructions that control the derivatives of the particular reconstructed polynomial considering the Total Variation Diminished (TVD) property. For more details about TVD methods we refer the reader to Chapters 13 and 14 of the book of Toro [95]. Based on TVD reconstruction, the Monotone Upstream Scheme for Conservation Laws (MUSCL) was proposed by van Leer in 1976 [104] which uses piece-wise linear interpolation and achieves second-order accuracy for smooth flow. Following this idea different MUSCL type schemes were proposed, for example the MUSCL-Hancock method in [107] and the Piece-Wise Linear method (PLM) of Colella [21], amongst others. The TVD approach introduced new improvements in constructing high-order schemes and avoided the theoretical restriction presented by the Godunov theorem. Strictly the TVD concept applies only to the scalar (linear and non-linear) equation without source term in one-space dimension. However one can extend the TVD idea in an empirical way to solve non-linear system in one and multiple space dimensions. But the order of these schemes is restricted to at most two. The solution to this limitation arrived with the introduction of Essentially Non-Oscillatory (ENO) schemes. ENO schemes reconstruct multiple high-order polynomials over different stencils, from cell averages, selecting the smoothest one. This selection is done by the oscillator indicator function. This form of high-order reconstruction is only Total Variation Bounded (TVB) and generalizes to 3

20 1. Introduction multidimensional problems. This approach was introduced in 1986 by Harten, Osher, Engquist and Chakravarthy in [44]. Further contributions on ENO schemes were presented by Harten and Osher in [47], Harten et al. in [45] and by Shu and Osher in [82], amongst others. A further extension from the Essentially Non-Oscillatory approach is called Weighted Essentially Non-Oscillatory (WENO). In it, the reconstructed polynomial is obtained from a convex combination of a number of reconstructed polynomials and non-linear weights are computed from the oscillator indicator. This approach was presented by Osher and Shu in 1988 [82] and seems to be the standard procedure up to day on high-order spatial non-oscillatory reconstruction. Improvements and generalizations appear in [62], [8], [1], [53], [49], [79], [86] and [28], amongst others. With the contributions on non-oscillatory high-order reconstruction, the spatial accuracy of the numerical methods for hyperbolic equations was theoretically arbitrary. Nevertheless, the global accuracy of the numerical method is governed by the lower order between the spatial and the temporal accuracy. One way to increase the accuracy in time is the Runge-Kutta time integration, however it needs some kind of limiting (in the sense of limiter functions), for example a TVD constraint [80], imposing an accuracy barrier, fifth order. In practice, fourth and fifth order methods are too complicated. The Runge-Kutta (RK) time integration is widely use in Discontinuous-Galerkin (DG) methods, giving form to the RKDG methods as presented by Cockburn and Shu in [18; 15; 19; 20]. See [17] for a review on RKDG methods. In resent years, a new approach has been proposed in order to match the temporal order of accuracy with the spatial order of accuracy. This approach is called Arbitrary accuracy DErivatives Riemann problem method (ADER). It was first proposed in 2001 by Toro et al.[98] and is a generalization of the Generalised Riemann Problem (GRP) method of Ben- Artzi and Falcovitz [5], which is only second order accurate. The ADER method allows the construction of arbitrarily high-order accurate schemes in space and time. The key ingredient of the ADER approach is the solution of a Derivative Riemann Problem (DRP). This allows the approximation, with arbitrary order of accuracy, of the time and space integrals involved in the construction of the numerical method. ADER numerical methods can be implemented in the framework of Finite Volumes (FV) and Discontinuous Galerkin Finite Element (DG) methods and in multidimensional non-linear problems. Extensions and improvements have been proposed in [99], [87], [77], [85], [76], [100], [88], [54], [29], [28], amongst others. This presentation is concentrated mainly on developments of the ADER methods. We propose new DRP solvers that can be applied on one dimensional and multidimensional 4

21 1. Introduction unstructured meshes. Assessment of the DRP solvers are presented for the Euler equations and the Baer-Nunziato equations. Implementation of the DRP solver in ADER schemes on unstructured triangular grids are presented with space-time accuracy up to fifth order for the Euler equations and the shallow water equations. We also present solutions of the classical Riemann problem that are used in the construction of the DRP solvers. Classical Riemann solvers are presented for the Euler equations, shallow water equations and the Baer-Nunziato equations. In particular, for the Baer-Nunziato equations, two new solvers are proposed and implemented in one dimension with a convergence test up to second order. All the methodology presented in this thesis can be used in constructing high-order Finite Volume and Discontinuous Galerkin FE numerical methods. All the tools developed originally in the DG FE framework, such as basis functions and reference coordinate system, are applied to the FV methods. We show both methods but concentrate on FV for numerical results. The rest of this thesis if organized as follows. In Chapter 2 we present some elements of the mathematical theory that allows us to work with first order hyperbolic partial differential equations. Balance laws, the characteristic curves, the definition of hyperbolic systems, characteristic variables, solution of a linear Riemann problem and characteristic fields are introduced. In Chapter 3 we present the numerical methods studied in this thesis. We start with a review about numerical methods for hyperbolic equations. Then, from the divergence form of the hyperbolic balance laws, the Finite Volume and the Discontinuous Galerkin Finite Element methods on triangular meshes are presented with a description of non-oscillatory reconstructions. Finally, the one-dimensional Finite Volume method is introduced. In this Chapter the space and time integrals that define the numerical flux between neighbours volumes are defined. In Chapter 4 we present the solution of the classical Riemann problem for three hyperbolic equations namely the Euler equations, the Baer-Nunziato equations and the shallow water equations. For the Euler and the shallow water equations we present the solution of the augmented one-dimensional problem, which is used in two-dimensional problems. For the Baer-Nunziato equations the solution is presented for the one-dimensional problem. For all three systems the solution of the non-linear Riemann problem is presented as well as the solution of the linear one. Both solutions, linear and non-linear, are used in the construction of DRP solvers. In Chapter 5 we present three DRP solvers. We start with a description of the Derivative Riemann Problem and a review of a well known method due to Toro & Titarev [99]. We 5

22 1. Introduction continue with the presentation of two new DRP solvers, the first one is a reinterpretation of ideas already proposed in the literature and called HEOC, for Harten, Engquist, Osher and Chakravarthy [45], while the second one is a novel approach to the solution of the DRP called CT for Castro & Toro [9]. In Chapter 6 we assess the proposed DRP solvers. We obtain a reference solution at the interface between two piece-wise polynomial initial data. We confront this reference solution with the approximation obtained for all three DRP solvers. We consider the Euler and the Baer-Nunziato equations. We finish this chapter with a discussion about the results. In Chapter 7 we describe the ADER approach on triangular meshes. We show how to evaluate with arbitrary high-order accuracy the space-time integrals involved in the numerical methods studied in this thesis, as well the space-time integrals of the normal numerical flux between two neighbouring volumes. Then we describe how each one of the DRP solvers is implemented on unstructured triangular grids. Finally in Chapter 8 we present results for the numerical methods presented in this thesis. There are two-dimensional convergence tests on unstructured triangular meshes for the Euler equation and the shallow water equations with source terms. Results of up to fifth order of accuracy for both systems are given. For the Baer-Nunziato equations a onedimensional convergence test is shown. There are also numerical results that demonstrate the applicability of this methods for more realistic problems. Some results related to this thesis have been submitted for publication in international journals and have been presented to international conferences: 1. C. E. Castro and E. F. Toro. Alternative solvers for the Derivative Riemann Problem for hyperbolic balance laws. J. Comput. Phys. (submitted). 2. C. E. Castro and E. F. Toro. New numerical approaches to multiphase flows modelling. In Advancements in Energetic Materials and Chemical Propulsion, edited by Kenneth K. Kuo and Juan de Dios Rivera, 6-ISICP, Begell House, Inc. of Redding, Connecticut 2007, ISBN: C. E. Castro and E. F. Toro. Godunov schemes for compressible multiphase flows. European Conference on Computational Fluid Dynamics, ECCOMAS CFD, Egmond aan Zee, The Netherlands, 2006, 5-8 September, 2006 (submitted). 4. C. E. Castro and E. F. Toro. ADER DG and FV schemes for shallow water flows. In proceeding The 14th European Conference on Mathematics for Industry, ECMI2006 Madrid Spain, July,

23 1. Introduction 5. E. F. Toro and C. E. Castro. The derivative Riemann problem for the Baer-Nunziato equations. In proceeding Eleventh International Conference on Hyperbolic Problems, HYP2006 Lyon France, July, C. E. Castro and E. F. Toro. High order numerical methods for the non-linear shallow water equations. Eleventh International Conference on Hyperbolic Problems, HYP2006 Lyon France, July, E. F. Toro, M. Käser, M. Dumbser and C. E. Castro. ADER Shock-Capturing Methods and Geophysical Applications. In proceeding The 25th International Symposium on Shock Waves, ISSW25 Bangalore India, July, Another scientific contribution generated by the author but is not part of this thesis is published in: 1. C. E. Castro and E. F. Toro. A Riemann solver and upwind methods for a twophase flow model in non-conservative form. Int. J. Numer. Meth. Fluids. 2006; 50:

24 1. Introduction 8

25 2 Hyperbolic systems of balance laws In this thesis we deal with the construction of high order numerical methods for hyperbolic balance laws. We intent to solve numerically, first order partial differential equations of hyperbolic type. In this chapter we present basic notions about this kind of partial differential equations (PDEs). 2.1 The balance laws We start with a system of first-order partial differential equations of the form q i p t + a ij (x, t, q 1,...,q p ) q j x = s i(x, t, q 1,...,q p ), (2.1) j=1 for i = 1,...,p. This system of p equations has p unknowns q i that depend on space x and time t. This system can be expressed in vectorial form as t Q + A x Q = S, (2.2) with Q = q 1. q p, A = a 11 a 1p a 21 a 2p... a p1 a pp, S =. s 1 s p. (2.3) System (2.1) is said to be linear with constant coefficients if all coefficients a ij and s i are constant. If a ij = a ij (x, t) and s i = s i (x, t) the system (2.1) is linear with variable coefficients. The system (2.1) is still linear if s i depends linearly on the vector of unknowns Q. System (2.1) is called quasi-linear if the coefficients a ij depend on the unknowns Q, writing A = A(Q). Note that quasi-linear means that in general (2.1) is a non-linear system. When p = 1 we said (2.1) is scalar. Scalar examples are the linear advection- 9

26 2. Hyperbolic systems of balance laws reaction equation and the inviscid Burgers equation q t + a q = s, (2.4) x q t + q q = 0. (2.5) x In the first one, a 11 = a is a constant wave propagation speed, while in the second one, a 11 = a(q) = q is variable and depends on the unknown q. An example when p = 2 is the one dimensional shallow water equations, for p = 3 we have the Euler equations. System (2.1) is written in differential, non-conservative form. Most systems of interest can be written in conservative form with source terms t Q + x F(Q) = S(Q). (2.6) We said that the system is a balance law. In the absence of source term, system (2.1) is said to be a homogeneous conservation law if can be written in the following form t Q + x F(Q) = 0, (2.7) where Q is the vector of conservative variables, F(Q) is the vector of fluxes, which depends on the components of Q. Expanding the space derivative of the vector F(Q), we can write the Jacobian matrix A(Q) = F(Q)/ Q where each component depend on the vector Q as follow t Q + A(Q) x Q = 0, (2.8) System (2.8) can also be expressed in terms of the physical or non-conservative variables, defining the vector W. In this case we can write t W + B(W) x W = 0. (2.9) In general matrix A(Q) B(W). For studying hyperbolicity and characteristic fields both forms, (2.8) and (2.9), of the equation can be used, see page 42 of [39] for example. 10

27 2. Hyperbolic systems of balance laws 2.2 Characteristic curves Now, we introduce the concept of characteristic curves using the simplest hyperbolic equation. Considering an initial value problem (IVP) for the scalar linear advection equation we have PDE: q t + a q x = 0, x [x L, x R ], t > 0. IC: with IC the initial condition. q(x,0) = q 0 (x), } (2.10) Considering curves x = x(t) and writing the total derivative of q = q(x(t), t) respect to the time we have dq dt = q t + q dx x dt. (2.11) Equation (2.11) is the rate of change of q along the curve x = x(t). In particular, if dq/dt = 0, q is constant along x = x(t). Moreover, if x = x(t) satisfies dx/dt = a then we can write dq dt = q t + a q = 0. (2.12) x In (2.12) x = x(t) is called a characteristic curve and a the characteristic velocity. Defining the initial condition q(x,0) = q 0 (x), the solution of (2.10) at (x, t), which is connected to (x 0, 0) through a characteristic line x = x 0 + at, is q(x, t) = q 0 (x 0 ) = q 0 (x at). (2.13) In other words, given an initial profile q 0 (x), it will travels at constant speed a unchanged, following the characteristic lines. For positive characteristic velocity a > 0, the left boundary x = x L is defined as an inflow boundary, where the solution is given by the value of q(x L, t). For the right boundary x = x R the information travels out, therefore the solution depends only on the data inside the domain. These properties of hyperbolic equations set the basis for constructing numerical methods; wave propagation velocities are finite and the solution depends on the domain of dependence, see [95], [39] for more details. See Fig. 2.1 where the family of curves for the linear advection equation is depicted. 11

28 2. Hyperbolic systems of balance laws t x=x 0 +at 1 a q(x R,t) q(x L,t) 0 0 x L x 0 x R x Figure 2.1: Family of curves for the linear advection equation on the x t plane for a characteristic velocity a > 0. The initial condition at t = 0 and the boundary conditions at x = x L (inflow) define the solution. At x = x R the solution depends on the initial conditions following characteristic lines. a is the slope of the characteristic curves. 2.3 The Riemann problem When the initial condition of the IVP is discontinuous across x = 0 we said it is a Riemann problem. When the initial conditions are constant we called classical Riemann problem. In Chapter 4 we present the solution for the classical Riemann problem for three hyperbolic equations. When the initial conditions are piece-wise polynomial we called Derivative Riemann Problem (DRP) and the solution is presented in Chapter 5. The classical Riemann problem for the linear advection equation is defined as follows, PDE: q t + a q x = 0, x (, ), t > 0. IC: q(x,0) = q 0 (x) = { q L if x < 0, q R if x > 0, (2.14) The solution of (2.14) is a special case of (2.13). Tracing the characteristic line x = at from x = 0 we can solve (2.14). The solution is q(x, t) = q 0 (x at) = { q L q R x if t < a, x if t > a. (2.15) In Fig. 2.2 is shown the classical Riemann problem for the linear advection equation and the characteristic line x = at form x = 0. Fig. 2.2(a) depicts the initial conditions at 12

29 2. Hyperbolic systems of balance laws t = 0. Fig. 2.2(b) depicts the structure of the solution with the characteristic line x = at which divide the half plane x t. The solution of (2.14) is q(x, t) = q L if x/t < a and q(x, t) = q R if x/t > a. q(x,0) t x=at q L x t <a q R q L x t >a q R x=0 x x=0 x (a) (b) Figure 2.2: The classical Riemann problem for the linear advection equation. (a): Initial condition at t = 0. (b): Structure of the solution on the x t plane. 2.4 Hyperbolicity In order to study hyperbolicity for a system we consider the following quasi-linear system t Q + A(Q) x Q = 0. (2.16) System (2.16) is said to be hyperbolic if A(Q) has p real eigenvalues λ 1,...,λ p and p linearly independent corresponding right eigenvectors r 1,...,r p. System (2.16) is strictly hyperbolic if the eigenvalues are all distinct. The scalar equations (2.4) and (2.5) are trivially hyperbolic. The right eigenvectors satisfy The left eigenvectors satisfy Ar k = λ k r k, 1 k p. (2.17) l k A = λ k l k, 1 k p. (2.18) Assuming that the system (2.16) is hyperbolic, the set {r k } forms a basis. Note that left and right eigenvectors are orthogonal and one can normalize them to have { l j r k 1 if j = k, = (2.19) 0 otherwise. 13

30 2. Hyperbolic systems of balance laws The matrix A is said to be diagonalisable if can be expressed as A = RΛR 1, where Λ is a diagonal matrix. The diagonal elements of Λ are the eigenvalues λ k of A and R is a matrix formed by the right eigenvectors r k. Hyperbolic systems sometimes are defined as systems with real eigenvalues and diagonalizable matrix. It can be shown that the inverse matrix R 1 is equivalent to the matrix L, where the rows of L are the left eigenvectors of A. 2.5 Characteristic variables Now we can define characteristic variables C = [c 1,...,c p ] T via the transformation C = R 1 Q = LQ. (2.20) At this stage we assume (2.16) to be linear with constant coefficient matrix A. We can multiply the system (2.16) from the left by L and introduce the identity matrix I = RL in order to obtain L t Q + LA RL x Q = 0, t (LQ) + LAR x (LQ) = 0. (2.21) The new vector C = LQ is the characteristic variable vector, the matrix Λ = LAR is a diagonal matrix with diagonal elements λ i,...,λ p. The new form of the system (2.16) is written as follow t C + Λ x C = 0. (2.22) System (2.22) is the characteristic or canonical form of system (2.16) and consists of p uncoupled scalar hyperbolic PDEs as follows t c i + λ i x c i = 0, for i = 1,...,p. (2.23) We can set a Riemann problem for each equation PDE: t c i + λ i x c i = 0, x (, ), t > 0. { IC: c i (x,0) = c 0i (x) = c il if x < 0, c ir if x > 0, (2.24) 14

31 2. Hyperbolic systems of balance laws with solution as (2.15) c i (x, t) = c 0i (x λ i t) = { c il if x λ i t < 0, c ir if x λ i t > 0, (2.25) Now we see that the solution in terms of the original variables is Q = RC, see (2.20). In other words, Q is a linear combination of the eigenvectors with coefficients given by the characteristic variables. The solution Q(x, t) = RC(x, t) results from the combination of the p characteristic waves. Q(x, t) = p r i c i (x, t) = i=1 p r i c 0i (x λ i t) (2.26) i=1 2.6 The linear Riemann problem The Riemann problem for a p p linear system reads, PDE: Q t + AQ x = 0, x (, ), t > 0. { IC: Q(x,0) = Q 0 (x) = Q L if x < 0, Q R if x > 0, (2.27) and we obtain the solution at (x, t). Assuming that the system in (2.27) is strictly hyperbolic we have the ordered eigenvalues λ 1 < λ 2 <... < λ p and corresponding eigenvectors R = [ r 1,...r p]. All p waves are represented in Fig Each wave i carries a discontinuity which travels with velocity λ i. To the left of the wave 1 the solution is the constant initial data Q L, while to the right of the wave p the solution is the constant initial data Q R. λ 2 t λ i λ p-1 λ 1 λp Q L Q R x=0 x Figure 2.3: Wave pattern generated from the Riemann problem for a linear system of p components. Because the eigenvectors are a base we can represent the initial condition as a linear 15

32 2. Hyperbolic systems of balance laws combination of r i as follows Q L = p α i r i, Q R = i=1 p β i r i. (2.28) The Riemann problem (2.27) can be solved in terms of the initial conditions for the characteristic variables, see (2.26). Comparing (2.28) with (2.26) we find c 0i (x) = { i=1 α i if x < 0, β i if x > 0, (2.29) for i = 1,.., p. Because left and right eigenvectors are orthogonal we can write the following relation C L = R 1 Q L = α 1., C R = R 1 Q R = β 1.. (2.30) α p β p From (2.26), the solution of (2.27) is found in terms of the initial condition of the characteristic variables, with p Q(x, t) = r i c 0i (x λ i t), (2.31) i=1 { c 0i (x λ i α i if x λ i t < 0, t) = β i if x λ i (2.32) t > 0. It is clear that the solution Q(x, t) depends on the initial condition. Which is more interesting is that the solution depends on the ratio x/t. The initial conditions (2.32) change as x/t is faster or slower than the characteristic speed λ i. This condition is known as the self-similarity property of the solution of the Riemann problem. We can express the solution (2.31) in terms of the coefficients α i and β i. For a particular point (x, t) there is an eigenvalue λ I such that λ I < x/t < λ I+1. The solution (2.31) considers the influence of the characteristic variables c 0i = β i i I and c 0i = α i i > I. The final expression is as follows I p Q(x, t) = r i β i + r i α i. (2.33) i=1 i=i+1 If we define the jump from the right to the left state = Q R Q L, we can write the 16

33 2. Hyperbolic systems of balance laws expansion p = δ i r i, (2.34) i=1 with δ i = β i α i. Numerically, we are interested in obtaining the solution at (0, t), along the t-axis. It can be easily shown that Q(0, t) = Q L + λ i 0 δ i r i or Q(0, t) = Q R λ i > 0δ i r i. (2.35) In this way, only one linear system (2.34) needs to be solved for the wave strengths δ i. 2.7 Characteristic fields for non-linear systems When we deal with hyperbolic non-linear PDE systems we should study the characteristic fields. The non-linear character of the equations generates different types of waves depending on the initial conditions. Knowing the properties of each field or wave, in some cases we can solve exactly the Riemann problem, in others, we can obtain a good approximation for numerical purposes, in all cases it is a fundamental information to understand the nature of the system. The reader is refers to [52], [39], [95] for details. Writing the quasi-linear form in non-conservative variables we present the basic ideas concerning the characteristic fields as follow t W + B(W) x W = 0, x R, t > 0 (2.36) We assume system (2.36) is strictly hyperbolic therefore we have the ordered eigenvalues λ 1 (W) < λ 2 (W) <... < λ p (W). For each eigenvalue a corresponding right eigenvector is obtained R = [ r 1 (W),...r p (W) ]. For each pair ( λ i (W),r i (W) ), for i = 1,...,p, an associated characteristic field exist. The ith characteristic field is said to be linearly degenerated if λ i (W) r i (W) = 0, W R p. (2.37) The ith characteristic field is said to be genuinely non-linear if λ i (W) r i (W) 0, W R p. (2.38) Is easy to see that linear systems generates only linearly degenerated fields. This type of waves generates the so called contact discontinuities. In the presence of genuinely non- 17

34 2. Hyperbolic systems of balance laws linear fields the waves generated can be one of the following two types: rarefaction waves or shock waves. t Q L t S i t Q L S i Q L Q R λ i (Q L ) λ i (Q R ) x λ i (Q L ) λ i (Q R ) Q R x λ i (Q L ) λ i (Q R ) Q R x (a) (b) (c) Figure 2.4: Elementary wave configuration of the i-th characteristic field. (a): Rarefaction wave. (b): Shock wave. (c): Contact discontinuity wave. Depending on the type of the characteristic field, different expressions can be used to connect the data on the left and on the right when solving the Riemann problem. We suppose that the i-th wave has data Q L = Q(W L ) and Q R = Q(W R ) immediately to the left and to the right respectively of the associated field λ i. In Fig. 2.4 the characteristic lines of the i-th field, evaluated on the data Q L and Q R, (a) diverge for a rarefaction, (b) converge for a shock and (c) travel parallel for a contact discontinuity Rarefaction waves Rarefaction waves are smooth continuous functions that connect two states without discontinuities. They are also called expansive waves. We can connect both states using the Generalized Riemann Invariants(GRIs) obtaining p 1 ordinary differential equations (ODEs). The Generalized Riemann Invariants applied to the i-th characteristic field is written as follows dw 1 r i 1 = dw 2 r i 2 = = dw p rp i. (2.39) Taking a pair of these ODEs, we integrated them across the rarefaction wave, from Q L to Q R, obtaining a Riemann Invariant, which is constant inside the wave. This constant value can be evaluated on the initial data Q L or Q R. dw k r i k = dw l rl i, 1 k, l p. (2.40) 18

35 2. Hyperbolic systems of balance laws rl i dw k rk i dw l = 0, rl i dw k rk i dw l = const. (2.41) In general it is not possible to evaluate the integrals (2.41) explicitly, as for example the Euler equations with general equation of state (EOS). Another property of rarefaction waves is that the entropy is preserved, in other words, the entropy is a Riemann Invariant. This is algebraically obtained by writing the system (2.36) considering the entropy in the physical variables and applying the Generalized Riemann Invariants. In this type of fields the divergence of characteristic relation is verified, see Fig. 2.4(a), this is λ i (Q L ) < λ i (Q R ). (2.42) Shock waves Shock waves connect the two states through a single discontinuous jump. We can connect the two states using the Rankine-Hugoniot Condition. Mathematically it is possible to find more than one solution that connect both states and obey the Rankine-Hugoniot Conditions. In these cases it is necessary to use physical considerations such as the entropy condition. Considering a system of hyperbolic conservation laws t Q + x F(Q) = 0, (2.43) we assume that has a discontinuous wave solution of speed S i associated to the field λ i. The Rankine-Hugoniot Conditions are expressed as F(Q) = S i Q, (2.44) with F(Q) = F(Q R ) F(Q L ) and Q = Q R Q L, (2.45) where Q L and Q R are the vectors immediately to the left and to the right of the discontinuity, see Fig. 2.4(b). In general, for non-linear systems, the solution of S i is not direct and must be found iteratively. The typical question is which values of the vector Q L connect through a shock wave with speed S i the data Q R. The entropy condition enforces that the velocity S i must be contained between the 19

36 2. Hyperbolic systems of balance laws characteristic wave velocity λ i (Q L ) and λ i (Q R ). λ i (Q L ) < S i < λ i (Q R ). (2.46) Contact discontinuity wave Contact discontinuity waves propagate a single jump discontinuity of velocity S i, see Fig. 2.4(c). Across this field we can apply Rankine-Hugoniot Conditions, Generalized Riemann Invariants and the parallel characteristic condition which is λ i (Q L ) = λ i (Q R ) = S i. (2.47) In general, for a non-linear hyperbolic PDE, we can study the eigenstructure and the characteristic fields of the system. Applying the Generalized Riemann Invariants and the Rankine-Hugoniot Conditions, the classical Riemann problem for the PDE system can be reduced to an algebraic non-linear system. This can be solved using an iterative procedure, see Sec. 4 in [95] for the Euler equations with ideal equation of state or Sec. 5 in [93] for the shallow water equations. In Fig. 2.5 we present the general wave structure for a 3 3 hyperbolic system. The general nomenclature considers a fan structure for the rarefaction wave, a segmented line for the contact discontinuity and a solid line for the shock wave. t Q L Q R x=0 x Figure 2.5: Wave structure for a 3 3 hyperbolic system. Solution of the Riemann problem. We assume a left rarefaction, a central contact an a right shock wave. In this chapter we introduced the basic notions about hyperbolic equations. We saw the characteristic curves, the Riemann problem, hyperbolicity, the characteristic variables, the solution of the linear Riemann problem and the characteristic fields. All this elements are used in the construction of numerical methods for hyperbolic equations studied in this thesis. In Chapter 4 we use this to obtain the solution of non-linear classical Riemann problems. 20

37 3 Numerical methods for hyperbolic equations 3.1 Review of numerical methods for hyperbolic equations A simple way of solving a PDEs is by direct approximation of the differential operator involved in the equation. The approximation process consist on the discretization and it transforms the differential operator in a difference operator. This approach is used in Finite Difference (FD) methods. The approximation is performed by truncating a Taylor series expansion. The differential operator can be approximated by different orders and using central, backward or forward representation. Consider a continuous function of (x, t), namely q(x, t), with all derivative defined at (x, t). The value of q at the point (x + x, t) can be approximate by a Taylor series expansion around (x, t), that is q(x, t) q(x + x, t) = q(x, t) + x + + ( x)n n q(x, t) x n! x n +. (3.1) Neglecting second and high order derivatives we obtain a first order approximation of the differential operator x q(x, t) as follows, q(x, t) x q(x + x, t) q(x, t) x. (3.2) Equation (3.2) is a forward first order representation of the derivative x q(x, t). Following the same idea we can construct, combining different Taylor expansions, central or backward difference operators. In a similar manner as we discretize the differential operator, we can discretize the physical domain where we intent to solve our PDEs. This consists on subdividing the full computational domain in small elements, volumes or points depending the methodology. The FD numerical methods apply the difference operators over the discrete points. In Fig. 3.1 we have the discrete domain at space positions x i 1, x i, x i+1 and time position t n and t n+1. Applying to the homogeneous equation (2.4) a FD scheme central in space 21

38 3. Numerical methods for hyperbolic equations t t n+1 x i n+1 t n n x i-1 x i n n x i+1 x i-1 x i x i+1 x Figure 3.1: Discrete domain for a Finite Difference numerical method. and forward in time, we obtain the following explicit numerical method, q(x i, t n+1 ) q(x i, t n ) t + a q(x i 1, t n ) q(x i+1, t n ) t = 0. (3.3) Besides FD is a very simple method, it presents some limitations. One is the assumption of continua and differentiable solutions, which is not valid specially near discontinuities as shock waves. The second drawback is that it needs structured grids which is unrealistic over complex geometries. Another well known method used to solve hyperbolic PDEs is the Finite Volume (FV). In this method, an average of the partial differential equations is obtained by integrating them inside the control volume. In this process surface integral along the edges of the volume are obtained, defining what is call the numerical flux. Because the flux leaving one volume is the same coming into the adjacent volume, FV are naturally conservative numerical methods. FV methods evolve cell averages data for each volume. The solution of the Riemann problem is introduced in the calculation of the numerical flux, and discontinuous solutions are admitted. The next and very interesting method used for solving hyperbolic PDEs is the Finite Element Discontinuous Galerkin (DG) method. This type of methods use all the knowledge developed in the Finite Element community but aloud the numerical solution between two neighbours cells to by discontinuous. Again, integration of the PDE is performed over finite elements. In this case, the numerical method considers a high order polynomial representation of the data. This method was introduced originally in 1973 by Reed and Hill [69] and applied to the scalar linear hyperbolic equation. The first mathematical analysis was presented by 22

39 3. Numerical methods for hyperbolic equations LeSaint and Raviart [59] in 1974 recognizing the good properties of the method. The first application for non-linear conservations law is due to Chavent and Salzano [13] in 1982 where time discretization was implicit. The first explicit Runge Kutta Discontinuous Galerkin (RKDG) method was introduced by Cockburn and Shu in [19]. Extensions to multi dimensional scalar cases was done in 1990 by Cockburn et al. [15] where new slope limiter where proposed. Further developments were presented in 1990 by Cockburn et al. [20]. An adaptive order of accuracy for RKDG version was presented in 1996 by Devine et al. in [24]. For a historical review of Finite Element Discontinuous Galerkin method and technical developments we suggest to see [16] and [17]. A new kind of DG methods was presented by Dumbser [27] in The ADER approach was satisfactorily introduced in the DG framework, creating the ADER-DG type methods. Another numerical method which presents very interesting properties is the Random Choice method (RCM). It was introduced by Glimm [36] in 1965 and used to construct a proof of existence of solutions for non-linear systems of hyperbolic conservation laws. Chorin [14] in 1976 successfully implemented a modified version of RCM to solve the Euler equations. This method needs the exact Riemann solver to solve the interaction between two adjacent volumes. It selects randomly one state from the wave pattern and defines it as the solution at the new time level. Probably is the only method that solves exactly all discontinuities from the first time step. The limitations are three: the stability condition is half of the Finite Volume method; it is only first order accurate; extensions to multidimensional problems are not possible. There are other kind of methods in which the approach is different, they do not use a grid or mesh. They are called mesh free or mesh less methods. In this approach the fluid elements are free to move inside the computational domain based on the fluid particle velocity. An example is the Front Tracking method, also called the method of Dafermos [23]. It consists on identify discontinuities in the initial data, place Riemann problems with piece-wise constant data and solve them exactly or approximately. At this point each wave generated from the locals Riemann problems is tracked until two of them collide. This condition defines a new time level where a piece-wise constant Riemann problems are set and the process starts again. One drawback is the treatment of rarefaction waves that needs to be dicretized in several piece-wise constant states. Another potential limitation is that the number of front waves may blow up in finite time for general systems. To minimize this, one may reduce the weak waves. Mono and multidimensional applications can be found in [43], [37], [38]. See the book of Holden [48] for a complete review of the method. 23

40 3. Numerical methods for hyperbolic equations Other mesh-less method is the Smoothed Particle Hydrodynamics (SPH). It was originally proposed for astrophysics simulations in [34] and [63]. It was used in many fields of research including astrophysics, ballistics, vulcanology and tsunami. It considers Lagrangian coordinates to follow the fluid elements. Each fluid element has an interaction radio defined by h where the properties of the fluid are represented by the kernel function. Typical kernel functions are the Gaussian functions or cubic spline. One drawback of the method is that it tends to smear out shock or contact discontinuities waves much more than standard mesh-based methods. For a modern review of this method see [61]. In this chapter we introduce the general methodology involved in constructing numerical methods for systems of hyperbolic equations. We are concerned with arbitrary high order explicit one-step numerical methods on non-structured grids. Two approaches are considered: Finite Volumes (FV) and Discontinuous Galerkin Finite Elements (DG) methods, where the first one can be seen as a particular case of the second. The extension to arbitrary high order accuracy in space and time is achieved by the use of the ADER approach (Arbitrary accuracy DEerivatives Riemann problem method) where an arbitrary accurate intercell flux function is constructed. This intercell flux is obtained by solving a Derivative Riemann Problem (DRP) in which the initial condition consists of two arbitrary but smooth vector fields. In what follows in this chapter we describe the methodology applied to a general twodimensional hyperbolic balance law: with t Q + x F(Q) + G(Q) = S(Q), (3.4) y Q = q 1. q p, F(Q) = f 1 (Q). f p (Q), G(Q) = g 1 (Q). g p (Q), S(Q) = s 1 (Q). s p (Q), (3.5) where Q is the unknown vector of conservative variables, F(Q) and G(Q) are the fluxes in x and y directions respectively and S(Q) is a source term, which does not involve derivatives of the unknown vector Q. System (3.4) can be written in quasi-linear form using the Jacobian matrices A(Q) = F(Q)/ Q and B(Q) = G(Q)/ Q as t Q + A(Q) x Q + B(Q) Q = S(Q). (3.6) y 24

41 3. Numerical methods for hyperbolic equations For the moment we assume that system (3.6) is hyperbolic in the sense that the eigenvalues of the matrix C(Q) = n x A(Q) + n y B(Q) formed by a linear combination of matrices A(Q) and B(Q) are real for any vector n = [n x, n y ] non identically zero. We should say that for particular hyperbolic systems one can write the equations in conservative form, in a mathematical sense, even if the unknowns are not the conservative variables in the physical sense. This issue must be considered with care in the presence of shock waves. For an example see section 3.9 of [93]. Further details about hyperbolic equations can be found in [52], [39], [95], [60]. The numerical method is constructed in a computational domain obtained by a conforming triangulation of the physical domain Ω R 2. The spatial control volume considered is a triangular element T m Ω. In what follows we use the divergence form of equation (3.4): where = [ x, y ] and H(Q) = [F(Q),G(Q)]. t Q + H(Q) = S(Q), (3.7) 3.2 Finite volume schemes on triangular meshes In this section we derive the finite volume numerical scheme integrating (3.7) in the control volume T m [t n, t n+1 ], being t n = n t the current time, as follow, t n+1 t n T m ( t Q(x, t) + H(Q(x, t)) S(Q(x, t))) dx dt = 0. (3.8) Using the divergence theorem, in which we replace the volume integral of H(Q(x, t)) by a surface integral, we obtain t n+1 t n T m t Q(x, t) dx dt = t n+1 t n T m H(Q(x, t)) ˆn dx dt + t n+1 t n T m S(Q(x, t)) dx dt (3.9) Now we introduce the cell average representation of the data, characteristic of the finite volume numerical method. The cell average of the vector Q(x, t) in the cell T m at a fixed time t = t n is defined as Q n m 1 Q(x, t n ) dx, (3.10) T m T m 25

42 3. Numerical methods for hyperbolic equations where T m is the volume (area in two space dimensions) of the cell T m. Integrating in time the left hand side of 3.9 and introducing 3.10 we obtain Q n+1 m = Q n m 1 T m t n+1 t n T m H(Q(x, t)) ˆn dx dt t n+1 t n S(Q(x, t)) dx dt (3.11) T m Equation 3.11 says that the cell average value at time t = (n + 1) t is equal to the cell average at time t = n t plus the normal fluxes trough the edges of the volume within the time interval [t n, t n+1 ] and the generation inside the control volume represented by the source. There is no simplification up to this point and equation (3.11) is exact. In practice, we do not know exactly the functions H(Q(x, t)) and S(Q(x, t)) and their respective integrals and therefore approximations must be introduced. If we assume the vector Q(x, t) to be piece-wise constant inside the element T m and the integrals are approximate using a first-order accurate quadrature rule we obtain the classical Godunov method. In this thesis we are concerned with high order evaluation of these integrals where ADER approach plays a crucial role, see Chapter Discontinuous Galerkin schemes Discontinuous Galerkin Finite Element numerical methods are constructed by considering that the vector function Q(x, t) is approximated numerically by the vector Q h (x, t). The approximate solution Q h (x, t) is sought in the finite element space of discontinuous functions V h. For simplicity we will go through the details considering only one component q(x, t) of the unknown vector Q(x, t), approximated by q h (x, t). q(x, t) q h (x, t) N 1 l=0 ˆq l (t)φ l (x) (3.12) The approximate solution q h (x, t) V h is a linear combination of spatial polynomial basis functions φ l (x) V h and temporal scalar degrees of freedom ˆq l (t) R, being N the number of degrees of freedom and basis functions. We consider orthogonal basis functions φ l (x) = φ l (x(ξ, η)) constructed in the reference space coordinate system ξ η over the unit triangle T E defined by 0 ξ 1, 0 η 1 ξ. In contrast to the Finite Volume scheme, where cell averages are evolved, the Discontinuous Galerkin scheme evolves the degrees of freedom ˆq l (t), in other words, it evolves the 26

43 3. Numerical methods for hyperbolic equations complete polynomial q h (x, t). Consider the j component of (3.7) with j = 1...p. Multiplying by the basis function φ k and integrating in space over the control volume T m we obtain ( t q φ k + h(q) φ k s(q) φ k ) dx = 0, (3.13) T m where we omit the space and time dependences for simplicity. In (3.13), the j component of H(Q) and S(Q) are defined as h(q) = [f j (Q), g j (Q)] and s(q) = s j (Q) respectively. Using the chain rule (φ k h(q)) = φ k h(q) + φ k h(q) and Gauss s divergence theorem we obtain t q φ k dx + φ k h(q) ˆn dx φ k h(q) dx = φ k s(q) dx. (3.14) T m T m T m T m Introducing (3.12) into (3.14) we project the continuous function q(x, t) to the finite element space V h, and considering the orthogonality of the basis functions, we obtain tˆq k φ k φ k dx + φ k h(q) ˆn dx φ k h(q) dx = φ k s(q) dx. (3.15) T m T m T m T m Integrating (3.15) in time, within time interval [t n, t n+1 ], and arranging the terms the following equation is obtained. ˆq n+1 k = ˆq k n 1 J m k t n+1 t n t n+1 t n T m φ k h(q) dx dt T m φ k h(q) ˆn dx dt t n+1 t n φ k s(q) dx dt. T m (3.16) In (3.16) we made use of the orthogonality property of the basis functions, introducing m k and J. Regarding that we approximate the integrals in (3.16), it gives an explicit one step evolution equation for the degree of freedom ˆq k from time level t = t n to time level t = t n+1. Applying (3.16) to each degree of freedom k = 0,...,N 1 and to each component of Q, the approximate solution Q n+1 m is obtained. The arguments of these integrals are space and time dependent functions of the vector Q in (3.4). These functions will be constructed using the ADER approach presented in 27

44 3. Numerical methods for hyperbolic equations Chapter 7. Depending on the order of the approximation q h in (3.12), the integrals in (3.16) will be computed by a suitable numerical quadrature. The particular case where only one basis function φ 0 = 1 is used, the second integral on the right hand side of (3.16) vanish, and the Finite Volume scheme is recovered, see (3.11). The basis functions considered here are obtained from the Jacobi polynomial. Defining the order of the numerical method as o we have N basis function, with N = 1 (o + 1)o, (3.17) 2 and polynomial of degree less or equal to o 1. A desirable property of these functions is orthogonality. Furthermore, because they are constructed in the reference coordinate system ξ η some integrals may be computed exactly, in advance. In Fig. 3.2 we have the triangular element T m defined by the vertexes x 1, x 2 and x 3. The element T m exist in the physical domain represented with the Cartesian coordinate system x y. It can be mapped to the reference triangle T E in the local reference domain defined with the Cartesian coordinate system ξ η. y x 3 η 1 T m x 1 x 2 x T E 0 1 ξ Figure 3.2: The triangular element T m is defined in the physical domain using the Cartesian coordinate system x y. The associated canonical triangular element T E is defined in the local reference domain with the Cartesian coordinate system ξ η. The base function φ k (ξ, η) is defined inside the canonical triangle T E { 0 ξ 1, φ k (ξ, η) T E 0 η 1 ξ. (3.18) 28

45 3. Numerical methods for hyperbolic equations The mapping function to transform coordinates is x(ξ, η) = x 1 + (x 2 x 1 )ξ + (x 3 x 1 )η, y(ξ, η) = y 1 + (y 2 y 1 )ξ + (y 3 y 1 )η, (3.19) and its inverse is ξ(x, y) = 1 J ((x 3y 1 x 1 y 3 ) + x(y 3 y 1 ) + y(x 1 x 3 )), η(x, y) = 1 J ((x 1y 2 x 2 y 1 ) + x(y 1 y 2 ) + y(x 2 x 1 )), (3.20) where J is the determinant of the Jacobian matrix J = (x, y)/ (ξ, η) and twice of the are of triangle T m. Matrix J and other useful relations are [ J = x ξ x η y ξ y η ] [, J 1 = ξ x ξ y η x η y ], dxdy = J dξdη, (3.21) J = (x 2 x 1 )(y 3 y 1 ) (y 2 y 1 )(x 3 x 1 ). (3.22) Finally we note that the space derivative of a function q(x, y) can be obtained from the following relation, [ q ξ q η ] = J T [ q x q y ]. (3.23) The orthogonality property of the basis functions is easily verified by integrating over the reference triangle T E, { 0 if k l φ k φ l = (3.24) T E m k R if k = l If we intend to approximate the continuous function q(x, t) with third order (o = 3) accuracy in space we need to use N = 6 base functions, that is φ = 0 to φ = 5. Note that base functions are constructed in blocks in the sense that when we increment the order of the approximation from o = 3 to o = 4 we added 4 (φ = 6 to φ = 9) new base functions, which has degree equal to 3. As an example, here we show the first 10 base functions for 29

46 3. Numerical methods for hyperbolic equations constructing a numerical method with spatial accuracy up to order 4. P 0 { P 1 P 2 P 3 φ 0 = 1, φ 1 = 1 + 2ξ + η, φ 2 = 1 + 3η, φ 3 = 1 6 ξ 2 η + 6ξ 2 + 6ξ η + η 2, φ 4 = 1 2 ξ 6 η + 10ξ η + 5η 2, φ 5 = 1 8 η + 10η 2, φ 6 = ξ + 3η 30 ξ 2 24 ξ η 3 η ξ ξ 2 η + 12ξ η 2 + η 3, φ 7 = 1 + 6ξ + 9η 6 ξ 2 48 ξ η 15 η ξ 2 η + 42ξ η 2 + 7η 3, φ 8 = 1 + 2ξ + 13η 24 ξ η 33 η ξ η η 3, φ 9 = η 45 η η 3. (3.25) 3.4 Reconstruction procedure The reconstruction procedure used in this thesis was presented by Dumbser and Käser in [28]. It follows the same ideas of WENO reconstructions. The main advantages are: the reconstructed polynomial is valid in the whole volume or element; they are constructed using the basis functions previously defined, therefore integrals may be computed in advance. Scaling effects are cancel out because integration in the reference space coordinates. Data reconstruction is necessary if we utilize Finite Volume numerical methods due to the cell average representation of the data. In Discontinuous Galerkin FE numerical methods the spatial high order representation of the data comes natural and reconstruction is not mandatory. Nevertheless, in both cases, near strong shocks non-oscillatory reconstruction is necessary. We do not intend to give all details about the reconstruction procedure here. For a review about ENO-WENO schemes see [81]. For technical details about this particular reconstruction procedure see [28]. Defining P k a set of polynomials of degree less than or equal to k. Given a function q(x), the reconstruction procedure computes a polynomial P m (x) P k for each element T m inside of the computational domain. The P m (x) polynomial has the same mean value of the function q(x) over the element T m and is a (k + 1) order approximation to q(x) on the element T m. The new polynomial P m (x) is valid over the whole element T m. 30

47 3. Numerical methods for hyperbolic equations The P k set of polynomials has K = k+1 and K = (k+1)(k+2)/2 degrees of freedom for one and two space dimensions respectively. In order to obtain these degrees of freedom we consider, in addition to the element T m itself, a set of at least K 1 neighbours elements. This set of K elements is called a stencil of the element T m. If we consider more than K 1 neighbours the solution of the degrees of freedom is overdetermined. In this case we impose strictly conservation of P m (x) over element T m and conservation over the rest of elements inside stencil in a least-square sense. For one dimensional problems we consider j = 1,...,k + 1 stencils and corresponding p j (x) P k polynomials that have the same mean value over element T m. In two dimensional problems over triangular meshes, we always consider j = 1,...,7 stencils where the first one is a central stencil, the second to the fourth are forward one side stencils, and the remaining three are backward one side stencils, see Käser et al.[54] for definition of forward and backward stencils. The reader has to be careful because not all stencils are admissible due to boundary conditions or geometrical criteria, for example if all elements inside the stencil are aligned. The reconstructed polynomial over element T m will be a combination of the stencil polynomials, with j an index over the different stencils and j = 1 the central one as follows, P m (x) = j ω j p j (x), (3.26) with the normalized non-linear weights ω j equal to ω j = ω j j ω, (3.27) j and the non-linear weights ω j computed from the oscillator indicator σ j and the linear weights λ j as follows, ω j = λ j (ǫ + σ j ) r, λ j = { λ 1 if j = 1 1 else (3.28) The oscillator indicator σ j is a measure of the gradients of the polynomial, and in reference coordinate system reads, σ j = K K α=1 β=1 T E ( ) α+β 2 ξ α η β p j(x(ξ, η)) dξdη. (3.29) In practice we use 50% more elements inside the stencil than degrees of freedom for 31

48 3. Numerical methods for hyperbolic equations triangular meshes, the constant positive number ǫ = 10 5 and the linear weight λ 1 = Finite Volume schemes in one space dimension In this section we present how to construct a one dimensional Finite Volume numerical scheme for hyperbolic equations. A general one dimensional non-linear hyperbolic balance law can be written as follows, t Q + x F(Q) + A(Q) x W(Q) = S(Q), (3.30) where Q is the unknown of conservative variables, W(Q) is the unknown vector of physical variables, F(Q) is the flux vector, A(Q) is a variable coefficient matrix and S(Q) is a source term which does not consider derivatives of the unknown vector. Definition of matrix A(Q) is somehow arbitrary depending on the particular equations. In our experience, the best choice is that the product A(Q) x W(Q) considers only the terms that can not be cast in conservative form. We assume that (3.30) can be cast in full quasi-linear form and the Jacobian matrix has real eigenvalues. Expression (3.30) is a general form which permits to introduce conservative and non conservative terms. When non-conservative and source terms are not present, (3.30) reduces to the classical one-dimension conservative form. Here we consider regular one dimensional grids, with x = x i the centre of the i-th cell and boundaries at x = x i 1/2 and x = x i+1/2 Integrating (3.30) over the control volume [x i 1/2, x i+1/2 ] [t n, t n+1 ] we have, where, Q n+1 i = Q n i t [ ] t Fi+1/2 F x i 1/2 x A [ ] i Wi+1/2 W i 1/2 + tsi, (3.31) xi+1/2 Q n i = 1 x Q(x, t) dx, x i 1/2 W i+1/2 = 1 t F i+1/2 = 1 t S i = 1 t t n+1 t n+1 t n W(Q(x i+1/2, t)) dt, t n+1 xi+1/2 1 x t n t n F(Q(x i+1/2, t)) dt, x i 1/2 S(x, t,q(x i+1/2, t)) dx dt, (3.32) 32

49 3. Numerical methods for hyperbolic equations with x = x i+1/2 x i 1/2 and t = t n+1 t n. Q n i is the cell average of the vector Q(x, t) inside the cell x i at time level t n. W i+1/2 and F i+1/2 are average time integrals in terms of Q(x i+1/2, t). Regarding that we provide approximations to integrals in (3.32) and define the linear coefficient matrix, we obtain an explicit Finite Volume numerical method to update the cell average from time level t n to time level t n+1. Linearization of A i may play an important role in the numerical solution. For FV and DG methods, a stability condition is necessary to compute the temporal step t. For one-dimensional FV methods the stability condition is C cfl 1.0, for twodimensional the condition is C cfl 0.5. For DG the stability condition is more restrictive as the order of the method increases. For two-dimensional DG methods we use C cfl 0.5/(2(o 1) + 1), with o the order of the method. Another element that is necessary to consider is the boundary conditions. In this thesis we solve Riemann problems on the boundary in order to set the correct boundary condition, therefore we need to define the external data for the Riemann problem. For transmissive boundary this data is a copy of the data inside. For reflective boundaries, or wall boundaries, we copy the data from inside modifying the normal velocity in order to obtain zero velocity normal to the boundary. For high-order numerical methods, these boundary conditions are implemented using the HEOC DRP solver presented in Chapter 5. Finite Volume and Discontinuous Galerkin FE methods have been presented in one and two dimensions. These are explicit one-step numerical methods where space and time integrals must be evaluated. In Chapter 5 we see how to solve with arbitrary order of accuracy these integral averages. Solving Q(x i+1/2, t) as the solution of the classical Riemann problem we obtain a first order numerical method of Godunov type. Finally, S i is an integral approximation within the whole control volume of the source term. High order evaluation of this term is presented in Chapter 7. 33

50 3. Numerical methods for hyperbolic equations 34

51 4 Classical Riemann solvers 4.1 Introduction One of the building blocks to construct upwind numerical methods for hyperbolic equations is the solution of the classical Riemann problem. The solution of this Initial Value Problem (IVP) involves the knowledge of the wave structure and the physics of the problem. In general the solution is more complex as more waves are involved and the physics is more realistic. There are cases where the physical model is simple, still very interesting, and the solution can be found exactly. On the other hand, there are cases where the model is too complex and one must approximate the solution of the Riemann problem. Solving the Riemann problem exactly gives us a reference solution which can be used to test our numerical methods. Furthermore, applying the solver locally in between two neighbouring cells a numerical flux can be obtained. Not all numerical methods utilize the exact solution of the Riemann problem. Sometimes the solver is too expensive (computational time) for numerical purposes and sometimes the solver does not exist. Nevertheless, some kind of approximation is considered in order to estimate the solution between two neighbour cells or the numerical flux. These solvers are known as approximate Riemann solvers. In this category we found the PVRS solver [90], the two-rarefaction solver (TRRS) and the two-shock solver (TSRS) [95], the classical HLL [46] and its improvement HLLC [94], the well known Roe s approximation originally presented in [70]. Recently, a new kind of approximate Riemann solvers have been presented using a predictor-corrector approach. These are the MUSTA [96] and EVILIN [97] solvers. These solvers can be used to construct first order Godunov-type methods. The Riemann solver (exact or approximate) is also a fundamental part in constructing high-order Godunov-type methods, such as ADER-type schemes, where the leading term must be computed. In addition, a linearized solver is also needed for computing the high order terms. We deal with these issues in Chapters 5 and 7. In this section we present Riemann solvers for three systems: the Euler equations, the Baer-Nunziato equations and the shallow water equations. We use the concepts presented in Chapter 2 applied to the specific equations. We start with the single phase gas dynamics 35

52 4. Classical Riemann solvers system, the Euler system. Then, the same approach is applied to the multiphase model of Baer-Nunziato. Finally, the shallow water system is treated. In what follows we refer to the unknown vector Q as a conservative variable vector and the unknown vector W as a non-conservative or physical variable vector. Transformation from one to the other is easily obtained from the respective balance laws and the equation of state, when needed. 4.2 Euler equations Introduction The Euler system is a system of non-linear hyperbolic conservation laws that governs the dynamics of compressible materials, such as gases but also liquids or granular materials at high pressures. The conservation laws are supplemented with a constitutive relation that relates the thermodynamic variables of the material. This constitutive relation is the equation of state (EOS) and physical principles of thermodynamics impose stringent constraints on it. A relevant publication on this is the one by Menikoff and Plohr [64]. Non-ideal EOS are used in order to approximate the real physics of a particular problem. In this case the solution of the Riemann problem becomes more complex. The first exact Riemann solver for the Euler equations is due to Godunov in [40], which was improved by himself in [41], by Chorin in [14] and van Leer in [107]. Later on, the work of Gottlieb and Groth [42] is credited for being one of the most efficient solvers. In [89], Toro presented the exact solution for the covolume gas, while Ivings et al. in [51] presented the solution for the stiffened EOS. In [95] details of the solution for the ideal EOS are found. All these solutions involve iterative procedures, which in the best case, solve one algebraic non-linear equation. The first work that we found related to the solution of the Riemann problem for general EOS is the one from Colella and Glaz [22]. In this work they made a local parametrisation of the equation of state. Later, Saurel et al. [74] presented the general methodology for a general EOS. A very similar approach was used by Quartapelle et al. in [68]. In general the method considering a general equation of state solves two non-linear algebraic equations. If one desires faster solvers, approximations can be introduced, as in the work of Dukowicz [26], Glaister [35] and Gallouët et al. [31]. In the following sections we present the equations and the eigenstructure of the Euler equations. Then, the exact Riemann problem is solved following the ideas of [74], [95] and [68]. Finally, the linear solution of the Riemann problem is presented. Here we neglect the effects of body forces, viscous stress and heat transfer. See the book of Toro [95] for a 36

53 4. Classical Riemann solvers comprehensive review about Riemann solvers Equations and eigenstructure Considering the homogeneous two dimensional Euler system we have t (ρ) + x (ρu) + y (ρv) = 0, t (ρu) + x (ρu 2 + p) + y (ρuv) = 0, t (ρv) + x (ρuv) + y (ρv 2 + p) = 0, t (E) + x (u(e + p)) + y (v(e + p)) = 0, (4.1) where the usual notation is used, ρ being the density, u and v the particle velocities in x and y directions respectively, p the pressure and E the total energy. The specific energy e is introduced in E = ρ ( 1 2 (u2 + v 2 ) + e ) and is defined by the equation of state, typically e = e(ρ, p). In consequence the sound speed a can be computed as follows, e = e(ρ, p), a = p ρ 2 e p e ρ. (4.2) e p In order to preserve hyperbolicity, the equation of state must obey the convexity property, that is, with ν = 1/ρ the specific volume. p ν < 0, 2 p ν 2 > 0, (4.3) Using vectors, system (4.1) can be cast in conservative form as follows, t Q + x F(Q) + y G(Q) = 0, (4.4) where Q = ρ ρu ρv E, F(Q) = ρu ρu 2 + p ρuv u(e + p), G(Q) = ρv ρuv ρv 2 + p v(e + p). (4.5) For solving the Riemann problem we can consider the following augmented one dimensional problem, t Q + x F(Q) = 0, (4.6) 37

54 4. Classical Riemann solvers which written in physical variables reads, W t + B(W)W x = 0, (4.7) with W = ρ u v p, B(W) = u ρ u 0 1/ρ 0 0 u 0 0 ρa 2 0 u. (4.8) System (4.7)-(4.8) has real eigenvalues λ 1 = u a, λ 2 = u, λ 3 = u, λ 4 = u + a, (4.9) with right eigenvectors r 1 = 1 a/ρ 0 a 2, r2 = , r3 = , r4 = 1 a/ρ 0 a 2, (4.10) and left eigenvectors l 1 = [ 0, 1, 0, 1 ] [, l 2 = 1, 0, 0, 1 ] [ ρa a 2, l 3 = [0, 0, 1, 0], l 4 = 0, 1, 0, ] 1. ρa (4.11) Studying the characteristic fields we know that the first and the fourth waves generate genuinely non-linear fields, while the second and the third waves generate linearly degenerate fields. Example: Considering the ideal equation of state e = p/(ρ(γ 1)), the sound speed becomes a = γp/ρ and the character of the field associated to λ 1 = u a is obtained evaluating (2.38) as follows, [ ] λ 1 ρ, λ1 u, λ1 v, λ1 [1, a/ρ,0,a 2] T = p(γ 2 + γ)/(2aρ 2 ) 0. (4.12) p This result confirms that the wave family associated with λ 1 = u a is genuinely nonlinear. A similar procedure can be applied to the remaining fields. We refer to the book of Godlewski and Raviart [39] for more details about the theory when general equations 38

55 4. Classical Riemann solvers of state are used. It is assumed that the only types of waves associated with the genuinely non-linear fields are rarefaction waves or shock waves, while the interior waves associated to families λ 2 and λ 3 generate only contact discontinuity waves. In the following section we see how to solve the classical Riemann problem, where the initial conditions are piece-wise constant data. In Chapter 5 we deal with the Derivative Riemann Problem (DRP) where the initial conditions are piece-wise polynomial data Non-linear Riemann solver The classical Riemann problem is, PDEs: t Q + x F(Q) { = 0, x (, ), t > 0, Q L = Q(W L ) if x < 0, IC: Q(x,0) = Q R = Q(W R ) if x > 0, (4.13) with W L = ρ L u L v L, W R = ρ R u R v R, (4.14) p L p R the initial constant data. The solution of (4.13) is depicted in Fig t λ 2 =λ 3 λ 1 λ 4 W* L W* R W L F L x=0 F R W R x Figure 4.1: Wave structure for the 2 dimensional Euler equations. Functions F L and F R connect the outer states W L and W R (constant initial conditions) with the unknown constant states WL and W R (Star Zone). As presented in section 2.7 we can connect the known data outside with the unknown values inside, called the star zone. In Fig. 4.1 the general wave structure of system (4.13) is presented. We have two genuinely non-linear fields, (λ 1, λ 4 ), and two linearly degenerate fields, (λ 2, λ 3 ). The initial conditions W L and W R are connected to the unknown values 39

56 4. Classical Riemann solvers WL and W R through the functions F L and F R. Then, a trivial condition through the contact waves connects both functions F L and F R. With this procedure we solve exactly the classical Riemann problem for the Euler equations. The unknown quantities in the star zone is denoted by the vectors WL = ρ L u L vl, W R = ρ R u R vr. (4.15) p L p R The final goal of the solution procedure is to obtain the two functions F L and F R as follows, F L (ρ L,W L ) = [ F ul (ρ L,W L) F pl (ρ L,W L) ], F R (ρ R,W R ) = [ F ur (ρ R,W R) F pr (ρ R,W R) ]. (4.16) In other words, to obtain algebraic functions that relate the initial data with the star zone. These functions depend on the type of waves they cross, either a shock or rarefaction wave. Moreover, they can be connected at the contact wave giving us a function, F = F (ρ L, ρ R,W L,W R ) = F L (ρ L,W L ) F R (ρ R,W R ) = 0, (4.17) which depends on the density on both sides of the contact and the initial data. This procedure reduce the solution of the Riemann problem in the star zone into an algebraic 2 2 non-linear system of equations that can be solved iteratively. Now we see how to obtain these functions depending on the waves that are present Contact waves The waves associated to λ 2 and λ 3 are linearly degenerate and apply the Generalized Riemann Invariants. Utilizing the physical variables formulation (4.7), the GRIs for field λ 2 give, The GRIs for field λ 3 give, dρ 1 = du 0 = dv 0 = dp 0. (4.18) dρ 0 = du 0 = dv 1 = dp 0. (4.19) 40

57 4. Classical Riemann solvers Equation (4.18) says that across λ 2 the pressure p and velocities u and v are constant, while (4.19) says that across λ 3 all variables are constant but v. We conclude that across the waves λ 2 = λ 3 = u = u we have u L = u R = u, p L = p R = p. (4.20) When we analyze the wave fields associated to λ 1 and λ 4 we need to distinguish between rarefaction waves and shock waves. The general criterion used is that the wave is a rarefaction if the density p p k, while it is a shock if p > p k for k = L, R. This criterion respects all the constraints mentioned in section Rarefaction waves Across a rarefaction wave we apply again the GRIs. For λ 1 = u a we obtain dρ 1 = du a/ρ = dv 0 = dp a 2. (4.21) The first conclusion is that the velocity v does not change. Taking the first equality in (4.21), and writing explicitly the dependence of the sound speed in term of density (isentropic path) we have, a(ρ) ρ dρ = du. (4.22) We can integrate (4.22) from the known data W L to the unknown value W L obtaining the second Riemann invariant (entropy was already known being a Riemann Invariant, see section 2.7), where (4.24) holds true for any point inside the rarefaction. ρ L a( ρ) u ρ L ρ d ρ = du, (4.23) u L ρ u L a( ρ) = u L d ρ, (4.24) ρ L ρ Finally, again using the isentropic path, we can express the pressure p = p(ρ) across a rarefaction wave. Moreover, we can find a function P s in such a way that p = P s (ρ L,W L). With these results, we write the vectorial function F L in (4.16) when the wave family 41

58 4. Classical Riemann solvers associated to λ 1 = u a is a rarefaction as follows, F ul (ρ L,W L) F pl (ρ L,W L) ρ u L a( ρ) = u L ρ L ρ d ρ, p = P s (ρ L,W L). (4.25) A similar result can be obtained for the field associated to λ 4 = u + a, and therefore for function F R in (4.16), we have, F ur (ρ R,W R) ρ u R a( ρ) = u R + ρ R ρ d ρ, F pr (ρ R,W R) p = P s (ρ R,W R). (4.26) The isentropic relations for the sound speed a = a(ρ) and for the pressure p = p(ρ) along the isentropic path are obtained from the particular equation of state which is chosen, see Appendix A for expressions for the stiffened EOS and for the polytropic van der Waals EOS Shock waves Shock waves are discontinuous waves as presented in section for which the Rankine- Hugoniot conditions are valid. Without loss of generality we can study a right travelling shock wave with speed S with known data ahead and unknown state behind as in Fig 4.2(a). For a proper transformation of the frame of reference we can obtain a stationary shock, where the quantities involved are modified by this transformation, see Fig 4.2(b). Here we apply the Rankine-Hugoniot conditions to the augmented one dimensional conservative form (4.6) where v can be interpreted as a passive scalar. S 0 ρ* k ρ k ρ* k ρ k u * u k u * u k v k * v k v k * v k p * p k p * p k (a) (b) Figure 4.2: Shock wave travelling to the right with known data ahead (right) of the shock and unknown state behind (left) of the shock. (a): Shock travelling with speed S in a stationary frame of reference. (b): After a transformation the shock is stationary with the new states modified by the transformation. 42

59 4. Classical Riemann solvers The proper transformation is given by û = u S and û k = u k S for k = L, R. To these new variables we can apply the Rankine-Hugoniot conditions with S = 0, therefore we have ρ k û k = ρ kû, ρ k û k v k = ρ kû vk, ρ k û 2 k + p k = ρ kû 2 + p, û k (Êk + p k ) = û (Ê k + p ). (4.27) A direct substitution of the first equation of (4.27) into the second gives that v k = v k, that means that the variable v does not change across shocks. Now, from (4.27), expressing the total energy in terms of the specific energy, the next relation is obtained, e e k = 1 [ ρ ] 2 (p + p k ) k ρ k ρ k ρ. (4.28) k Defining the mass flux M ρ k û = ρ kû and using equations (4.27) we obtain, M M 2 = (u u k ) ρ k ρ k ρ k ρ, k = (p k p ) ρ k ρ k ρ k ρ. k (4.29) Now, replacing in (4.28) the internal energy e = e(ρ k, p ) obtained from the particular equation of state, and using (4.29), the functions that connect the known state W L to the unknown state WL across the shock wave read, F ul (ρ L,W L) u = u L (p L p )( 1 ρ L 1 ρ L ), F pl (ρ L,W L) p = P RH (ρ L,W L), (4.30) and for connecting the state W R to the unknown state W R, F ur (ρ R,W R) u = u R + (p R p )( 1 ρ R 1 ρ R ), F pr (ρ R,W R) p = P RH (ρ R,W R). (4.31) In (4.30) and (4.31), F ul and F ur depend on ρ k. The function P RH(ρ k,w k) relates the pressure and the density in the star zone. This function depends on the particular equation of state. See Appendix A for details on this function when stiffened and van der Waals EOS are used. It is important to remark that equations (4.25),(4.26),(4.30) and (4.31) do not consider 43

60 4. Classical Riemann solvers any particular equation of state up to this point Iterative procedure to solve the star values With this approach, we transformed the solution of the IVP (4.13) in the star zone into a 2 2 algebraic non-linear system that can be solved iteratively. We use the general Newton-Raphson procedure to solve the system. Writing the vector function F defined in (4.17) we have, [ ] F (ρ L, ρ F ul (ρ L R,W L,W R ) =,W L) F ur (ρ R,W R) F pl (ρ L,W L) F pr (ρ R,W = 0. (4.32) R) Function F (ρ L, ρ R,W L,W R ) depends non-linearly on the two densities, ρ L and ρ R. The dependence on the initial data W L and W R is through the type of wave associated to wave fields λ 1 and λ 4. Once the type of wave fields are defined we have a system of two non-linear equations F (ρ L, ρ R ) = 0 with two unknowns ρ L and ρ R. The Newton-Raphson procedure solves iteratively (4.32). At each iteration, the corrector value X is obtained from solving the following m-th linear system, The Jacobian matrix J m is obtained as follows, J m = The corrected values read, F ul (ρ L,W L) ρ L F pl (ρ L,W L) ρ L [ XJ m = F (ρ m L, ρ m R ). (4.33) ρ m+1 L ρ m+1 R F ur(ρ R,W R) ρ R F pr(ρ R,W R) ρ R ] = [ ρ m L ρ m R ]. (4.34) /(ρ L,ρ R )=(ρ m L,ρ m R ) + X. (4.35) The generalization of this method gives the final solution ρ L and ρ R. At each iteration we must verify if the non-linear waves are shocks or rarefactions in order to compute correctly the functions F L and F R. This iterative method provides a very good convergence rate, if the initial guess is good enough. Before using it, it is necessary to know the behaviour of the vector function F. We know, for example, that near low densities the procedure may fail, due to square roots of negative values, see (4.2). Another difficulty appears when shock waves are present, which is a discontinuity in the function P RH (ρ k,w k), see (4.30) and (4.31). For example 44

61 4. Classical Riemann solvers for the ideal EOS with γ = 1.4 the discontinuity is located at ρ k = 6ρ k. For a guess value ρ 0 k 6ρ k the iterative procedure converges to a wrong solution. With this procedure we obtain the solution of the densities in the star zone. It remains to determine the velocity and pressure. This is done with the use of the functions defined in the previous section, see for example (4.25) and (4.26) for rarefactions waves, or (4.30) and (4.31) for shock waves. Finally, for the complete solution in the half plane x t the velocities of the waves needs to be evaluated, defining the complete wave structure, see [95] for details. For numerical purpose we need to sample this wave structure and identify the state along the t axis. This state is used to compute the numerical flux in one of the numerical methods presented in Chapter 3. On constructing high-order numerical methods, and in this thesis the solution of the Derivative Riemann Problem, see Chapter 5, is also necessary to solve linear Riemann problems. This is the subject of the following section Linear Riemann solvers for conservative variables Here we present the solution of the linear Riemann problem. This solution will be used later in Chapter 5. We start with equations (4.6). t Q + x F(Q) = 0. (4.36) Computing the Jacobian matrix we obtain the quasi-linear form, t Q + A(Q) x Q = 0. (4.37) We linearize (4.37) and pose the following classical linear Riemann problem, PDE: t Q + à x Q = 0, x (, ), t > 0. { Q L if x < 0, IC: Q(x, 0) = Q R if x > 0, (4.38) where à = A( Q) is frozen at the state Q = 1 2 (Q L + Q R ). Provided that we have the right eigenvector matrix of à we can solve (4.38) as in section 2.6. For completeness we recall the eigenvalues: λ 1 = u a, λ 2 = u, λ 3 = u and λ 4 = u+a. We define the jump = Q R Q L and solve the following linear system 4 δ i r i =, (4.39) i=1 45

62 4. Classical Riemann solvers where r i are the right eigenvectors and the unknowns are the wave strengths δ i. The main difficulty is that the Jacobian matrix Ã, and therefore the right eigenvectors, depends on the equation of state. We present the solution in terms of the eigenvectors and wave strengths considering the stiffened-gas EOS, and the politropic van der Waals EOS Solution for the stiffened-gas EOS The right eigenvectors are: r 1 = 1 u a v, r2 = V a2 γ 1 ua 1 u v V 2 2 0, r3 = 0 1, r4 = v 1 u + a v, V a2 γ 1 + ua (4.40) with V 2 = (u 2 + v 2 ). The components and all vectors in (4.40) are known, as are the components of. Solving (4.39) for the unknown wave strengths δ i, we obtain, δ 1 = V2 (γ 1)+2ua 4a 2 δ 2 = V2 (γ 1) 2a 2 2a 2 δ 3 = v 1 + 3, δ 4 = V2 (γ 1) 2ua 4a 2 1 u(γ 1)+a 2a u(γ 1) a 2 1 u(γ 1) a 2a 2 2 v(γ 1) 2a (γ 1) 2 + v(γ 1) a 2 2a 2 4, 3 (γ 1) a 2 4, 2 v(γ 1) 2a (γ 1) 2a 2 4. (4.41) 46

63 4. Classical Riemann solvers Solution for the van der Waals EOS The right eigenvectors are: r 1 = r 2 = r 4 = 1 u a v Cv(β (ρ a2 +p+3 ρ 2 α) 2 ρ α a 2 ) R + V2 2 ua 2 ρ α 1 u v Cv (β (p+3 ρ2 α) 2 ρ α) R + V2 2 2 ρ α 1 u + a, 0, r3 = 0 1, v v Cv(β (ρ a2 +p+3 ρ 2 α) 2 ρ α a 2 ) R + V2 2 + ua 2 ρ α, (4.42) with V 2 = (u 2 + v 2 ). The components and all vectors in (4.42) are known, as are the components of. The constant α, β, R and C v are parameters of the van der Waals equation of state. See the Appendix A. Solving (4.39) for the unknown wave strengths δ i we obtain, δ 1 δ 2 = ( ( R Cv = 1)ρα ( 1+ρβ)a 2 + ( 2( R Cv 1)ρα ( 1+ρβ)a 2 + V 2 R C v( 1+ρβ)4a + βp+3βρ2 α 2 ( 1+ρβ)2a 2 u 2a V 2 R C v( 1+ρβ)2a + βp+3βρ2 α ( 1+ρβ)a 2 ) ) ( ) 1 + R u 2 +v 3 4 C v ( 1+ρβ)2a 2 2 2a, ( ) 1 R u 2 +v 3 4 C v, ( 1+ρβ)a 2 δ 3 = v ( 1 + 3, ) ( R Cv δ 4 = 1)ρα ( ) V + 2 R ( 1+ρβ)a 2 C v( 1+ρβ)4a + βp+3βρ2 α + u 2 ( 1+ρβ)2a 2 2a 1 + R u 2 +v 3 4 C v + ( 1+ρβ)2a 2 2 2a, (4.43) The solution of the linear Riemann problem (4.38) on the t axis is obtained from (2.35) as follows, Q(0, t) = Q L + λ i 0 δ i r i or Q(0, t) = Q R λ i > 0δ i r i. (4.44) 47

64 4. Classical Riemann solvers 4.3 The Baer-Nunziato equations Introduction For nearly 30 years multiphase models have been widely studied and a number of models have been proposed, see for example Baer and Nunziato [4], Saurel and Abgrall [73], Romenski and Toro [71], Stewart and Wendroff [83], Drew [25], Ishii [50] or Gidaspow [33], amongst others. Here we deal with the multiphase model presented by Baer and Nunziato in [4]. This model was proposed to describe the combustion behaviour of granular energetic materials. In particular the deflagration to detonation transition (DTT) phenomena. It is a twopressure model and agrees with the known fact that having two pressure models preserves hyperbolicity. Each phase obeys the single phase Euler equations plus non-conservative terms (called nozzling terms) which take into account the interphase exchange of pressure and velocity. Considering numerical methods for hyperbolic equations the experience is vast for simple problems. Now this experience is coming into multiphase models and a major effort has been put into it. See for example the work of Andrianov and Warnecke with their inverse problem [3], Chang and Liou [12], Abgrall [2] or Schwendeman et al. [78], where an iterative two-step Riemann solver is proposed. The work [78] is very interesting, where they manage to solve the complete Riemann problem iteratively, in a similar way as in the Riemann solver presented in Castro and Toro [10] for the Saurel-Abgrall isentropic model. Here the key point is to find proper jump relations across the solid contact, making use of an auxiliary variable and then solving the decoupled system for the solid and gas phases. As Schwendeman et al. [78] say in the article, this procedure is expensive in computational time and a less expensive procedure is still needed. In this section we deal with solution methods for the classical Riemann problem using three different approaches: the first one solves the exact Riemann problem as proposed in [78], the second one applies the EVILIN Riemann solver [97] and the third one uses a stratification hypothesis [11] prior to the application of the EVILIN [97] or the exact Riemann solver [51]. Because interaction between the two phases takes place in the solution through a contact wave, complete Riemann solvers are needed, that is Riemann solvers that account for all characteristic fields in their structure. 48

65 4. Classical Riemann solvers Equations and eigenstructure The multiphase model proposed by Baer-Nunziato [4] represents the interaction between two compressible fluids considering non-equilibrium pressure. Originally presented as a deflagration to detonation transition model (DDT) was analized by Embid and Baer [30]. The two compressible fluids are denoted by suffixes k = 1, 2. The interface velocity and pressure are respectively denoted by u 1, p 2. Due to the presence of the nozzle terms the system cannot be cast in conservative form. Neglecting exchange terms such as chemical reactions and drag forces, the homogeneous one dimensional model is written as t (α 1ρ 1 ) + x (α 1ρ 1 u 1 ) = 0 t (α 1ρ 1 u 1 ) + x (α 1[ρ 1 u p 1]) p 2 (α x 1 ) = 0 t (α 1E 1 ) + x (α 1u 1 [E 1 + p 1 ]) p 2 u 1 x (α 1) = 0 t (α 2ρ 2 ) + x (α 2ρ 2 u 2 ) = 0 t (α 2ρ 2 u 2 ) + x (α 2[ρ 2 u p 2]) + p 2 (α x 1 ) = 0 t (α 2E 2 ) + x (α 2u 2 [E 2 + p 2 ]) + p 2 u 1 (α x 1 ) = 0 t (α 1) + u 1 (α x 1 ) = 0 (4.45) In (4.45), the usual notation is used where ρ k denotes density, u k is particle velocity, p k is pressure, E k = ρ k ( 1 2 u2 k + e k) is total energy, e k is specific energy and α k is the void fraction, assuming α k = 1. The first three equations in (4.45) represent the space and time evolution of fluid 1, as the classical gas dynamic Euler system, plus the inclusion of the interaction terms between both phases. An analogous description applies for the fourth to sixth equations for fluid 2. The seventh equation is a closure relation that represents the advection of the interface between the fluids. Each fluid is governed by a general EOS of the form e k = e k (ρ k, p k ) with the sound speed a k computed as, e k = e k (ρ k, p k ), a k = p k ρ 2 k e k p k System (4.45) can be written in vectorial notation as follows, e k ρ k e k p k. (4.46) t Q + x F(Q) + A(Q) x W(Q) = S(Q), (4.47) where vector Q is the conservative unknown vector, F(Q) is the flux vector, A(Q) is a coefficient matrix with variable entries and W(Q) is a non-conservative variables vector. Equation (4.47) represents a general hyperbolic system with conservative and non- 49

66 4. Classical Riemann solvers conservative terms whose choice is somewhat arbitrary. In our experience the following represents the most desirable combination for system (4.45). α 1 ρ 1 α 1 ρ 1 u 1 α 1 E 1 Q = α 2 ρ 2 α 2 ρ 2 u 2 α 2 E 2 α 1 α 1 ρ 1 u 1 α 1 [ρ 1 u p 1] α 1 u 1 [E 1 + p 1 ] F(Q) = α 2 ρ 2 u 2 α 2 [ρ 2 u p 2] α 2 u 2 [E 2 + p 2 ] 0 (4.48) p p 2 u 1 A(Q) = p p 2 u 1 W(Q) = ρ 1 u 1 p 1 ρ 2 u 2 p 2 (4.49) u 1 α 1 An alternative formulation for system (4.45) is the fully non-conservative one where the unknowns are the physical variables. This formulation will allow us to extract very useful information from the eigenstructure of the system which in quasi-linear form reads, t W + B(W) x W = 0, (4.50) where W is the vector of physical variables and B(W) is the Jacobian matrix, namely W = ρ 1 u 1 p 1 ρ 2 u 2 p 2 α 1 The real eigenvalues are u 1 ρ u 1 ρ a 2 1 ρ 1 u B(W) = u 2 ρ u 2 ρ 2 0 p 1 p 2 α 1 ρ 1 ρ 2 (u 1 u 2 ) α a 2 2 ρ 2 u 2 a 2 2ρ 2 (u 1 u 2 ) α u 1 (4.51) λ 1 = u 1 a 1, λ 2 = u 2 a 2, λ 3 = u 2, λ 4 = λ 5 = u 1, λ 6 = u 2 + a 2, λ 7 = u 1 + a 1, (4.52) 50

67 4. Classical Riemann solvers with corresponding right eigenvectors r 1 = 1 a 1 /ρ 1 a , r 2 = a 2 /ρ 2 a 2 2 0, r 3 = , r 4 = , (4.53) r 5 = 0 0 α 2 ( (u1 u 2 ) 2 a 2 2 ) (p1 p 2 ) α 1 ρ 2 (u 1 u 2 ) 2 a 2 2 (u 1 u 2 ) ρ 2 a 2 2 (u 1 u 2 ) 2 α 2 ( (u1 u 2 ) 2 a 2 2 ), r 6 = a 2 /ρ 2 a 2 2 0, r 7 = 1 a 1 /ρ 1 a , (4.54) and left eigenvectors l 1 = [0, α 1 ρ 1 a 1, α 1, 0, 0, 0, p 1 p 2 ], l 2 = [ 0, 0, 0, 0, α 2(u 1 u 2 +a 2 ) a 2 (u 1 u 2 ), α 2(u 1 u 2 +a 2 ) a 2 2 ρ 2(u 1 u 2 ), 1 ], l 3 = [ 0, 0, 0, a 2 2, 0, 1, 0], l 4 = [ a 2 1, 0, 1, 0, 0, 0, p 1 p 2 α 1 ], l 5 = [0, 0, 0, 0, 0, 0, 1], l 6 = [ 0, 0, 0, 0, α 2(u 1 u 2 a 2 ) a 2 (u 1 u 2 ), α 2(u 1 u 2 a 2 ) a 2 2 ρ 2(u 1 u 2 ), 1 ], l 7 = [0, α 1 ρ 1 a 1, α 1, 0, 0, 0, p 1 p 2 ]. (4.55) From (4.52) we see that all eigenvalues of B(W) are real; however they are not distinct and therefore this system is not strictly hyperbolic. Possible singular points are: (a) u 1 = u 2, (b) u 1 = u 2 ± a 2, (c) u 1 ± a 1 = u 2, and (d) u 1 ± a 1 = u 2 ± a 2. Condition (b), called sonic condition or chocked flow condition, represents a real problem in which the eigenvectors do not form a complete set of linear independent vectors, leading to a parabolic degeneracy [30], [3]. From the work of Embid and Baer [30] we know that characteristic fields associated with λ 1, λ 2, λ 6 and λ 7 are genuinely non-linear fields, as a result shock or rarefaction waves are 51

68 4. Classical Riemann solvers produced. Waves associated to λ 3, λ 4 and λ 5 are linearly degenerated fields and therefore contact waves are produced. More over, λ 3 is a contact wave associated to fluid 2, λ 4 is a contact wave associated to fluid 1 and λ 5 is a contact wave that connects both fluids and coalesces with λ 4. We present this wave structure in Fig 4.3 for the case λ 3 < λ 5. λ 1 λ 3 t λ 4 =λ 5 λ 2 λ 6 λ 7 W L α 1L x=0 α1r W R x Figure 4.3: Wave structure of the system (4.45). Wave fields associated to λ 1, λ 2, λ 6 and λ 7 are genuinely non-linear. Wave fields associated to λ 3, λ 4 and λ 5 are linearly degenerated. The void fraction jumps discontinuously only across wave field λ 5. In this case λ 3 < λ 5. If we intent to solve exactly system (4.45), we need to be able to define jump relations across shock waves. This conditions, Rankine-Hugoniot conditions, are well defined only for PDEs in conservative form for physical conservative variables. See proposition in [93] for an example of conservative laws in a mathematical sense but not in a physical sense. Andrianov et al. in [3] and later Schwendeman et al. in [78] considered the fact that the void fraction does not jump away from the contact discontinuity associated to λ 5, a property that was identified by Embid et al. in [30]. This is observed when we consider the characteristic variables, see last equation in (4.45) where α 1 is a characteristic variable. This condition allow us to consider two uncoupled conservative Euler equations away from the field λ 5 = u 1. Rankine-Hugoniot conditions can be applied to each one of the Euler single phase equations. One possible difficulty pointed in [30] exists when a shock wave travels at velocity S = u 1. In this case, the shock and the contact wave λ 5 collide and the simplification of two uncoupled conservative Euler equations is not possible. Moreover, if we have a non-linear field λ = u 2 ± a 2 = u 1, condition b) is verified and the hyperbolicity is lost for system (4.45). This is still an open problem. Assuming that condition (b) is not reached and that no shock waves travel at the particle velocity of fluid 1, we can use all methodology presented in 2. 52

69 4. Classical Riemann solvers Non linear Riemann solver The classical Riemann problem that we want to solve is, PDEs: t Q + x F(Q) { + A(Q) x W = 0, x (, ), t > 0, Q L = Q L (W L ) if x < 0, IC: Q(x,0) = Q R = Q R (W R ) if x > 0, (4.56) with initial data given by the vectors W L = ρ L1 u L1 p L1 ρ L2 u L2 p L2, W R = ρ R1 u R1 p R1 ρ R2 u R2 p R2, (4.57) α L1 α R1 We define five vector unknown states of three components, similar as the Euler one dimensional vectors, which are connected through the different wave fields. Two unknown states for fluid 1 on each side of λ = λ 4 = λ 5 and three unknown states for fluid 2 which are divided by λ = λ 3 and λ = λ 4 = λ 5. These unknowns are as follows, W L2 = W L1 = ρ L2 u L2 ρ L1 u L1 p L1, WR1 =, WC2 = ρ C2 u C2 ρ R1 u R1 p R1, (4.58), WR2 = ρ R2 u R2. (4.59) p L2 p C2 p R2 Note that these vectors have the sub index 1 or 2 making reference to the respective fluid and are not as the ones defined in (4.57). As mentioned before, we can see the system (4.45) as two Euler equation systems connected by a contact discontinuity. In Fig. 4.4(a) we show the fluid 1 wave structure with two non-linear waves and two coincident contact waves. Function F L1 connects the initial data W L with the unknown state W L1. In the same way function F R1 connects the initial data W R with the unknown state WR1. In Fig. 4.4(b) we show the fluid 2 wave structure with two non-linear waves and two contact waves, one related to fluid 2 and one related to the full system. Function F L2 connects the initial data W L with the unknown state 53

70 4. Classical Riemann solvers W L2, as function F R2 connects the initial data W R with the unknown state W R2. The unknown state W C2 is defined between contact waves λ3 and λ 5, in this case λ 3 < λ 5. λ 1 W * L1 t λ 4 =λ 5 W * R1 λ 7 λ 3 t λ 2 λ 6 * W * C2 * W L2 λ 5 W R2 W L F L1 x=0 F R1 W R x W L F L2 x=0 F R2 W R x (a) Fluid 1 (b) Fluid 2 Figure 4.4: Uncoupled representation of the wave structure of system (4.45) with λ 3 < λ 5. (a): Fluid 1 wave structure. Functions F L1 and F R1 connect the initial data with the unknown states WL1 and W R1 respectively. (b): Fluid 2 wave structure. Functions F L2 and F R2 connect the initial data with the unknown states WL2 and W R2 respectively. Unknown state W C2 is defined between λ3 and λ 5. The functions F Lk and F Rk, with k = 1, 2, are defined identically as in (4.16) for each fluid. They read F Lk (ρ Lk,W L) = [ F ulk (ρ Lk,W L) F plk (ρ Lk,W L) ], F Rk (ρ Rk,W R) = [ F urk (ρ Rk,W R) F prk (ρ Rk,W R) ]. (4.60) Across left rarefaction waves we have F ulk (ρ Lk,W L) F plk (ρ Lk,W L) Across right rarefaction waves we have ρ u Lk = u Lk a( ρ) Lk ρ Lk ρ d ρ, p Lk = P s(ρ Lk,W L). (4.61) F urk (ρ Rk,W R) ρ u Rk = u Rk a( ρ) Rk + ρ Rk ρ d ρ, F prk (ρ Rk,W R) p Rk = P s(ρ Rk,W R). (4.62) Across left shock waves we have F ulk (ρ Lk,W L) F plk (ρ Lk,W L) u Lk = u Lk (p Lk p Lk )( 1 ρ 1 Lk ρ Lk ), p Lk = P RH(ρ Lk,W L). (4.63) 54

71 4. Classical Riemann solvers Across right shock waves we have F urk (ρ Rk,W R) u Rk = u Rk + (p Rk p Rk )( 1 ρ 1 Rk ρ Rk ), F prk (ρ Rk,W R) p Rk = P RH(ρ Rk,W R). (4.64) Now we need to analyze the contact waves. The GRIs across λ 3 are trivial and give, ρ 1 u 1 p 1 u 2 p 2 α 1 = constant, = constant, = constant, = constant, = constant, = constant. (4.65) The GRIs across λ 4 are trivial and give, u 1 p 1 ρ 2 u 2 p 2 α 1 = constant, = constant, = constant, = constant, = constant, = constant. (4.66) It is important to remark that across λ 4 there is no jump in velocity or pressure due to this wave field, nevertheless λ 4 coincides with λ 5 which generates discontinuous jumps. Embid and Baer in [30] presented the GRIs for the wave family λ 5. They read, u 1 η 2 = constant, = constant, α 2 ρ 2 (u 1 u 2 ) = constant, α 1 p 1 + α 2 p 2 + α 2 ρ 2 (u 1 u 2 ) 2 = constant, 1 2 (u 1 u 2 ) 2 + h 2 = constant. (4.67) In (4.67) η is the entropy and h the enthalpy. From (4.67) it follows that particle velocity of fluid 1 and entropy of fluid 2 are conserved across the contact wave. The other 3 equations conserve mass, momentum and energy of the mixture across the contact discontinuity. Now we identify two possible configurations. They are λ 3 < λ 5 and λ 5 < λ 3. 55

72 4. Classical Riemann solvers Fluid-2 contact to the left of fluid-1 contact In this configuration we suppose that λ 3 < λ 5. See Fig.4.5, where the unknown state W C2 has the wave λ 3 on the left and the wave λ 5 on the right. In Fig. 4.5(a) the unknowns for fluid 1 are shown. Observe that the particle velocity is constant across the contact waves. In Fig. 4.5(b) the unknowns for fluid 2 are shown. We emphasize the fact that across the contact wave of fluid 2 pressure and particle velocity are constant. t λ 4,λ 5 λ 1 λ 7 ρ L1 u L1 p L1 ρ * L1 u* 1 p* L1 ρ * R1 u* 1 p* R1 x=0 x (a) Fluid 1 ρ R1 u R1 p R1 λ 3 t λ 5 λ 2 * * * λ 6 ρ L2 u L2 p L2 ρ L2 u* L2 p* L2 ρ C2 u* L2 p* L2 ρ R2 u* R2 p* R2 x=0 x (b) Fluid 2 ρ R2 u R2 p R2 Figure 4.5: Possible structure of the Riemann problem. In this case λ 3 < λ 5. Initial conditions and unknown variables in the star zone for each fluid. We consider the constant values across contact waves replacing u C2 = u L2 and p C2 = p L2. Across contact discontinuity λ 5 we have η C2 = η R2 u L1 = u R1 α L2 ρ C2 (u L1 u C2 ) = α R2ρ R2 (u R1 u R2 ) α L1 p L1 + α L2p C2 + α L2ρ C2 (u L1 u C2 )2 = α R1 p R1 + α R2p R2 + α R2ρ R2 (u R1 u R2 )2 1 2 (u L1 u C2 )2 + h C2 = 1 2 (u R1 u R2 )2 + h R2 (4.68) From (4.65) we know the Riemann invariants across λ 3 and therefore from (4.68) we can write the following five relations F 1 η(ρ C2 ) η(ρ R2 ) F 2 u(ρ L1 ) u(ρ R1 ) F 3 α L2 ρ C2 (u(ρ L1 ) u(ρ L2 )) α R2ρ R2 (u(ρ R1 ) u(ρ R2 )) F 4 α L1 p(ρ L1 ) + α L2p(ρ L2 ) + α L2ρ C2 (u(ρ L1 ) u(ρ L2 ))2 α R1 p(ρ R1 ) α R2p(ρ R2 ) α R2ρ R2 (u(ρ R1 ) u(ρ R2 ))2 F (u(ρ L1 ) u(ρ C2 ))2 + h(ρ C2 ) 1 2 (u(ρ R1 ) u(ρ R2 ))2 h(ρ R2 ) (4.69) 56

73 4. Classical Riemann solvers In (4.69) we use u C2 = u L2 and p C2 = p L2 from (4.65). We also express the entropy and the enthalpy in terms of density which can be obtained from the specific equations of state, see Appendix A Fluid-2 contact to the right of fluid-1 contact In this configuration we suppose that λ 5 < λ 3. See Fig.4.6, where the unknown state W C2 has the wave λ 5 on the left and the wave λ 3 on the right. In Fig. 4.6(a) the unknowns for fluid 1 are shown. Observe that the particle velocity is constant across the contact waves. In Fig. 4.6(b) the unknowns for fluid 2 are shown. We emphasize the fact that across the contact wave of fluid 2 pressure and particle velocity are constant. λ 4,λ 5 t λ 1 λ 7 ρ L1 u L1 p L1 ρ * L1 u* 1 p* L1 ρ * R1 u* 1 p* R1 x=0 x (a) Fluid 1 ρ R1 u R1 p R1 λ 5 t λ 3 λ 2 * * * λ 6 ρ L2 u L2 p L2 ρ L2 u* L2 p* L2 ρ C2 u* L2 R2 p* L2 R2 ρ R2 u* R2 p* R2 x=0 x (b) Fluid 2 ρ R2 u R2 p R2 Figure 4.6: Possible structure of the Riemann problem. In this case λ 5 < λ 3. Initial conditions and unknown variables in the star zone for each fluid. We consider the constant values across contact waves replacing u C2 = u R2 and p C2 = p R2. Across contact discontinuity λ 5 we have η L2 = η C2 u L1 = u R1 α L2 ρ L2 (u L1 u L2 ) = α R2ρ C2 (u R1 u C2 ) α L1 p L1 + α L2p L2 + α L2ρ L2 (u L1 u L2 )2 = α R1 p R1 + α R2p C2 + α R2ρ C2 (u R1 u C2 )2 1 2 (u L1 u L2 )2 + h L2 = 1 2 (u R1 u C2 )2 + h C2 (4.70) From (4.65) we know the Riemann invariants across λ 3 and therefore we can write the 57

74 4. Classical Riemann solvers following five functions F 1 η(ρ L2 ) η(ρ C2 ) F 2 u(ρ L1 ) u(ρ R1 ) F 3 α L2 ρ L2 (u(ρ L1 ) u(ρ L2 )) α R2ρ C2 (u(ρ R1 ) u(ρ R2 )) F 4 α L1 p(ρ L1 ) + α L2p(ρ L2 ) + α L2ρ L2 (u(ρ L1 ) u(ρ L2 ))2 α R1 p(ρ R1 ) α R2p(ρ R2 ) α R2ρ C2 (u(ρ R1 ) u(ρ R2 ))2 F (u(ρ L1 ) u(ρ L2 ))2 + h(ρ L2 ) 1 2 (u(ρ R1 ) u(ρ R2 ))2 + h(ρ C2 ) (4.71) In (4.71) we use u C2 = u R2 and p C2 = p R2 from (4.65). We also express the entropy and the enthalpy in terms of density which can be obtained from the specific equations of state, see Appendix A. In (4.69) and (4.71) the relations for velocity and pressure are computed in terms of density using expressions (4.61) to (4.64) with u(ρ L1 ) F ul1(ρ L1,W L), u(ρ R1 ) F ur1(ρ R1,W R), u(ρ L2 ) F ul2(ρ L2,W L), u(ρ R2 ) F ur2(ρ R2,W R), p(ρ L1 ) F pl1(ρ L1,W L), p(ρ R1 ) F pr1(ρ R1,W R), p(ρ L2 ) F pl2(ρ L2,W L), p(ρ R2 ) F pr2(ρ R2,W R). (4.72) For both configurations described in section and section the system of equations F 1 (ρ L1, ρ R1, ρ L2, ρ C2, ρ R2 ) F 2 (ρ L1, ρ R1, ρ L2, ρ C2, ρ R2 ) F = F 3 (ρ L1, ρ R1, ρ L2, ρ C2, ρ R2 ) = 0, (4.73) F 4 (ρ L1, ρ R1, ρ L2, ρ C2, ρ R2 ) F 5 (ρ L1, ρ R1, ρ L2, ρ C2, ρ R2 ) is solved by an iterative procedure for the densities in the star zone. In our experience the normal Newton-Rhapson method does not guarantee convergence, even if we consider the discontinuities of functions P RH introduced in (4.63) and (4.64), and when density is near zero. We solve (4.73) considering a globally convergent method, see Numerical Recipes [67]. After the system (4.73) is solved we can compute velocities and pressures in the star zone using relations (4.61) to (4.64), taking into account the presence of rarefaction or shock waves. After all unknowns in the star zone are obtained, velocities for all seven waves can be 58

75 4. Classical Riemann solvers obtained and the solution in the half-plane x t is complete, defining the complete wave structure. For numerical purpose we need to sample this wave structure and identify the state along the t axis. This state is used to compute the numerical flux in one of the numerical methods presented in Chapter 3. This Riemann solver can be used in Chapter 5 to solve the Derivative Riemann Problem. In Chapter 6 is used to construct the solution to the Derivative Riemann Problem. Alternative solvers can be constructed. The following one is the EVILIN Riemann solver and avoid the solution of the original non-linear Riemann problem (4.56) EVILIN Riemann solver In this section we present the EVILIN Riemann solver for the Baer-Nunziato system. The EVILIN approach was presented by Toro [97] and the idea is to evolve the initial data by a simple scheme (predictor step) and then solve a linear Riemann problem with evolved data as initial condition (corrector step). EVILIN follows the framework of the MUSTA [96] approach to solve the classical Riemann problem, but in the corrector step includes upwind information. In Fig. 4.7 the initial data Q 0 Q i and Q 1 Q i+1 is evolved to Q L and Q R using a non conservative step of equations. Q 1 i + 2 λ 2 λ 3 λ 4 λ 6 τ λ 5 λ 1 λ 7 Q L Q R Q 0 Q 1 d Figure 4.7: EVILIN Riemann solver for the non conservative formulation. Q 0 and Q 1 are the initial data which are evolved to Q L and Q R, this is the predictor step. A linear Riemann solver in the corrector step is used to obtain Q i+ 1. The 2 complete procedure is performed in local space d τ. 59

76 4. Classical Riemann solvers The non-conservative step is defined as follows, Q 1 2 L = Q 0 Q 1 2 C = 1 2 (Q 0 + Q 1 ) 1 τ 2 d [F(Q 1 ) F(Q 0 )] 1 τ dã1 2 2 C [W(Q 1 ) W(Q 0 )] Ã 1 2 C = A ( 1 2 (Q 0 + Q 1 ) ) Q 1 2 R = Q 1 [ Q L = 1 2 (Q1 2 L + Q 1 2 C ) 1 τ 2 d Q R = 1 2 (Q1 2 C + Q 1 2 R ) 1 τ 2 d ( ) 1 Ã L = A 2 (Q1 2 L + Q 1 2 C ) ( ) 1 Ã R = A 2 (Q1 2 C + Q 1 2 R ) ] F(Q 1 2 C ) F(Q 1 2 L ) [ ] F(Q 1 2 R ) F(Q 1 2 C ) [ 1 τ 2 dãl 1 τ 2 dãr ] W(Q 1 2 C ) W(Q 1 2 L ) [ ] W(Q 1 2 R ) W(Q 1 2 C ) (4.74) (4.75) Once the initial data have been evolved to Q L and Q R using (4.74) and (4.75), a linear Riemann solver is used in order to find the sought intercell state Q i+1/2 solving the following classical linear Riemann problem. t W + ˆB x W = { 0, W L W L ( W(x, 0) = Q L ) if x < 0 W R W R ( Q R ) if x > 0 Q i+1/2 is used in the numerical scheme (3.31)-(3.32) for updating cell averages. (4.76) Stratified Riemann solver The stratified formulation for two-phase flows was reported in [11], whereby the two-phase Riemann problem is reduced to a set of single phase Riemann problems. These simpler problems obey the single-phase gas dynamic Euler system, for which more choices to solve the Riemann problem are available. Here we couple this formulation with the exact and EVILIN Riemann solvers. In general multifluid models are constructed from averaging techniques [50] where the fraction of the volume occupied by each fluid is known but not the spatial distribution. Applying the stratified hypothesis this spatial distribution is constructed based on the void fractions α k producing the configuration depicted in Fig. 4.8: two adjacent cells containing a mixture of two phases are transformed to the stratified representation by applying the 60

77 4. Classical Riemann solvers stratified hypothesis. In this new representation three fluid interfaces are produced at x i+1/2. Four interactions are possible depending on the void fraction in each cell, namely: fluid 1 - fluid 1, fluid 1 - fluid 2, fluid 2 - fluid 1 and fluid 2 - fluid 2. Each interaction has a weight ω l with l = 1, 2, 3, 4 defined as follow ω 1 = min{α 1i, α 1i+1 } ω 4 = min{α 2i, α 2i+1 } ω 2 = max{0, α 1i α 1i+1 } ω 3 = max{0, α 2i α 2i+1 }, (4.77) with the conditions ω 2 ω 3 = 0 and ω 1 +ω 2 +ω 3 +ω 4 = 1. With these weights it is possible α 1 α2 ω 4 α 2 0 ω 3 α 1 α 1 ω 1 x x 1 i i + x i+1 2 x Figure 4.8: Stratified approach: The upper figure shows two adjacent cells with two fluid mixtures. The bottom figure shows the representation of the data after applying the stratification hypothesis. to compute the intercell state where Q (l) i+ 1 2 Q i+ 1 2 = ω 1 Q (1) + ω i+ 1 2 Q (2) + ω i+ 1 3 Q (3) + ω i+ 1 4 Q (4), (4.78) i is the solution of the Riemann problem for a single phase Euler system. t Q + x F(Q) = 0 { QL Q k,i if x < x i+ 1 Q(x, 0) = 2 Q R Q k,i+1 if x > x i+ 1 2 (4.79) 61

78 4. Classical Riemann solvers The initial data Q k,i and Q k,i+1 with k = 1, 2 are computed for cell i as Q i = α 1 ρ 1 α 1 ρ 1 u 1 α 1 E 1 α 2 ρ 2 α 2 ρ 2 u 2 α 2 E 2 α 1 i Q 1,i = Q 2,i = ρ 1 ρ 1 u 1 E 1 ρ 2 ρ 2 u 2 E 2 i i (4.80) The Riemann problems (4.79) have a well defined solution. The velocity u between each pair of non-linear waves (the velocity of the contact wave) plays a very important role. We define the pair {k L, k R } with k L = 1, 2 and k R = 1, 2 for all combinations between fluids 1 or 2 into cell i(l) or i + 1(R). Following Table 4.1 the solution of the Riemann problem is correctly assigned to Q (l). i+ 1 2 After all Q (l) i+ 1 2 intercell state Q i+ 1 2 averages. k L k R l if u > if u < if u < if u > 0 Table 4.1: Assignment for intercell vectors Q (l) i+ 1 2 are correctly computed, we used (4.78) to obtain the seven-component. Q i+1/2 is used in the numerical scheme (3.31)-(3.32) for updating cell The three solvers presented in sections 4.3.3, and can be used to solve the classical non-linear Riemann problem. They also can be used to construct the solution of the Derivative Riemann Problem of Chapter 5. In the following section we present the solution of the linear Riemann problem for the Baer-Nunziato equations which is used in the corrector step of the EVILIN solver Linear Riemann solver Here we present the solution of the linear Riemann problem formulated in terms of the physical variables. This solution can be used in the corrector step in the EVILIN solver as 62

79 4. Classical Riemann solvers presented in section 4.3.4, and in the ADER framework for solving the high order terms, in the construction of the Derivative Riemann Problem solver. We solve the following linearized Riemann problem, PDE: t W + B x W = 0, x (, ), t > 0. IC: W(x, 0) = { W L if x < 0, W R if x > 0, (4.81) with W and B = B( W) as in (4.51), and W frozen at W = 1 2 (W L+W R ). The eigenvalues and right eigenvector of B were given in (4.52), (4.53) and (4.54). We define the jump = W R W L and solve the following linear system, 7 δ i r i =, (4.82) i=1 with r i being the right column eigenvectors (4.53) and (4.54) and δ i being the unknown wave strengths. The wave strengths are found to be, δ 1 = ρ 1 2 2a a 2 + (p 1 p 2 ) 7 1 2α 1 a 2 1 δ 2 = ρ 2 5 2a δ 3 = 4 6 a 2 2 δ 4 = 1 3 δ 5 = a 2 1 2a 2 2 ρ 2(u 1 u 2 ) 7 2α 2 (u 1 u 2 + a 2 ) (p 1 p 2 ) 7 α 1 a 2 1 ) 7 α 2 ( (u1 u 2 ) 2 a 2 2 δ 6 = ρ 2 5 2a a 2 ρ 2(u 1 u 2 ) 7 2α 2 2 (u 1 u 2 a 2 ) δ 7 = ρ 1 2 2a a 2 + (p 1 p 2 ) 7 1 2α 1 a 2 1 (4.83) The solution of the linear Riemann problem along the t axis is obtained from (2.35) as follows, W(0, t) = W L + λ i 0 δ i r i or W(0, t) = W R λ i > 0δ i r i. (4.84) 63

80 4. Classical Riemann solvers With the solvers presented on this section we can solve the classical Riemann problem. These solvers are used to obtain the intercell state between two neighbouring cells and, introduced in (3.31), update the cell average in time. They can also be used in the construction of the Derivative Riemann Problem solver, as is shown in Chapter Shallow water equations In this section we develop solutions to the classical Riemann problem for the shallow water equations Introduction The shallow water equations are a system of non-linear hyperbolic conservation laws that governs water flows with a free surface under the force of gravity. The principal feature of the shallow water equations comes from assuming the hydrostatic hypothesis over the general conservations principles. In addition, proper boundary conditions, constant density and negligible viscous stress are further simplifications. Typical applications of this type of equations are dam break modelling, flood waves in rivers, tides in oceans and tsunami waves. See [93] for a review of the system. See also [84] and [58] for general background. The main difficulties arise from the complex geometries involved and when wet/dry fronts are considered. Also variable topography and friction terms represent non trivial numerical issues. Another difficulty with shallow water equations arise when numerical schemes are unable to correctly balance fluxes and sources in the steady state. In the next section we present the solution of the Riemann problem following the description presented in the book of Toro [93]. The reference coordinate system is depicted in Fig The free surface has elevation z = b(x, y) + h(x, y, t), where b(x, y) is the prescribed space dependent bottom elevation, and h(x, y, t) is the unknown space-time dependent depth of water. We consider the free surface as the sum of the water depth and the bottom elevation. The zero hight reference level is typically located below the minimum of the bottom elevation. 64

81 4. Classical Riemann solvers z Free surface z=b(x,y)+h(x,y,t) h(x,y,t) y Bottom b(x,y) z=b(x,y) x Figure 4.9: Reference coordinate system for bed b(x, y) and free surface elevation b(x, y) + h(x, y, t) Equations and eigenstructure Considering the two dimensional shallow water equations with non-horizontal bottom elevation we have, t (h) + x (hu) + y (hv) = 0, t (hu) + x (hu gh2 ) + y (huv) = ghb x, t (hv) + x (huv) + y (hv gh2 ) = ghb y, (4.85) where h is the water depth, u and v are the particle velocity components in x and y directions respectively, g is the gravity force and b is the bottom elevation. In (4.85), b x and b y are the partial derivatives in x and y directions. System (4.85) can be written in vectorial form as follows, t Q + x F(Q) + y G(Q) = S(Q), (4.86) where Q = h hu, F(Q) = hu hu gh2, hv huv G(Q) = hv huv, S(Q) = 0 ghb x. (4.87) hv gh2 ghb y For solving the classical Riemann problem we consider the following augmented homo- 65

82 4. Classical Riemann solvers geneous one dimensional problem, t Q + x F(Q) = 0, (4.88) which written in quasi linear form for the physical variables reads, t W + B(W) x W = 0, (4.89) with W = h u v, B(W) = u h 0 g u u. (4.90) The eigenvalues of matrix B(W) are with respective right eigenvectors, and left eigenvectors, r 1 = λ 1 = u a, λ 2 = u, λ 3 = u + a, (4.91) h a 0, r 2 = 0 0 1, r 3 = h a 0, (4.92) l 1 = [a, h, 0], l 2 = [0, 0, 1], l 3 = [a, h, 0], (4.93) with a = a(h) = gh, the celerity. The shallow water equations are a hyperbolic system and is strictly hyperbolic if the depth is strictly positive h > 0. Studying the characteristic fields we observe that wave families associated to λ 1 and λ 3 are genuinely non-linear, while the wave family associated to λ 2 is linearly degenerate Non linear Riemann solver We want to solve the following classical Riemann problem, PDEs: t Q + x F(Q) { = 0, x (, ), t > 0, Q L = Q(W L ) if x < 0, IC: Q(x,0) = Q R = Q(W R ) if x > 0, (4.94) 66

83 4. Classical Riemann solvers with W L = h L u L, W R = h R u R, (4.95) v L v R the initial constant data. t λ 2 λ 1 λ 3 W* L W* R W L F L x=0 F R W R x Figure 4.10: Wave structure for the 2 dimensional shallow water equations. Functions F L and F R connect the outer states W L and W R (constant initial conditions) with the unknown constant states WL and W R (Star Zone). The unknown quantities in the star zone are given by the vectors W L = h L u L, WR = h R u R. (4.96) v L v R For constructing the exact Riemann solver, we use the same methodology as presented in the previous sections. Applying the GRIs we know that across the contact wave λ 2, dh 0 = du 0 = dv 1, (4.97) the depth h and the velocity u remain constant, that is, h L = h R = h and u L = u R = u. This condition suggests to find two expressions F L and F R, one on the left and one on the right of the contact wave associated to λ 2 = u, that connects the star zone with the initial data in terms of the depth h and the velocity u. In this form we can use the velocity u to obtain one non-linear algebraic equation that dependent only on h. These two functions are F L (h,w L ) and F R (h,w R ) and are defined depending on the type of non-linear waves that they cross, named rarefaction waves or shock waves. 67

84 4. Classical Riemann solvers The GRIs across λ = u ± a for rarefaction waves read, dh h = du ±a = dv 0. (4.98) From (4.98), is concluded that v is constant across rarefactions. The first equality of (4.98) gives Functions F L and F R for rarefaction waves read, u 2a = constant. (4.99) F L (h,w L ) u = u L + 2 (a L + a(h )), F R (h,w R ) u = u R 2 (a R a(h )). (4.100) For shock waves we use Rankine-Hugoniot conditions. Utilizing the transformation variable in order to obtain a stationary shock, as in section , the Rankine-Hugoniot conditions apply to the PDEs in (4.94). Here we only show the final result. For details see Chapter 3 in [93]. Functions F L and F R for shock waves read, ( F L (h,w L ) u = u L (h h L ) 2 g h +h L, ( ) F R (h,w R ) u = u R + (h 1 h R ) 2 g h +h R h h R. 1 h h L ) (4.101) Note that (4.100) and (4.101) express the velocity u in terms of the know data and the unknown depth h. The first part of the Riemann problem (4.94) is solved finding the roots of the following non-linear algebraic equation, F(h,W L,W R ) = F L (h,w L ) F R (h,w R ) = 0, (4.102) Functions F L (h,w L ) and F R (h,w R ) are selected considering the non-linear waves associated to λ 1 and λ 3 from (4.100) and (4.101). We select the non-linear waves, and therefore the functions, using the entropy condition (2.46), which apply to (4.94). The wave family associated to λ = u ± a is a rarefaction wave if h h k, while it generates a shock wave if h > h k, with k = L, R. Once the particular waves are selected, the non-linear equation (4.102) has as unknown the depth h and can be solved using an iterative procedure such as Newton-Raphson. For one non-linear equation this method is highly efficient, regarding that we identify proper limits, such as negative depth. 68

85 4. Classical Riemann solvers After solving (4.102) for h, the velocity u is obtained from (4.100) or (4.101). With this information we compute the wave speeds and the complete wave structure is defined. For numerical purpose we need to sample this wave structure and identify the state along the t axis. This state is used to compute the numerical flux in one of the numerical methods presented in Chapter Linear Riemann solver Here we present the solution of the linearized version of the Riemann problem (4.94). We start with equations (4.88) as follows, t Q + x F(Q) = 0. (4.103) Computing the Jacobian matrix we obtain the quasi-linear form, t Q + A(Q) x Q = 0. (4.104) We linearise (4.104) and pose the following linear Riemann problem, PDE: t Q + à x Q = 0, x (, ), t > 0. { Q L if x < 0, IC: Q(x, 0) = Q R if x > 0, (4.105) where à = A( Q) is frozen at Q = 1 2 (Q L + Q R ). We can solve (4.105) as in section 2.6. Recall that the eigenvalues are: λ 1 = u a, λ 2 = u, λ 3 = u + a. We define the jump = Q R Q L and solve the following linear system for the unknown wave strengths δ i, 3 δ i r i =, (4.106) i=1 where r i are the right column eigenvectors of Ã, r 1 = 1 u a v, r 2 = 0 0 1, r 3 = 1 u + a v. (4.107) 69

86 4. Classical Riemann solvers The solution of the linear problem (4.106) is δ 1 = (u + a) 1 2 2a, δ 2 = v 1 + 3, δ 3 = (u a) 1 2 2a The expected solution over the t axis is obtained from (2.35) as follows,. (4.108) Q(0, t) = Q L + λ i 0 δ i r i or Q(0, t) = Q R λ i > 0 δ i r i. (4.109) In this chapter we presented solutions of the classical Riemann problem where the initial data are constant values. We presented solutions of the linear and non-linear cases. Fundamental knowledge to solve these classical Riemann problems are the waves structure and the characteristic fields involved. These solutions can be used to evaluate the numerical flux between two neighbouring cells and therefore to construct Godunov-type numerical methods. In the next chapter these solvers are used to construct the solution of the Derivative Riemann Problem. The Chapter 5 is dedicated to the solution of the Derivative Riemann Problem (DRP). In this type of Riemann problems, the initial data are not constant, instead of this, they are polynomial vectors on both sides of the initial discontinuity. Solving the DRP allow us to construct ADER schemes of arbitrary accuracy in space and time. 70

87 5 The Derivative Riemann Problem 5.1 Introduction The classical Riemann problem is the Cauchy problem for a system of conservation laws, with initial condition consisting of two constant states separated by a discontinuity. The solution of this Riemann problem was first used by Godunov to construct his first-order upwind numerical scheme [40]. Methods to solve the classical Riemann problem are studied, for example, in [95]. In Chapter 4 of this thesis we studied methods for solving the classical Riemann problem for three systems. The so-called generalized Riemann problem, in which the initial condition consists of two polynomials of first degree (vectors) separated by a discontinuity at the interface, has been used to construct Godunov-type schemes of second order of accuracy [66], [102], [5], [21], [7], [65], [92], [95], [6]. The more general Cauchy problem with initial conditions consisting of two smooth functions separated by a discontinuity at the origin has also been studied in recent years; this is the theme of this chapter. We call this Cauchy problem the Derivative Riemann Problem, or DRP, to distinguish it from the terminology Generalized Riemann problem, mostly associated with the simpler problem with piece-wise linear initial conditions. Theoretical aspects regarding the DRP are found in [56], [57] and references therein. A method to solve the Derivative Riemann Problem was presented in [99], which is a generalization to arbitrary order of the MGRP approach communicated in [91], [92] for second-order schemes, to solve non-linear hyperbolic systems. The method of [99] is applicable to non-linear hyperbolic systems with source terms. The technique expresses the time-dependent solution at the interface as a power series expansion of order K. The leading term of the expansion is the solution of a classical, usually non-linear, Riemann problem. The determination of the higher order terms involves the Cauchy-Kowalewski method to express time derivatives in terms of functions whose arguments are spatial derivatives. In order to define the spatial derivatives one first constructs new evolution equations for these and then solves additional classical Riemann problems. The solutions of these classical Riemann problems define all spatial derivatives at the interface. The complete solution is then built up by evaluating the functions of spatial derivatives and 71

88 5. The Derivative Riemann Problem assembling the complete series. In this manner the method of solution of order K boils down to solving 1 classical non-linear Riemann problem for the leading term and K classical linear Riemann problems for spatial derivatives. An extended version of the method to deal with more general hyperbolic systems is presented in [101]. The present chapter is motivated by number of issues. First, it appears necessary to examine more closely the quality of the approximate solutions produced by the existing DRP solver of Toro and Titarev [99], for the case of non-linear systems. In addition, we have recently identified a class of problems for which this DRP solver may experience some difficulties. The problems in question include, locally, a stationary discontinuity, a shock wave or a contact wave, for which the DRP expansion of [99] may be non-unique, giving rise to a non-unique choice of intercell numerical flux. At the level of the firstorder scheme the choice is unique due to the Rankine-Hugoniot conditions that ensure the continuity of the flux; that is, the flux is the same whether it is taken from the left or from the right of the interface. For the high-order schemes the fluxes are different. We also present new DRP solvers and discuss their relative performance, at the local level, as well as a means to provide a numerical flux for high-order methods. The high-order method first proposed by Harten, Engquist, Osher and Chakravarthy [45], after a minor modification, may be interpreted in the frame of the ADER methods. That is, we could define an associated derivative Riemann problem with a corresponding method to solve it. We call the resulting method the HEOC solver. We also propose a new solver that is a modification of the Toro-Titarev solver. The main feature of this DRP solver is that the high-order terms are computed by solving linearized classical Riemann problems for the high-order time derivatives. This is motivated by the fact that for a linear system, all-order time derivatives obey the original system of PDEs. We note that for the case of a linear system with constant coefficients all three methods studied here coincide and their solution is identical to the exact solution of the DRP problem. 5.2 The Derivative Riemann Problem Here we first state the mathematical problem and then briefly review an existing semianalytical method to compute the solution at the interface as a function of time. 72

89 5. The Derivative Riemann Problem The Mathematical Problem The Derivative Riemann Problem is the initial-value problem PDEs: t Q + x F(Q) = S(Q), x (, ), t > 0, Q L (x) if x < 0, IC: Q(x,0) = Q R (x) if x > 0. (5.1) The partial differential equations (PDEs), with source terms, are assumed to be a general system of hyperbolic balance laws. The initial condition (IC) consists of two vectors Q L (x), Q R (x), the components of which are assumed to be smooth functions of x, with K continuous non-trivial spatial derivatives away from zero. We denote by DRP K the Cauchy problem (5.1). In the DRP 0 all first and higher-order spatial derivatives of the initial condition away from the origin vanish identically; this case corresponds to the classical piece-wise constant data Riemann problem, associated with the first-order Godunov scheme [40]. Similarly, in the DRP 1 all second and higher-order spatial derivatives of the initial condition for the DRP away from the origin vanish identically; this case corresponds to the piece-wise linear data Riemann problem, or the so-called generalized Riemann problem (GRP), associated with a second-order method of the Godunov type [66], [102], [5], [21], [7], [65], [92], [6]. q L q(x,0) t q R Q L Q R x=0 x x=0 x (a) (b) Figure 5.1: The classical Riemann problem for a typical 3 3 non-linear system. (a): Initial condition at t = 0 for a single component q of the vector of unknowns Q. (b): Structure of the solution on the x t plane. Fig. 5.1 depicts the classical Riemann problem DRP 0 for a typical 3 3 non-linear system for which it is assumed that the left wave is a rarefaction, the right wave is a shock and the middle wave is a contact. Fig. 5.1(a) shows the initial condition for a single component q of the vector of unknowns Q. Fig. 5.1(b) depicts the structure of the corresponding solution on the x t plane; characteristics curves are straight lines. Fig. 5.2 illustrates 73

90 5. The Derivative Riemann Problem q L (x) q(x,0) Q LR (τ) t q R (x) Q L (x) Q R (x) x=0 x x=0 x (a) (b) Figure 5.2: The Derivative Riemann Problem for a typical 3 3 non-linear system. (a): Initial condition at t = 0 for a single component q of the vector of unknowns Q. (b): Structure of the solution on the x t plane. the Derivative Riemann Problem DRP K ; Fig. 5.2(a) depicts the initial condition for a single component q and consists of two smooth functions (space dependent) separated by a discontinuity at the origin. Fig. 5.2(b) depicts the corresponding structure of the solution on the x t plane. Now characteristics are no longer straight lines. Compare Figs. 5.1 and 5.2. The aim of a DRP solver is to find the solution of (5.1) at the origin x = 0 and for t > 0, as a function of time and represented by Q LR (τ) in Fig. 5.2(b). Recall that for the classical Riemann problem the solution is self-similar, it depends on the ratio x/t and is constant at x = 0 (the interface) for t > 0. In many situations of interest one can find the solution everywhere in the half plane x (, ), t > 0, although for the purpose of computing a numerical flux, knowing the solution along the interface is sufficient. For the derivative Riemann problem DRP K, with K > 0, finding the solution in the half plane x (, ), t > 0, is a formidable task that is possible only in special cases. See [65] for the complete solution of the DRP 1 for the Euler equations for ideal gases. To construct high-order numerical methods of the ADER type [98] it is sufficient to find the solution Q LR (τ) at the interface position x = 0, as a function of time τ alone. Q LR (τ) will provide sufficient information to compute a numerical flux to construct a numerical scheme of (K +1)-th order of accuracy in both space and time. The corresponding intercell numerical flux, denoted by F LR, is the time-integral average F LR = 1 t t 0 F(Q LR (τ))dτ, (5.2) where t is the time step of the scheme. Numerical methods based on this framework were called ADER methods in [98]. Early versions of the approach were communicated in 74

91 5. The Derivative Riemann Problem [91], [92]. Note that the conventional case of piece-wise constant data reproduces the classical first-order upwind method of Godunov [40] A Known Method of Solution Here we briefly review the method proposed by Toro and Titarev [99], [101], whereby a semi-analytical solution of the Derivative Riemann Problem (5.1) is obtained. Their method, as in [5] for second order and in [56] for the general case, first expresses the sought solution Q LR (τ) at the interface x = 0 as the power series expansion in time where Q LR (τ) = Q(0, 0 + ) + K k=1 Q(0, 0 + ) = lim t 0 + Q(0, t). [ ] (k) τ k t Q(0, 0 + ) k!, (5.3) The solution contains the leading term Q(0, 0 + ) and higher-order terms, with coefficients determined by the time derivatives (k) t Q(0, 0 + ). The determination of all terms in the expansion includes the following steps: Step (I): The leading term. To compute the leading term one solves exactly or approximately the conventional Riemann problem PDEs: t Q + x F(Q) = 0, Q L (0 ) if x < 0, ICs: Q(x, 0) = Q R (0 + ) if x > 0, (5.4) with Q L (0 ) = lim x 0 Q L (x), Q R (0 + ) = lim x 0 + Q R (x). (5.5) The similarity solution of (5.4) is denoted by D (0) (x/t) and the leading term in (5.3) is Q(0, 0 + ) = D (0) (0). (5.6) Step (II): Higher order terms. There are three sub-steps here. 1. Time derivatives in terms of spatial derivatives: Use the Cauchy-Kowalewski method to express time derivatives in (5.3) in terms of functions of space deriva- 75

92 5. The Derivative Riemann Problem tives (k) t Q(x, t) = G (k) ( (0) x Q, (1) x Q,..., (k) x Q). (5.7) The source terms S(Q) in (5.1) are all included in the arguments of the functions G (k). The problem now is that of determining the arguments of G (k), namely the spatial derivatives at the interface. 2. Evolution equations for derivatives: Construct evolution equations for spatial derivatives t ( (k) x Q(x, t)) + A(Q) x ( (k) x Q(x, t)) = H (k), (5.8) where A(Q) is the Jacobian matrix of the PDEs in (5.1). 3. Riemann problems for spatial derivatives: Simplify (5.8) by neglecting the right-hand side terms H (k) and linearizing the evolution equations. Then pose classical, homogeneous linearized Riemann problems for spatial derivatives PDEs: ICs: t ( (k) x Q(x, t)) + A (0) LR x( (k) x Q(x, t)) = 0, x (k) Q(x,0) = (k) x Q L (0 ) if x < 0, (k) x Q R (0 + ) if x > 0, (5.9) with A (0) LR = A(Q(0, 0 +)). Solve these Riemann problems to obtain similarity solutions D (k) (x/t) and set (k) x Q(0, 0 + ) = D (k) (0). (5.10) Step (III): The solution. Form the solution as the power series expansion: with C 0 as in (5.6) and for k = 1,...,K. Q LR (τ) = C 0 + C 1 τ + C 2 τ C K τ K, (5.11) C k (k) t Q(0, 0 + ) = G(k) (D (0) (0),D (1) (0),...,D (k) (0)), (5.12) k! k! This solution technique for the Derivative Riemann Problem DRP K reduces the problem to that of solving K + 1 classical homogeneous Riemann problems, one (generally non- 76

93 5. The Derivative Riemann Problem linear) Riemann problem to compute the leading term and K linearized Riemann problems to determine the higher order terms. The leading term requires the availability of a Riemann solver, exact or approximate. The K linearized Riemann problems (5.9) for most well-known systems associated with the higher order terms can be solved analytically and no choice of a Riemann solver is necessary. Moreover, all of these linearized problems have the same eigenstructure, as the coefficient matrix is the same for all Riemann problems for derivatives. In principle, the technique can be applied to calculate the early-time solution of advectionreaction equations with piece-wise smooth initial conditions. One can set up a derivative Riemann problem at any desired position, taking care that at each point x = x d of discontinuity in the initial condition one sets a corresponding derivative Riemann problem centred at x d. The solution at each point x d, for a small time τ, can be used to check the results of numerical schemes. 5.3 Other Methods of Solution Here we study two alternative methods for solving the DRP (5.1). The first results from a re-interpretation of the high-order numerical method first proposed by Harten, Engquist, Osher and Chakravarthy [45]. Consequently we call this derivative Riemann solver, the HEOC solver. The second method we study results from a modification of both the Toro- Titarev solver [99] of section and the HEOC solver. We denote this method of solution as the CT solver Interaction of Power-Series Expansions Here we re-interpret the method proposed by Harten, Engquist, Osher and Chakravarthy [45] to compute numerical fluxes for their high-order methods, as a technique to provide an approximate solution to the derivative Riemann problem (5.1) at the interface x = 0, as a function of time. They proposed power series expansions in space and time for the solution in each control volume, or cell. There followed the application of the Cauchy-Kowalewski method to convert all time derivatives in the expansions to space derivatives, which in turn could be computed on the initial data. In our re-interpretation we include source terms in the equations and consider power series expansions in time on each side of the interface Q L (τ) = Q L (0 ) + K k=1 [ ] (k) τ k t Q(0, 0) k! (5.13) 77

94 5. The Derivative Riemann Problem and with and Q R (τ) = Q R (0 + ) + K k=1 [ ] (k) τ k t Q(0 +, 0) k!, (5.14) Q(0, 0) = lim x 0 Q(x,0) Q L (0 ) (5.15) Q(0 +, 0) = lim x 0 + Q(x,0) Q L (0 + ). (5.16) The Cauchy-Kowalewski method allows us to use the PDEs in (5.1) to express all time derivatives in (5.13), (5.14) as functions of space derivatives and of the source terms S(Q), namely (k) t Q(x, t) = G (k) ( (0) x Q, (1) x Q,..., (k) x Q). (5.17) These expressions are well defined to the left and right of the interface, given that the initial conditions in (5.1) are assumed to be smooth away from 0. We can also define the limiting values from left and right, at t = 0, of the spatial derivatives of the initial conditions Thus we have Q (k) L (0 d k ) lim x 0 dx kq L(x), (5.18) Q (k) R (0 d k +) lim x 0 + dx kq R(x). (5.19) (k) t Q(0, 0) = G (k) (Q (0) L (0 ),Q (1) L (0 ),...,Q (k) L (0 )), (5.20) and (k) t Q(0 +, 0) = G (k) (Q (0) R (0 +),Q (1) R (0 +),...,Q (k) R (0 +)). (5.21) We define the solution of the DRP (5.1) at the interface x = 0, at time t = τ as Q LR (τ) = D(τ, 0), (5.22) where now D(τ, x/(t τ)) is the similarity solution of the classical, homogeneous Riemann 78

95 5. The Derivative Riemann Problem t D(τ,0) Q L (τ) Q R (τ) t=τ L R x t=0 L R x=0 x Figure 5.3: Illustration of the HEOC Derivative Riemann Problem solver. The limiting values of the initial data from left and right (circles) are time evolved separately to any time τ (rhombuses). The desired solution results from solving the classical Riemann problem with these evolved states as data. problem PDEs: t Q + x F(Q) = 0, Q L (τ) if x < 0, ICs: Q(x, 0) = Q R (τ) if x > 0. (5.23) Note that here D(τ, x/(t τ)) depends on the parameter τ. We call this re-interpretation of the method proposed by Harten et al. [45] as a derivative Riemann solver, the HEOC solver. Fig. 5.3 gives an interpretation of the HEOC solution method for the DRP (5.1). At time t = 0 one performs a Taylor series expansion in time on the limiting values of the data left and right of the interface (circles). Upon the application of the Cauchy-Kowalewski method one evolves the data in time on each side of the interface to yield time-evolved states Q L (τ) and Q R (τ), at any chosen time t = τ (rhombuses in Fig. 5.3). These (constant) states at t = τ form the initial conditions for a classical Riemann problem, as depicted on the top part of Fig. 5.3 by the self-similar wave pattern. The sought solution is that given by (5.22), which is constant along the t-axis associated with the self-similar wave pattern. As the method applies to any time τ one has a time-dependent solution at the interface. We remark that, just as in the Toro-Titarev solver [99] reviewed in Sect , the 79

96 5. The Derivative Riemann Problem HEOC solution method as presented here applies to in-homogeneous non-linear conservation balance laws. The influence of the source term enters via the Cauchy-Kowalewski method, but note that at no point in the method it becomes necessary to solve Riemann problems, explicitly accounting for the influence of the source terms Interaction of Time Derivatives Another method of solution for the DRP (5.1) results from a modification of both the Harten et al. (HEOC) and the Toro-Titarev (TT) solvers. The sought solution at the interface is again expressed as in (5.3), with the leading term computed as in (5.6). This part is identical to the TT solver. To compute the higher order terms we solve timederivative Riemann problems, that is, for any index k > 0 we compute (k) t Q(0, 0) and Q(0 +, 0) as in (5.20) (left) and (5.21) (right). To find (k) Q(0, 0 + ) right at the interface (k) t we solve the classical linearized homogeneous Riemann problem PDEs: ICs: t ( (k) t Q(x, t)) + A (0) LR x( (k) t Q(x, t)) = 0, (k) t Q(x,0) = (k) t Q(0, 0) if x < 0, (k) t Q(0 +, 0) if x > 0. The similarity solution is denoted by T (k) (x/t) and the sought value is t (5.24) (k) t Q(0, 0 + ) = T (k) (0). (5.25) The final solution has the form (5.11) with C 0 as in (5.6) and for k = 1,...,K. C k (k) t Q(0, 0 + ) = T(k) (0), (5.26) k! k! Note the analogy between (5.9) and (5.24). Both are motivated by the fact that for a linear homogeneous system with constant coefficient matrix  all temporal and spatial partial derivatives of the vector of unknowns, if defined, obey the original system, namely t ( (k) t Q(x, t)) +  x( (k) t Q(x, t)) = 0, t ( (k) x Q(x, t)) +  x( (k) x Q(x, t)) = 0. (5.27) 80

97 5. The Derivative Riemann Problem Note also that for the DRP for the linear advection equation PDE: t q + λ x q = 0, x (, ), t > 0, q L (x) if x < 0, IC: q(x,0) = q R (x) if x > 0, (5.28) with λ a constant wave propagation speed, all the proposed solutions (TT, HEOC, CT) coincide with the exact solution and is given by with q LR (τ) = q L (0 ) + [ ] K k=1 ( 1) k λ k q (k) τ k L (0 ) k! q R (0 + ) + [ ] K k=1 ( 1) k λ k q (k) τ k R (0 +) k! if λ > 0, if λ < 0, (5.29) q (k) L (0 ) dk dx k q L(0 ), q (k) R (0 +) dk dx k q R(0 + ). (5.30) One can show that this is also true for a linear system with constant coefficients. For non-linear systems the theoretical justification of the proposed solution method remains an issue, as we shall point out in Chap. 6. Obviously, for the special case of piece-wise constant data, the classical Riemann problem for non-linear systems is reproduced by all the methods studied. All these DRP solvers can be used to evaluate accurately the surface integrals in the Finite Volume or in the Discontinuous Galerkin approach to compute numerical fluxes and thus define high order numerical methods. See Chapter 7. In the next chapter we assess the performance of the varies DRP solvers studied in this chapter. 81

98 5. The Derivative Riemann Problem 82

99 6 Assessment of the Derivative Riemann Problem solvers 6.1 Reference solution In this section we assess the performance of the Derivative Riemann Problem solvers studied in the previous chapter via a series of test problems for two different non-linear hyperbolic systems. As no exact solutions are known for the class of test problems of interest here, we obtain reference solutions by computing solutions numerically. To this end we use three numerical methods, the first-order Godunov method, the second-order MUSCL-Hancock (MH) method and the Random Choice Method (RCM) [36], all of them applied on a very fine mesh. We note that RCM has the unique property of being able to resolve the very-early time evolution of the solution in a way that no other method known to us can do. This is important, as the proposed DRP solvers are assessed in their domain of validity, namely for short times. For test problems involving an initial discontinuity, most methods will require a fairly large number of time steps to begin to gradually establish the structure of the true solution. Moreover, the early-time numerical results may exhibit large unphysical oscillations, even when monotone (for the scalar case) schemes are used. To illustrate this point we solve a simple shock-tube problem for the Euler equations in the domain [ 1, 1], with initial data ρ L = 1, u L = 3/4, p L = 1 for x < 0 and ρ R = 1/8, u R = 0, p R = 1/10 for x > 0. Fig. 6.1 shows the exact (full line) and numerical solutions (symbols) at time t = using the the Godunov first-order method (circles), the MUSCL-Hancock method (squares) and the RCM method (triangles). For all three numerical methods we use the exact Riemann solver. For the first two methods we use C cfl = 0.9 and for RCM we use C cfl = Fig. 6.1 shows only the region close to the position of the discontinuities at time t = 0. For the output time considered the Godunov and MUSCL-Hancock methods do only four time steps and RCM nine time steps. The first two methods are unable to resolve the wave structure correctly. RCM finds all intermediate states correctly. This property of RCM is useful to our purpose. 83

100 6. Assessment of the Derivative Riemann Problem solvers 1 1 Density (Kg/m^3) Pressure (Pa) Position x (m) Position x (m) Velocity u (m/s) Energy (J/Kg) Position x (m) Position x (m) Figure 6.1: Shock-tube test problem. Exact (full line) and numerical solutions (symbols) at time t = using the Godunov first-order method (circles), the MUSCL- Hancock method (squares) and the random choice method (triangles). Recall that the DRP solution is valid precisely at the interface x = 0, as a function of time. The numerical methods give the approximate solution in every cell of the mesh that discretizes the domain [ 1, 1]. For any mesh used one always has, at any time, one value (vector) immediately to the left of x = 0 and one immediately to the right of x = 0. To extract the sought reference solution at a given time we solve the classical Riemann problem for these two neighbouring states and pick up the solution right at the interface x = 0. This is the numerical solution that we compare with the DRP solutions. 84

101 6. Assessment of the Derivative Riemann Problem solvers 6.2 The Derivative Riemann Problem for the Euler equations In this section we assess the Derivative Riemann Problem solvers applied to the Euler equations of gas dynamics. The series of test problems includes a simple test (Test 1) with smooth initial condition throughout, no discontinuities in the data are present. A second test (Test 2) has no jump discontinuities in the state variables but admits discontinuities in derivatives at the interface. Other more demanding test problems are constructed from Test 2, by adding a discontinuity in pressure. Four new cases are thus generated by varying the strength of the initial pressure jump p = (p L p R )/p R at the interface, namely p = 0.01, p = 0.1, p = 1.0 and p = For these four cases with an initial jump discontinuity the reference numerical solution used is that obtained by the Random Choice Method, on a very fine mesh Test 1: smooth initial conditions The chosen initial conditions are smooth throughout, there are no jumps in state or derivatives at x = 0 namely ρ(x,0) = sin(πx 2 ) sin(5πx 2 ) u(x,0) = 1 2 (x 1 2 )4 p(x,0) = x 4 (6.1) In this particular case all three DRP solvers give, algebraically, the same solution. In Fig. 6.2 we present the solution of the DRP problem up to fifth order (DRP 4) for all three components of the vector Q = [ρ, ρu, E] [Q(1),Q(2),Q(3)], where ρ is density, u is particle velocity and E is total energy. As expected, by increasing the order, the DRP solution approximates the reference solution very well. The DRP 0 solution is constant in time and the DRP 1 solution is linear in time. We note that the approximation improves with the order, which is verified for all three components Q(1), Q(2) and Q(3). For Q(2) the DRP 1 solution is practically identical to the DRP 0 solution. This is correct in the sense that at the time τ = 0 + the slope of the reference solution is close to zero and the linear characteristic of the DRP 1 solution will not modify this slope. Table 6.1 shows the error in the L 2 norm at different times. The main feature of these errors is that as time decreases the error decreases and as the order of accuracy increases the error decreases, as expected. 85

102 6. Assessment of the Derivative Riemann Problem solvers Order t = t = t = t = DRP DRP DRP DRP DRP Table 6.1: Errors in L 2 norm for the vector Q for Test Test 2: Initial data with discontinuous derivatives Test 2 has piece-wise smooth initial conditions that are continuous at x = 0 but with discontinuous derivatives at x = 0, namely ρ L (x,0) = sin(π(x 0.3) 2 ) sin(5π(x 0.3) 2 ) u L (x,0) = 1 2 (x 4 5 ) p L (x,0) = (x 0.3) 4 ρ R (x,0) = sin(πx 2 ) sin(5πx 2 ) u R (x,0) = 1 2 (x 1 2 )4 p R (x,0) = x 4 (6.2) For this test problem all three DRP solvers (TT, CT and HEOC) agree quite well for the very early times but differ quite visibly for larger times. Figure 6.3 shows the fifthorder (DRP 4 ) solution for the three solvers, for each component of the vector Q. For the first and third components the solver CT is the most accurate, followed by HEOC. For the second component of Q, the TT solver gives better results. More comprehensive information about the relative merits of the three solvers is given in Tables 6.2 to 6.4, where errors measured in the L 2 norm are displayed. For time t = , the error of the DRP 4 solution for the TT solver is , for the HEOC solver is and for the CT solver is A general conclusion is that for all three solvers the error diminishes as the order increases, while the solution is more accurate for small times, and for which they all tend to agree Tests with discontinuous initial conditions The tests of this section are generated from the initial conditions (6.2) of Test 2 by adding a term in p L (x,0) and thus generating a jump p = (p L (0, 0) p R (0, 0))/p R (0, 0) in pressure at x = 0. We consider the four cases: p = 0.01, 0.1, 1.0, Results are shown in Figures 6.4 to Figure 6.7. Figure 6.4 displays results for p = 0.01, 86

103 6. Assessment of the Derivative Riemann Problem solvers Order t = t = t = t = DRP DRP DRP DRP DRP Table 6.2: Toro-Titarev solver. Errors for Test 2. Order t = t = t = t = DRP DRP DRP DRP DRP Table 6.3: HEOC solver: Errors for Test 2. Order t = t = t = t = DRP DRP DRP DRP DRP Table 6.4: Castro-Toro solver. Errors for Test 2. 87

104 6. Assessment of the Derivative Riemann Problem solvers 1.05 Q(1) Reference solution (MH) DRP 0 DRP 1 DRP 2 DRP 3 DRP Time (s) 0.20 Q(2) Reference solution (MH) DRP 0 DRP 1 DRP 2 DRP 3 DRP Time (s) Reference solution (MH) DRP 0 DRP 1 Q(3) DRP 2 DRP 3 DRP Time (s) Figure 6.2: Test 1: DRP solution for Q(1) = ρ, Q(2) = ρu and Q(3) = E. Thick line is the reference solution. with a small pressure jump; the DRP solution improves as the order increases, for all three solvers. In Fig. 6.5, for p = 0.10, the solution from the TT solver improves as the order increases. The situation is different for the CT and HEOC solvers, whose solutions cross the reference solution. Results for p = 1.00 are shown in Fig Here the Toro-Titarev solver seems to perform better but note that when the order increases to DRP 4, it misrepresents the curvature and thus crosses the reference solution. The solutions of the present solvers CT and HEOC show wrong initial slopes. As the order increases the curvature seems to approximate the curvature of the reference solution better, with the HEOC solution being 88

105 6. Assessment of the Derivative Riemann Problem solvers 1.20 Q(1) Reference solution (MH) TT 4 CT 4 HEOC Time (s) 0.80 Q(2) Reference solution (MH) TT 4 CT 4 HEOC Time (s) Q(3) Reference solution (MH) TT 4 CT 4 HEOC Time (s) Figure 6.3: Fifth order DRP solutions for Test 2, using TT, CT and HEOC for the three components of Q. 89

106 6. Assessment of the Derivative Riemann Problem solvers 0.20 Q(2) TT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) 0.20 Q(2) CT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) 0.20 Q(2) HEOC solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) Figure 6.4: DRP solution for TT, CT and HEOC solvers for p = Thick full line is the reference solution obtained with RCM method. closer to the reference solution than that of CT. For both the CT and HEOC solvers the second order solution crosses the reference solution. Figure 6.7 shows the DRP solutions for p = All three solvers give the wrong initial slope. Their failure to agree with the reference solution increases dramatically as the initial pressure jump becomes larger. Moreover, they fail to capture the initial slope and the behaviour of the reference solution. 90

107 6. Assessment of the Derivative Riemann Problem solvers 0.30 Q(2) TT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) Q(2) CT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) Q(2) HEOC solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) 0.04 Figure 6.5: DRP solution for TT, CT and HEOC solvers for p = Thick full line is the reference solution obtained with RCM method. 91

108 6. Assessment of the Derivative Riemann Problem solvers 1.02 Q(2) TT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) Q(2) CT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) Q(2) HEOC solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) 0.03 Figure 6.6: DRP solution for TT, CT and HEOC solvers for p = Thick full line is the reference solution obtained with RCM method. 92

109 6. Assessment of the Derivative Riemann Problem solvers 3.60 Q(2) TT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) Q(2) CT solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) Q(2) HEOC solver Reference solution (RCM) DRP 4 DRP 3 DRP 2 DRP Time (s) 0.01 Figure 6.7: DRP solution for TT, CT and HEOC solvers for p = Thick full line is the reference solution obtained with RCM method. 93

110 6. Assessment of the Derivative Riemann Problem solvers 6.3 The Derivative Riemann Problem for the Baer-Nunziato equations In this section we assess the Toro&Titarev DRP solver for the multiphase Baer-Nunziato equations (4.45). We present three test problems. The first one is an advection test for the void fraction, the second test considers discontinuities in the pressures and the void fraction, and the third test is more demanding where pressures have large discontinuities. All tests consider ideal gases. Tests 1 and 2 consider gases characterized with γ = 1.4 while test 3 considers two different fluids with γ 1 = 4.4 and γ 2 = 1.4. The reference solution for each test is obtained with the MUSCL-Hancock numerical method and the EVILIN Riemann solver presented in section on a very fine mesh. The computational domain is big enough such that the boundary conditions do not affect the solution at x = Test 1: Advection of the void fraction The chosen initial conditions for Test 1 have constant density, velocity and pressure for both fluids and a continuous distribution for the void fraction, namely ρ 1 (x,0) = 5 ρ 2 (x,0)= 1 u 1 (x,0) = 1 u 2 (x,0)= 1 p 1 (x,0) = 10 p 2 (x,0)= 10 α 1 (x,0) = x x x3 (6.3) Fig. 6.8 shows the solution of the Derivative Riemann problem for Test 1 at x = 0. The DRP solution approximates the reference solution as the order increases. Moreover, for the DRP 3, the solution looks very close to the reference one. Fig. 6.9 shows the error of the DRP solution respect to the reference solution. As the order of the approximation increases the error decreases as expected. This behaviour remains up to a certain time, after which the approximation looses accuracy. 94

111 6. Assessment of the Derivative Riemann Problem solvers void fraction Reference solution (MH) DRP 0 DRP 1 DRP 2 DRP Time (s) Figure 6.8: Test 1 for the DRP for the Baer-Nunziato equations. Solutions DRP k (k=0,1,2,3) for the void fraction are compared with the reference solution void fraction Error DRP 0 Error DRP 1 Error DRP 2 Error DRP Time (s) Figure 6.9: Test 1 for the DRP for the Baer-Nunziato equations. Errors DRP k (k=0,1,2,3) for the void fraction are compared with the reference solution. 95

112 6. Assessment of the Derivative Riemann Problem solvers Test 2: Mild jump in fluid-1 pressure Test 2 considers constant densities if 1 and 0.9 for fluid 1 and 2 respectively. Velocities for both fluids are set equal to 0. The pressures and void fractions are given by polynomials to the left (L) of x = 0 and to the right (R) of x = 0: p 1L = x x 2 p 1R = x x 2 p 2L = x + 0.9x 2 p 2R = x 0.9x 2 α 1L = x x 3 α 1R = x x 3 (6.4) Fig shows a comparison of the solution of the derivative Riemann problem at x = 0, as a function of time. Results are compared with the reference solution (broken line). Fig shows plots of the corresponding errors. It is seen that for short times our solution method provides accurate results, and these get better as the order of accuracy of the DRP increases. Fluid-1 pressure Reference solution (MH) DRP 0 DRP 1 DRP 2 DRP Time (s) Figure 6.10: Test 2 for the DRP for the Baer-Nunziato equations. Solutions DRP k (k=0,1,2,3) for the fluid-1 pressure are compared with the reference solution. 96

113 6. Assessment of the Derivative Riemann Problem solvers Errors in fluid-1 pressure 0.02 Error DRP 0 Error DRP 1 Error DRP 2 Error DRP Time (s) Figure 6.11: Test 2 for the DRP for the Baer-Nunziato equations. Errors DRP k (k=0,1,2,3) for the fluid-1 pressure are compared with the reference solution Test 3: Large jumps in pressures and densities. Test 3 considers discontinuous initial conditions for densities, pressures and the void fraction. Velocities for both fluids are set equal to 0. The initial conditions for densities, pressures and the void fraction are given by polynomials to the left (L) of x = 0 and to the right (R) of x = 0, namely ρ 1L = 10.0 ρ 1R = 1.0 p 1L = x + 1.1x 2 + x 3 p 1R = 5.0 ρ 2L = 1.0 ρ 2R = 1.0 p 1L = x + 1.1x 2 + x 3 p 2R = 5.0 α 1L = 0.8 α 1R = 0.2 (6.5) Fig shows the solution of the Derivative Riemann problem for Test 3. In this test, where the initial discontinuity has a large jump, the DRP solver finds difficulties to reduce the error with respect to the reference solution. The solution improves as the order is increases, however, the level of approximation does not improve visibly. Fig shows the error of the DRP solution respect to the reference solution. 97

114 6. Assessment of the Derivative Riemann Problem solvers Fluid-2 velocity Reference solution (MH) DRP 0 DRP 1 DRP 2 DRP Time (s) Figure 6.12: Test 3 for the DRP for the Baer-Nunziato equations. Solutions DRP k (k=0,1,2,3) for the fluid-2 velocity are compared with the reference solution. Errors in fluid-2 velocity 0.1 Error DRP 0 Error DRP 1 Error DRP 2 Error DRP Time (s) Figure 6.13: Test 3 for the DRP for the Baer-Nunziato equations. Errors DRP k (k=0,1,2,3) for the fluid-2 velocity are compared with the reference solution. 98

115 6. Assessment of the Derivative Riemann Problem solvers 6.4 Discussion of results Recall that the main purpose of solving the Derivative Riemann Problem (DRP) is to provide a time-dependent solution at each cell interface, from which a corresponding numerical flux can be found and used in the context of finite volume methods or discontinuous Galerkin finite element methods. However, before considering numerical methods we focus the discussion on the solution of the particular Cauchy problem, the DRP. There are a number of aspects of the solution procedure of the DRP that warrant a detailed discussion. One issue concerns the time evolution of the initial data, as done in the HEOC solver, and of the solution, as done in the TT and CT solvers. We have observed that even when the initial condition consists of physically admissible data, it is possible that the time evolution yields unphysical values, such as negative densities. This appears to be more crucial for the HEOC solver, because it could happen that, at a given time, the time evolved data contains unphysical values, which then the appropriate (classical) Riemann solver rejects, leading to a failure of the scheme. The TT and CT solvers appear to be less sensitive to this problem. These two solvers evolved in time the sought solution right at the interface. The corresponding time-dependent solution may still include unphysical values. However, since these are then only used in the numerical integration to obtain the flux it is possible that the scheme may continue to function. Stationary discontinuities in the solution of the DRP represent another situation where differences between the various solvers exist. The TT and CT solvers expand the solution at the interface starting from a leading term computed a time t = 0 + that dominates the evolution. In the presence of a stationary discontinuity at t = 0 + there are two possible choices for the leading term. For the first-order mode of the methods, it does not matter which of the two states is taken, as these satisfy the Rankine-Hugoniot conditions and therefore the respective fluxes are identical. For the higher-order version of the methods the situation is not clear. There will be two different time expansions, depending on which side is taken as the leading term. This non-uniqueness remains so even if the discontinuity moves for times t > 0. We have performed some numerical tests on the effect of choosing from the two available expansions. There is an observable numerical difference but, at least for the tests performed, it is very small and as time evolves it virtually vanishes. Still, this is an aspect of the TT and CT methods that would benefit from further investigations. On the other hand, the HEOC method is less sensitive to this problem. In particular, if the discontinuity positioned at the origin, at the early times, then moves as time increases. The the HEOC has the mechanism to capture this behaviour. In general, boundary conditions are a challenging problem in the context of high order 99

116 6. Assessment of the Derivative Riemann Problem solvers numerical methods. For example, for reflecting boundary conditions we solve an inverse Riemann problem, in the sense that we need to identify appropriate initial conditions such that the Riemann problem solution at the boundary reproduces what is physically sought, for example, zero velocity. To this purpose the HEOC solver appears more attractive than the TT and CT solvers, as it is very simple to create, at each time, in the time evolution process, the appropriate data states to match the desired condition. Some comments regarding computational cost are in order. The HEOC solver needs a robust Riemann solver for each time-integration point τ, within the time step 0 τ t. This can be time-consuming as a robust Riemann solver will in general be a non-linear Riemann solver. In addition, the HEOC solver requires the development of two series expansions, one on each side of the interface. The TT solver, on the other hand, requires a single expansion right at the interface. Moreover, in the TT and CT solvers, one uses a non-linear Riemann solver only once, in order to compute the leading term reliably. A rather surprising observation resulting from the present work is that, from the evidence available, all three DRP solvers are unable to resolve correctly the DRP problem for the case of non-linear systems with large jumps. This is different from the case of scalar nonlinear problems. In fact, it has been demonstrated that the TT solver [101] gives the correct solution for the non-linear Burgers equation, even with a non-linear source term, for initial jumps of any size. This property does not seem to carry to systems (non-linear). Recall that for linear systems all methods are equivalent and all give the correct solution. On the other hand, the available experience shows that the high order ADER schemes based on the solution of the Derivative Riemann Problem are capable of producing the theoretically expected orders of accuracy. Obviously, the corresponding convergence rate tests are performed for smooth solutions. However, even for smooth solutions the local reconstruction procedure will necessarily produce jumps at the interfaces. Obviously these jumps are small, and possibly within the range that allows the existing DRP solvers to resolve correctly. In this chapter approximations to the Derivative Riemann Problem have been presented for the Euler and Baer-Nunziato equations. These solutions provide a time-dependent function at each cell interface, from which a corresponding numerical flux can be found and used in the context of Finite Volume methods or Discontinuous Galerkin finite element methods. These functions are used to accurately evaluate the surface integrals in the numerical schemes presented in Chapter 3. In the next chapter we see how to construct ADER schemes on unstructured triangular meshes using the DRP solvers studied in Chapter 5 to accurately evaluate the numerical fluxes. 100

117 7 ADER schemes on triangular meshes 7.1 Introduction ADER approach was first introduced in 2001 by Toro, Millington and Nejad in [98] and the name comes from Arbitrary accuracy DErivatives Riemann problem method. The idea is to accurately evaluate the integrals in space and time that are used in the formulation of the numerical methods presented in Chapter 3. For this end we use the Cauchy-Kowaleswski method and the DRP solvers presented in Chapter 5. Technically, we construct a power series expansion of the unknown vector Q(x, t). Different approaches have been presented between which we can mention the state expansion in [87] and the flux expansion in [85] and [100]. Extensions and improvements of the ADER approach have been reported in [99], [87], [77], [85], [76], [100], [88], [54], [29], [28], amongst others. In order to accurately evaluate the integrals we use the following elements: A piece-wise polynomial spatial representation of the data over the triangular cells. In the case of ADER-DG the data are represented by a piece-wise polynomial. In the case of ADER-FV we need to do a non-oscillatory reconstruction from the cell averages. In both cases, non-oscillatory reconstructions near strong gradients are needed to prevent spurious oscillations. Here we use the procedure presented in section 3.4. In this chapter we assume that it is available a piece-wise polynomial spatial representation of the data up to order r. The Cauchy-Kowalewski method to solve initial value problems with continuous initial data. This method is used when space-time integrals have to be computed inside the triangular elements. The solution of the Derivative Riemann problem presented in Chapter 5 together with the Cauchy-Kowalewski procedure applied to the balance law in (3.4). Solving the DRP involves the solution of linear and non-linear classical Riemann problems presented on Chapter 4. The solution of the classical non-linear Riemann problem is used to obtain the leading term, while the solution of the linear Riemann problem is 101

118 7. ADER schemes on triangular meshes used to obtain the high order terms. We remark that these Riemann problems are augmented one dimensional problems and have to be solved in the direction normal to the edge of the triangular element. Taking as an example the balance law (3.4) and the Finite Volume numerical scheme presented in Chapter 3 we see the ADER approach on triangular meshes. The balance law reads, with t Q + x F(Q) + y G(Q) = S(Q), (7.1) Q = q 1. q p, F(Q) = f 1 (Q). f p (Q), G(Q) = The Finite Volume numerical methods reads, g 1 (Q). g p (Q), S(Q) = s 1 (Q). s p (Q). (7.2) Q n+1 m = Q n m 1 T m t n+1 t n T m H(Q(x, t)) ˆn dx dt t n+1 t n S(Q(x, t)) dx dt. (7.3) T m In order to update the cell averages on (7.3) we need to compute time, volume and surface integrals. In general these integrals can not be evaluated exactly, therefore we use an approximation by a quadrature rule of a desire order. Fig. 7.1 shows the element T m, the integration point x h and the function Q(x, t) at local time t = 0. In Fig. 7.1(a) is depicted the element T m where the integration point x h is inside the element, that is x h T m / T m, therefore the function Q(x h, 0) is continuous and all its spatial derivatives are defined. In this case we use the Cauchy-Kowalewski method to evaluate the time integrals. In Fig. 7.1(b) is depicted the j-th edge of the element T m and the corresponding neighbouring element T j. The integration point x h is on the boundary, that is x h T m, and the function Q(x h, 0) and all its spatial derivatives jump discontinuously. In this case we solve a Derivative Riemann Problem, as in Chapter 5, with the data from the element T m and the data from the neighbouring element T j, which are called for convention Q L (x h, 0) and Q R (x h, 0) respectively. 102

119 7. ADER schemes on triangular meshes Q y Q y Q R x h,0) x Q m (x,0) x Q L ( h,0) v 3 x h T v m 1 v 2 T m x h T j (a) (b) Figure 7.1: Triangular element T m and integration point x h. (a): The integration point x h is inside the element T m. (b): The integration point x h is on the j-th boundary of the element T m. 7.2 Evaluation of the volume integral In order to compute the space-time integrals in (7.3) with high-order accuracy we make use of the Cauchy-Kowalewski method to solve the following initial value problem, PDEs: t Q + x F(Q) + y G(Q) = S(Q), x T m, t > 0, IC: Q(x,0) = Q(x h, 0) The approximate solution of (7.4) reads k=1 (7.4) r 1 [ ] Q(x h, τ) = Q(x h, 0) + (k) τ k t Q(x h, 0) k!. (7.5) Solution (7.5) is a power series expansion around the leading term Q(x h, t). This term is evaluated from the piece-wise polynomial spatial representation of the data. The high-order terms or derivative terms (k) t Q(x h, 0) are time derivatives obtained from the Cauchy- Kowalewski procedure and the balance law. Applying the Cauchy-Kowalewski procedure 103

120 7. ADER schemes on triangular meshes to (7.1) the time derivatives are expressed as follows, t Q = ( ) F x Q Q ( 2 ) F tx Q = Q 2 ( x Q) 2 ( ) S + ( x Q), Q ( ) G y Q + S, Q ( 2 ) F ty Q = Q 2 ( x Q)( y Q) ( ) S + ( y Q), Q ( 2 ) F tt Q = Q 2 ( t Q)( x Q) ( ) S + ( t Q). Q ( ) F xx Q Q ( ) F xy Q Q ( ) F tx Q Q ( 2 ) G Q 2 ( x Q)( y Q) ( 2 ) G Q 2 ( y Q) 2 ( 2 ) G Q 2 ( t Q)( y Q) ( ) G xy Q Q ( ) G yy Q Q ( ) G ty Q Q (7.6) The first two time derivatives of the unknown vector Q(x, t) are reported in (7.6). The same procedure is applied to obtain third and higher order time derivatives. This procedure can be computed beforehand using an algebra package, for example MAPLE. The Cauchy-Kowalewski procedure expresses time derivatives of Q(x, t) of order r in terms of the fluxes F(Q) and G(Q), the source S(Q) and the spatial derivatives of order equal and lower than r. These spatial derivatives are obtained from the piece-wise polynomial spatial representation of the data. Constructing (7.5) over the spatial integration points inside the triangle, see Fig. 7.1(a), and evaluating it at the temporal integration points, we compute the volume-time integrals in (7.3) with the desire order of accuracy. This can be see in the results presented in Chapter 6 for test problem 1 for the Euler equation, where the initial conditions of the DRP are continuous and smooth. 104

121 7. ADER schemes on triangular meshes 7.3 Evaluation of the surface integral When we need to compute the surface-time integrals in (7.3), t n+1 t n T m H(Q(x, t)) ˆn dx dt, (7.7) is necessary to define the value of the function Q(x h, 0) and its derivatives, with x h T m, see Fig. 7.1(b). This is solved by applying one of the DRP solvers presented in Chapter 5 in the direction normal to the interface. This is done placing the following augmented one dimensional Derivative Riemann Problem, PDEs: t Q + n F(Q) = S(Q), x (, ), t > 0, Q L (x h, 0) if x T m, IC: Q(x,0) = Q R (x h, 0) if x T j. (7.8) where the partial derivative n is the derivative in direction normal to the interface. The initial values are obtained from the piece-wise polynomial data Q L (x h, 0) and Q R (x h, 0) rotated in the direction normal to the interface as follows, Q L (x h, 0) = Q R (x h, 0) = lim x x h x T m Q m (x, t n ) ˆn, lim x x h x T j Q j (x, t n ) ˆn, (7.9) and all their spatial derivatives x (α,β) Q L (x h, 0) and x (α,β) Q R (x h, 0), x (α,β) Q L (x h, 0) = lim (α+β) Q x x h x α y β m (x, t n ) ˆn, x T m x (α,β) Q R (x h, 0) = lim (α+β) Q x x h x α y β j (x, t n ) ˆn, x T j (7.10) from the piece-wise polynomial representation of the data. Q m (x, t n ) is the piece-wise polynomial data on triangle T m and Q j (x, t n ) is the piece-wise polynomial data on triangle T j. Q L and Q R are called the extrapolated values of the polynomials functions. The integers α 0 and β 0. The spatial derivatives are used to obtain the high-order terms. The solution of the DRP (7.8) is a power series expansion constructed over the spatial 105

122 7. ADER schemes on triangular meshes integration point x h T m as follows, r 1 [ ] Q(x h, τ) = Q(x h, 0) + (k) τ k t Q(x h, 0) k!. (7.11) Then, the evaluation of the surface-time integral (7.7) is obtained with a quadrature rule of the desire order. In Chapter 5 we present three forms of solving (7.8) named TT-DRP, CT-DRP and HEOC-DRP. k=1 t y x τ=t n+1 τ=t n x h F(x( h,τ k ) n T j y x v 3 x h n T j T m τ k T m v 1 v2 (a) (b) Figure 7.2: Surface-time integration over the j-th edge of the normal flux. (a): the flux function is evaluated at (x h, τ k ) for the surface-time integral. (b): outward normal vector to the j-th edge over the spatial integration point x h. The spatial integration point x h over the edge of T m is shown in Fig Fig. 7.2(a) depicts the space-time integration domain over the j-th edge. The numerical flux normal to the edge is evaluated at every space-time Gaussian integration point. Fig. 7.2(b) depicts the outward unitary normal vector to the j-th edge and the spatial integration point x h where the power series expansion (7.5) has to be constructed. When we implement one of the DRP solvers over triangular meshes we have to take into account that every Riemann problem, linear and non-linear, need to be solve with the left and right data rotated in the normal direction. Moreover, the computation of the numerical flux has to be done in the normal direction. Nevertheless, the Cauchy- Kowalewski procedure is presented on the x y coordinates. These two aspects make necessary to rotate and rotate back the vector function Q and the numerical flux F(Q). This is possible because of the rotational invariance property. The rotation is necessary because the velocity components of the vector Q, u and v, are defined as x and y particle velocities respectively. 106

123 7. ADER schemes on triangular meshes 7.4 DRP over triangular meshes Here we summarize the fundamental steps used to compute the numerical flux considering the extrapolated values and all their extrapolated derivatives defined in (7.9) and (7.10) respectively ADER TT-DRP scheme over triangular meshes These are the steps needed to implement ADER TT-DRP. The extrapolated values are rotate in the direction normal to the edge. With the rotated extrapolated values a non-linear Riemann problem is placed as (5.4). The solution of this non-linear problem is the leading term. Using the leading term the evolution equation for the extrapolated spatial derivatives (5.8) is linearized. These linear Riemann problems are solved to find the spatial derivatives at the interface. The leading term and the spatial derivatives at the interface are rotated back and the Cauchy-Kowalewski procedure is used to obtain the time derivatives. Now we construct the power series expansion in time (5.3). This time expansion is evaluated at the desire time integration point, rotated in the direction normal to the edge. With this state the numerical flux normal to the edge is evaluated. The numerical flux is rotated back to the x y coordinate system, obtaining H(Q(x, t)) ˆn in (7.7) ADER CT-DRP scheme over triangular meshes These are the steps needed to implement ADER CT-DRP. Two set of time derivatives are obtained using the Cauchy-Kowalewski procedure applied to the extrapolated values and extrapolated derivatives from the left and from the right as in (5.20) and (5.21). The two set of time derivatives are rotated in the direction normal to the edge. The rotated values define a non-linear Riemann problem as in (5.4). The solution of this non-linear problem is the leading term. Time derivatives at the interface are computed from the solution of linear Riemann problems as (5.24) with initial data from the rotated time derivatives and linearized on the leading term. We construct the power series expansion in time (5.3) with the leading term and the time derivatives at the interface. The numerical flux normal to the edge is computed from the evaluation of the time expansion. The numerical flux is rotated back to the x y coordinate system, obtaining H(Q(x, t)) ˆn in (7.7). 107

124 7. ADER schemes on triangular meshes ADER HEOC-DRP scheme over triangular meshes These are the steps needed to implement ADER HEOC-DRP. Two set of time derivatives are obtained using the Cauchy-Kowalewski procedure applied to the extrapolated values and extrapolated derivatives from the left and from the right as in (5.20) and (5.21). Two power series expansion in time are constructed with the extrapolated values and the time derivatives as in (5.13) and (5.14). We evaluate the time expansions at any desire time integration point defining the left and the right evolved states. These states are rotated in the direction normal to the edge and a non-linear Riemann problem is set as (5.23). We compute the numerical flux normal to the edge with the solution of the non-linear problem. The numerical flux is rotated back to the x y coordinate system, obtaining H(Q(x, t)) ˆn in (7.7). The ADER approach presented in this Chapter for the Finite Volume method is equivalent to the used on Discontinuous Galerkin Finite Element method. In the next Chapter we present numerical results that validate the numerical method presented in this thesis. 108

125 8 Numerical results Here we show numerical results for the numerical methods presented in this thesis. Three hyperbolic equations are used: Euler, Baer-Nunziato and shallow water. Two dimensional convergence test are presented for the Euler and shallow water equations using ADER schemes. A one dimensional convergence test is presented for the Baer-Nunziatio equations using MUSCL_Hancock reconstruction. Also shock tube problems are presented. Full colour version of the thesis is available on-line at: Euler equations Two dimensional convergence test: Vortex evolution For the purpose of studying the convergence rates of the schemes we adopt the test problem proposed in [49], which consists of a convected isentropic vortex computed in a square domain with periodic boundary conditions. The initial condition consists of a mean constant flow modified by an isentropic perturbation. The initial mean flow is given by ρ = 1, p = 1 and (u, v) = (1, 1) and the perturbation is given by δu = ǫ 2π e1 2 (1 r2)y, δv = ǫ 2π e1 2 (1 r2)x, δρ = (1 + δt) 1 γ 1 1, δp = (1 + δt) γ γ 1 1, (γ 1)ǫ2 δt = 8γπ 2 e 1 r2, (8.1) where r 2 = x 2 + y 2, ǫ = 5 (the vortex strength) and γ in the ideal EOS is taken as γ = 1.4. The computational domain is [ 5, 5] [ 5, 5] discretized by an unstructured mesh of triangles. Tables 8.1 to 8.4 give errors and convergence rates for the FV ADER schemes using the CT derivative Riemann problem solver. Schemes up to fifth order of accuracy are considered, on four levels of mesh refinement. Errors are measured in three norms L 1, L 2, 109

126 8. Numerical results L and the corresponding empirical orders of accuracy are O 1, O 2 and O. The expected orders of accuracy are reached in all cases. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.1: Convergence rates test: second order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.2: Convergence rates test: third order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.3: Convergence rates test: fourth order method Shock wave reflection problem The purpose of this test is to illustrate the potential of the presented methods to solve realistic problems to high accuracy on complicated domains discretized with unstructured meshes. To this end we consider the reflection of a shock wave from a solid body of triangular shape. The two-dimensional computational domain is the region [ 0.65, 0.5] [ 0.5, 0.5], with a triangular solid body defined by the positions of its vertexes v 1 = ( 0.2, 0), v 2 = (0.1, 1/6) and v 3 = (0.1, 1/6). The incident shock wave has shock Mach number Ms = 1.3 and at time t = 0 is placed at x = 0.55, with initial conditions 110

127 8. Numerical results Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.4: Convergence rates test: fifth order method. ahead of the shock given by ρ = 1.225(Kg/m 3 ), p = (Pa) and zero velocity. Conditions behind are calculated from the Rankine-Hugoniot conditions. The mesh consist of triangles. Boundary conditions are: left boundary at x = 0.65 is defined as inflow condition with the corresponding state defined by the Rankine- Hugoniot conditions; at the right boundary at x = 0.5 we set an outflow condition. The remaining boundaries are solid reflecting boundaries. For the shown results we used the FV third order ADER scheme along with the CT solver for the Derivative Riemann Problem and Courant number C cfl = Figures 8.1 to 8.4 display Schlieren images for the density at times t = , t = , t = and t = respectively. The shock wave propagates to the right, reflects from the solid triangle and generates a circular reflection shock. The two expansion waves created by the vertexes v 2 and v 3 produce regions of low density and pressure. See Fig. 8.5 for a three dimensional image of the density at time t = The main physical features of the flow looks reasonable, as compared with analogous problems for which there are experimental results, see [75]. 111

128 8. Numerical results Figure 8.1: Shock wave reflection problem. Schlieren image for density at time t = The shock wave begins the interaction with the triangle. Figure 8.2: Shock wave reflection problem. Schlieren image for density at time t = The shock wave reflects from the edges of the triangle and generates a single Mach reflection pattern. 112

129 8. Numerical results Figure 8.3: Shock wave reflection problem. Schlieren image for density at time t = The shock wave generates two expansive waves over the corners of the triangle. A slip surface is produced from the interaction of the original shock wave and the reflected one. Figure 8.4: Shock wave reflection problem. Schlieren image for density at time t = Two vortexes evolve behind the triangle. The expansion waves interact with the shock and with the boundaries. 113

130 8. Numerical results Figure 8.5: Shock wave reflection problem. Schlieren image for density at time t = Three dimensional image for the density. 114

131 8. Numerical results CPU cost of Riemann solvers When we develops numerical methods, one of the goals is to reduce the cost in terms of CPU time. However, a high degree of optimization of the code reduces the range of problems that can solves. When we develop high-order numerical methods in the framework of ADER schemes, the high-order reconstruction of the data and the Cauchy-Kovalewski procedure consume much of the total CPU time. In this case, the use of an exact non-linear Riemann solver may not represent a big disadvantage in term of computational cost. Here we compare the computational time used to solve an initial value problem. In one case we use the exact general-eos Riemann solver presented in Chapter 4. In the other case, the very efficient one developed for the ideal EOS, see Chapter 4 of [95]. We solve the one-dimensional Euler equation with ideal equation of state in a spatial domain [0, 1]. The initial data consist of two constant states on both sides of the initial discontinuity at x = 0.3. For the left state we have ρ = 1.0, u = 0.75 and p = 1.0. For the right state we have ρ = 0.125, u = 0.0 and p = 0.1. We discretize the domain with 500 cells and consider C cfl = 0.9. The final time is t = 0.2. Using the FV numerical method and the CT DRP solver we obtain the followings computational time. Ideal-EOS RS General-EOS RS Order CPU time % 100 CPU time % Table 8.5: CPU computational cost. We measure the total time used to solve an initial value problem considering FV numerical method with the CT DRP solver up to fifth order, the Ideal-EOS Riemann solver and the general-eos Riemann solver. From table 8.5 we see that the computational cost increases with the order of the method and when we use the general-eos Riemann solver. However, the relative cost decreases as the order of the method increases. For the first order method we see that the general-eos Riemann solver is more than twice time expensive than the ideal-eos Riemann solver. When the solution is computed with the fifth order numerical method this gap reduces to The fact that the type of high-order numerical methods studied in this thesis introduce reconstruction and evaluation of the Cauchy-Kowalewski procedure, produces a relatively small impact in the full computational time of the Riemann solver. 115

132 8. Numerical results 8.2 Baer-Nunziato In this section we present two test problems to assess the numerical solutions. The first test is a shock tube problem for liquid and gas, while the second one is a convergence test. For each test three numerical solutions are shown, EVILIN, StraEv and StraEx. EVILIN comes from applying the EVILIN Riemann solver to the full seven equation system. StraEv comes from applying the stratification hypothesis plus EVILIN Riemann solver for the interaction between identical fluids and the exact Riemann solver for different fluids. Finally, StraEx consists of the stratification hypothesis plus the exact Riemann solver for all interactions. For test 1, first and second order numerical solutions are presented with two meshes, 100 and 800 cells. For test 2 a smooth initial condition is set and evolved until t = 0.1. The second order scheme used in this test problems is the MUSCL_Hancock approach [108]. Data reconstruction is performed using piece wise linear functions. Boundary extrapolated values are evolved by half a time step and then used as initial data for a piece-wise constant data Riemann problem. These Riemann problems are solved using the Riemann solvers presented in Chapter 4. Non oscillatory properties come from TVD slope limiters applied to the data reconstruction step. See [95] for more details Test 1: Shock tube problem for liquid and gas This test problem generates the interaction between one liquid and one gas with the following constants for the stiffened equation of state: γ 1 = 4.4, p o 1 = , γ 2 = 1.4 and p o 2 = 0.0. The initial discontinuity is at x = 0.5, the CFL coefficient is C cfl = 0.5 and the output time is t = s. The wave structure of the solution is shown in figure 8.6. We observe two left-travelling rarefaction waves, one for each fluid, and two right-travelling shock waves, one for each fluid. All three contact waves travel slowly to the right and are indistinguishable from each other. Initial conditions are given in table 8.6. ρ 1 u 1 p 1 α 1 ρ 2 u 2 p 2 W L W R Table 8.6: Initial conditions for shock tube problem for liquid and gas Numerical results are presented in figures 8.7 and 8.8. In each figure first and second order numerical methods are presented, with the first order method shown on top. For the results shown in figures 8.7 and 8.8 we used 100 and 800 cells, respectively. The figures show the density of fluid 1 and the velocity of fluid 2. The exact solution obtained with the exact Riemann solver of section is shown by the black line. It is observed that 116

133 8. Numerical results t Red: Fluid 1 Blue: Fluid 2 Figure 8.6: Wave pattern for test 1 with two left rarefactions, 2 right shocks and 3 contact waves. x results may be improved by going from a first order to a second order scheme or by refining the mesh from 100 to 800 cells. The EVILIN approach is more accurate in the presence of shocks and rarefactions. All three methods produce spurious oscillations near the contact waves, which tend to disappear with mesh refinement. A special comment on slowly moving contacts is in order: these waves are badly smeared by non-complete Riemann solvers, whereas our method recognizes all of them and resolves them properly. As mentioned above, computational cost can be a limitation when exact Riemann solvers are used. The exact solution presented in section is computed iteratively in two steps. In table 8.7 we compare the computational cost of the exact, EVILIN, StraEv and StraEx Riemann solvers. It is clear that the exact solver is more expensive than the others. In the first order code the exact solver is 3.1 times more expensive than EVILIN solver and 5.3 times than StraEx solver. In the second order code the relations change to 2.2 for the EVILIN solver and 2.9 for the StraEx solver because of the additional reconstruction procedure. An important aspect is that when the order of the method increases, the exact Riemann solver pays off. In our computations we observe that the cost increase is about 20 percent for the exact solver, while the others show an increase close to 50 percent. 117

134 8. Numerical results Fluid 1 Density (Kg/m 3 ) EVILIN StraEV StraEX Fluid 2 Velocity u (m/s) Position x (m) Position x (m) Fluid 1 Density (Kg/m 3 ) Fluid 2 Velocity u (m/s) Position x (m) Position x (m) Figure 8.7: Numerical results for Test 1: First order method on the top of the figure. Second order solution is below. 100 cell are used Test 2: Numerical convergence With this test the numerical convergence rate is tested where the initial condition generates a smooth solution with no discontinuities. An exact solution for this test does not exist, thus a numerical solution with very fine mesh is used in order to have a reference solution. The initial data was taken from Schwendeman et al. [78] and consists of constant density and pressure for both fluids, constant velocity for fluid 2 and smooth transition for void fraction and for the velocity of fluid 1, as follows. ρ 1 (x,0) = ρ 2 (x,0) = 1.0, p 1 (x,0) = p 2 (x,0) = 1.0, v 2 (x,0) = 0.0, (8.2) v 1 (x,0) = tanh(20x 10), α 1(x,0) = tanh(20x 8) (8.3) 5 Parameters for the equation of state are γ 1 = 4.4, p o 1 = 0.0, γ 2 = 1.4 and p o 2 = 0.0. The computational domain is [0, 1] for times t [0, 0.1]. Transmissive boundary conditions are 118

135 8. Numerical results Fluid 1 Density (Kg/m 3 ) EVILIN StraEV StraEX Fluid 2 Velocity u (m/s) Position x (m) Position x (m) Fluid 1 Density (Kg/m 3 ) Fluid 2 Velocity u (m/s) Position x (m) Position x (m) Figure 8.8: Numerical results for Test 1: First order method on the top of the figure. Second order solution is below. 800 cell are used. used and four mesh sizes are employed: 100, 200, 400 and 800 cells. The convergence rate and errors are computed using L 1, L 2 and L norms. Convergence rates for the first order scheme are not shown but the expected order is reached. For second order schemes using TVD limiters the order of convergence is around 1, as expected. If no limiter is used and MUSCL reconstruction is performed, the order 2 is reached in all norms and for all three numerical schemes, see table 8.8, 8.9 and

136 8. Numerical results Method 1 st cpu time (s) ratio 2 nd cpu time (s) ratio Exact EVILIN StraEv StraEx Table 8.7: Computational cost for numerical methods normalized to the exact Riemann solver for Test 1 with 800 cells and first and second order. L 1 L 2 L Cells Error r Error r Error r Table 8.8: L 1, L 2 and L inf norm for EVILIN method with MUSCL_Hancock reconstruction. L 1 L 2 L Cells Error r Error r Error r Table 8.9: L 1, L 2 and L inf norm for StraEv method with MUSCL_Hancock reconstruction. L 1 L 2 L Cells Error r Error r Error r Table 8.10: L 1, L 2 and L inf norm for StraEx method with MUSCL_Hancock reconstruction. 120

137 8. Numerical results 8.3 Shallow water Convergence test We do a convergence test for studying the convergence order of ADER schemes using DRP solvers on unstructured grids. In this test we compare the numerical solution with the exact analytical solution. Because only very simple test problems have analytical solution we propose the construction of the test starting with the exact solution and then finding a new source term that balances the balance law. with The exact solution Q(x, y, t) reads, Q(x, y, t) = h(x, y, t) h(x, y, t)u(x, y, t) h(x, y, t)v(x, y, t), (8.4) b(x, y) = exp( 8(x 2 + y 2 ))/5, h(x, y, t) = exp(t/10) b(x, y), u(x, y, t) = (1 + sin(xπ))/10, v(x, y, t) = (1 + sin(yπ))/10, (8.5) Introducing this exact solution in the balance law (4.86) we obtain the new source term, S(x, y, t) = t Q + x F( Q) + y G( Q) S( Q). (8.6) The convergence test problem is then defined as follows, PDEs: t Q + x F(Q) + y G(Q) = S(Q) + S(Q), IC: Q(x, y,0) = Q(x, y,0), } (8.7) The computational domain is [ 1, 1] [ 1, 1] discretized by an unstructured mesh of triangles with periodic boundary conditions, and the final time considered is t = 1. Schemes up to fifth order of accuracy are considered on two levels of mesh refinement. Errors are measured in three norms L 1, L 2, L and the corresponding empirical orders of accuracy are O 1, O 2 and O. Tables 8.11 to 8.14 give errors and convergence rates for the FV ADER scheme using the TT derivative Riemann problem solver. Tables 8.15 to 8.18 give errors and convergence rates for the FV ADER scheme using the CT derivative Riemann problem solver. Tables 8.19 to 8.22 give errors and convergence rates for the FV ADER scheme using the HEOC 121

138 8. Numerical results derivative Riemann problem solver. Tables 8.23 to 8.26 give errors and convergence rates for the DG ADER scheme using the CT derivative Riemann problem solver. The expected orders of accuracy are reached in all cases. Numerical Method: FV ADER with the TT DRP solver Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.11: Shallow water convergence rates test: FV TT-DRP second order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.12: Shallow water convergence rates test: FV TT-DRP third order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.13: Shallow water convergence rates test: FV TT-DRP fourth order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.14: Shallow water convergence rates test: FV TT-DRP fifth order method. 122

139 8. Numerical results Numerical Method: FV ADER with the CT DRP solver Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.15: Shallow water convergence rates test: FV CT-DRP second order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.16: Shallow water convergence rates test: FV CT-DRP third order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.17: Shallow water convergence rates test: FV CT-DRP fourth order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.18: Shallow water convergence rates test: FV CT-DRP fifth order method. 123

140 8. Numerical results Numerical Method: FV ADER with the HEOC DRP solver Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.19: Shallow water convergence rates test: FV HEOC-DRP second order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.20: Shallow water convergence rates test: FV HEOC-DRP third order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.21: Shallow water convergence rates test: FV HEOC-DRP fourth order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.22: Shallow water convergence rates test: FV HEOC-DRP fifth order method. 124

141 8. Numerical results Numerical Method: DG ADER with the CT DRP solver Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.23: Shallow water convergence rates test: DG CT-DRP second order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.24: Shallow water convergence rates test: DG CT-DRP third order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.25: Shallow water convergence rates test: DG CT-DRP fourth order method. Mesh L 1 error O 1 L 2 error O 2 L error O Table 8.26: Shallow water convergence rates test: DG CT-DRP fifth order method. 125

142 8. Numerical results Idealised circular dam This is a circular dam-break problem. At the beginning of the simulation we assume a circular wall that contains the water. The circular wall collapses instantaneously generating a circular shock wave front. The computational domain is a square of m. The circular wall has centre at (x c, y c ) = (20, 20) and a radius R = 2.5 m. The initial conditions are given by constant velocities u = v = 0 throughout the domain and the following free surface distribution, { h = 2.5 m if (x x c ) 2 + (y y c ) 2 R 2, h(x, y,0) = h = 0.5 m if (x x c ) 2 + (y y c ) 2 R 2 (8.8). The computational domain is discretized with triangles dividing each boundary in 100 elements. The boundary conditions are set to be transmissive conditions. We solve this problem with the Finite Volume ADER numerical method and the TT DRP solver up to third order. The reference solution is obtained by solving a one dimensional problem in radial coordinates with a very fine mesh Water height h (m) Position x (m) Figure 8.9: Circular dam-break problem. Radial cut of the free surface elevation at t = 4.7 s. Reference solution in solid line. The first (red square), second (green triangle) and third (blue circle) order solution using FV ADER scheme with the TT DRP solver. In Fig. 8.9 and 8.10 we present a radial cut of the two dimensional solutions. In Fig. 8.9 the numerical solution for the first, second and third order numerical method is shown for the free surface. In Fig the numerical solution for the first, second and third order numerical method is shown for the particle velocity u. We see from Fig. 8.9 and 8.10 that the solution improves with the order of the method. The first order solution 126

143 8. Numerical results 0.5 Velocity u (m/s) Position x (m) Figure 8.10: Circular dam-break problem. Radial cut of the velocity u at t = 4.7 s. Reference solution in solid line. The first (red square), second (green triangle) and third (blue circle) order solution using FV ADER scheme with the TT DRP solver. is very diffusive. The second and third order solutions give similar results in the smooth part of the flow. Near the main outward shock wave, the third order solution shows better agreement with the reference solution. In Fig we present the solution for the free surface. We cut half of the solution in order to show the inner structure. Observing Fig. 8.11(a) we see that immediately after the circular wall collapses, a circular shock wave travels outward while a rarefaction wave travels inward. In Fig. 8.11(b) the shock wave is clearly visible and the rarefaction wave is reaching the centre. We see that the free surface presents high gradients which are not easily to solve numerically. The inward travelling rarefaction wave implodes in the centre reflecting back. In Fig. 8.11(c), the free surface goes below the original level outside of the dam. This is the effect of the implosion of the rarefactions. This is clear in Fig. 8.11(d) where the water height reaches a minimum and a secondary shock is formed. In Fig. 8.11(e) the primary shock keep travelling outward followed by a rarefaction and the secondary shock is reaching the centre. Due to the collapse of the secondary shock in the centre it reflects and the free surface increases. 127

144 8. Numerical results (a) t = 0.1 s (b) t = 0.4 s (c) t = 0.8 s (d) t = 1.6 s (e) t = 3.8 s (f) t = 4.7 s Figure 8.11: Circular dam-break problem. Half plane of the free surface solution at different times. (a): after the dam collapse, a circular shock wave travels outward while a rarefaction wave travels inwards. (b): the rarefaction wave implodes at the centre and reflects back. (c)-(d): due to the reflected inner circular rarefaction the free surface goes below the initial level outside the dam. (d): a secondary shock formed travelling inward. (e): the primary shock continues to travel outward while the secondary shock almost collapses in the centre. (f): the secondary shock collapses in the centre generating a strong elevation of the free surface. 128

145 8. Numerical results Dam with channel with 45 o bend We solve a two dimensional dam break problem performed in a laboratory. The computational domain consists of a square reservoir of dimensions m connected to a channel with a bend of 45 degrees. We discretize the domain using an unstructured grid with 9228 triangle, see Fig y(m) 4 Reservoir Channel open boundary x(m) Figure 8.12: Unstructured grid with 9228 elements. From the left to the right, the square reservoir connects to the channel which has a bend of 45 o. The initial condition, as depicted in Fig. 8.13, is b(x, y) + h(x, y,0) = 0.58 m inside the reservoir, b(x, y) + h(x, y, 0) = 0.34 m in the channel and zero velocities. The bottom elevation inside the reservoir is b(x, y) = 0, while in the channel is b(x, y) = 0.33 m. All the boundaries are reflective walls except the end of the channel which is open. Figure 8.13: Dam with channel. The initial conditions present a discontinuity in the free surface. The free surface (transparent blue) over the bottom elevation (red). Note that the bottom elevation presents a discontinuity which cannot be solved by this shallow water model, see Fig However, we approximate the physical domain considering a small transition area between x = 2.39 m and x = 2.49 m. This transition area connects the two levels of the bottom elevation. Inside this area a number of triangles 129

146 8. Numerical results are defined. Considering the cell average of the bottom inside each triangle we reconstruct the bottom elevation taking into account the order of the numerical method used. Using this reconstruction procedure we transform the discontinuity of the bottom elevation in a smooth transition. In Fig we show the reconstruction of the bottom elevation. In Fig. 8.14(a) the bottom is represented by piece-wise constant polynomial. The reconstruction procedure approximates the transition between both levels. In Fig. 8.14(b) the bottom is reconstructed with piece-wise parabolic polynomial. Z Z Y X Y X (a) First order approximation. (b) Third order approximation. Figure 8.14: Dam with channel. Reconstruction of the bottom elevation. (a): data inside each triangle is constant and the transition area has discontinuous steps. (b): data inside each triangle is represented by a second order polynomial. In this manner the discontinuous jumps of the bottom reduce with the order of the method. The solution of this problem initially consists of a right travelling shock wave and a left travelling rarefaction wave. The shock flows into the channel until reaches the bend. Because the 45 o bend, the chock reflects forming a characteristic wave pattern. In the exterior side of the bend the free surface increases while in the interior side of the bend the water level decreases almost to generate a dry zone. On the other hand, the left travelling rarefaction wave expands inside the reservoir and interacts with the solid walls. This interaction generates a very complicate wave pattern that needs highly accurate numerical methods to be revealed. Eventually, the rarefaction wave reflects and goes back into the channel. 130

147 8. Numerical results y(m) 4 Free surface H (m) x(m) (a) First order FV y(m) 4 Free surface H (m) x(m) (b) Second order FV y(m) 4 Free surface H (m) x(m) (c) Third order FV Figure 8.15: Dam with channel. Free surface contour plot. (a): First order Finite Volume. (b): Second order Finite Volume. (c): Third order Finite Volume. 131

148 8. Numerical results In Fig to 8.16 we show the numerical result at the final time t = 4.0 s. The solution is presented as a contour plot of the free surface for the Finite Volume and Discontinuous Galerkin schemes. First to third order FV numerical solution are presented in Fig In Fig the second order DG solution is presented. As the order increases we can observe more details in the wave patterns inside the reservoir, where many waves are already reflected, see Fig. 8.15(a) to 8.15(c). In the channel, the shock wave already interacts with the bend, reflecting back and forward from the solid walls. Both second order methods, Fig. 8.15(b) and Fig. 8.16, show similar results. The FV solution seems to reveal more details in the wave pattern inside the reservoir. For this type of problem, where strong discontinuities are present, the DG scheme has to reconstruct the data near the discontinuities introducing more computational time to the numerical method. y(m) 4 Free surface H (m) x(m) Figure 8.16: Dam with channel. Free surface contour plot. Second order Discontinuous Galerkin Malpasset This simulation is presented to show the potentialities of the numerical methods studied in this thesis. The simulation is about a real dam-break problem. The physical domain is the narrow Reyran river valley where the Malpasset dam failed in 1959 causing more than 400 casualties. We do not intent to present an accurate simulation, in particular because friction is not included and the numerical results were obtained with a first order FV numerical method but the use of unstructured meshes allow us to discretize complex geometries. Fig depicts the computational domain and the location of the dam. The computational domain consists of triangles. The size of each triangle varies giving more resolution where it is more demanding. To the left of the dam we set a free surface elevation b(x, y)+h(x, y,0) = 120m and everywhere else is dry. At time zero the dam collapses 132

149 8. Numerical results instantaneously and the water starts to flow through the valley. In Fig are shown the numerical results of the simulation. In Fig. 8.18(a) the initial condition are shown, Fig. 8.18(b) to 8.18(d) show the simulation every 30 seconds. Comparing this solution with field data we observe that the method overestimates the water velocity. This is principally due to the absence of friction. Considering that the first order method is able to run the simulation with wet/dry fronts, we are optimistic in the possibility to couple the high-order results presented in this thesis with dry fronts. Figure 8.17: Malpasset dam: Computational domain formed by triangles. On the left of the dam the free surface is 120 m over the sea level. Everywhere else is dry. At time zero the dam collapses instantaneously and the water flows through the valley. 133

150 8. Numerical results (a) (b) (c) (d) Figure 8.18: Numerical solution of the Malpasset dam simulation. Contour plot for the water depth. The red zones show a high level of water while the blue zones low levels. (a): initial condition. (b): Numerical simulation after thirty seconds. (c): Numerical simulation after sixty seconds. (d): Numerical simulation after ninety seconds. 134