A Hybridized Discontinuous Galerkin Method for Time-Dependent Compressible Flows

Transcription

1 A Hybridized Discontinuous Galerkin Method for Time-Dependent Compressible Flows Master s Thesis by Alexander Jaust Institut für Geometrie und Praktische Mathematik Rheinisch-Westfälische Technische Hochschule Aachen December 9th, 2013 Supervisor: Prof. Dr. Sebastian Noelle Co-Examiner: Dr. Jochen Schütz

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3 Eidesstattliche Versicherung Hiermit erkläre ich, dass ich die vorliegende Masterarbeit selbstständig verfasst habe. Es wurden keine anderen als die in der Arbeit angegebenen Quellen und Hilfsmittel benutzt. Die wörtlichen oder sinngemäß übernommenen Zitate habe ich als solche kenntlich gemacht. Aachen, den 9. Dezember 2013 Alexander Jaust

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5 Acknowledgments I want to thank my supervisor Prof. Dr. Sebastian Noelle for giving me the opportunity to work at this thesis in an interesting field of computational fluid dynamics. I thank my advisor Dr. Jochen Schütz who introduced me to hybridized discontinuous Galerkin methods and supported me during my thesis. Thanks to Michael Woopen who was always giving helpful advises about the code. Further thanks go to Tobias Leicht from the DLR for providing me the opportunity to participate at the 2nd International Workshop on High-Order CFD Methods in Cologne. I also want to thank my family for their support. Without them I could not have pursued my studies.

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7 Contents 1. Introduction 1 2. Governing Equations Notation Scalar Convection-Diffusion Equation Euler Equations Navier-Stokes Equations Dimensionless Numbers Spatial Discretization The Hybridized Discontinuous Galerkin Method Hybridization of the Nonlinear Convection-Diffusion Equation Discretization of the Nonlinear Convection-Diffusion Equation Time Integration Methods General Setting Consistency and Convergence Stability A- and A(α)-Stability L-Stability Diagonally Implicit Runge-Kutta Methods Stability Time Step Adaption Backward Differentiation Formulas Stability Time Integration for the Hybridized Discontinuous Galerkin Method Notes on Differential-Algebraic Equations Backward Differentiation Formulas Singly Diagonally Implicit Runge-Kutta Methods Shock-Capturing An Artificial Viscosity Model for the Hybridized Discontinuous Galerkin Method Numerical Results Rotating Gaussian Convergence Study Time Step Adaptation Radial Expansion Wave Kármán Vortex Street Sod s Shock Tube Constant Viscosity Adaptive Artificial Viscosity Conclusions and Outlook 58 A. Additional and Tabulated Results 60 A.1. Rotating Gaussian A.2. Radial Expansion Wave A.3. Sod s Shock Tube

8 Contents ii Bibliography 68

9 List of Figures 3.1. Numerical method: Domain Ω split into two parts by an edge Γ Stability region: Examples of the stability regions of an A-stable method (left) and an A(α)-stable method with α = π 4 (right) are shown Stability region: The stability regions of all SDIRK methods. Stable regions with R(z) 1 are indicated white while unstable regions are red Initialization of the time integration for the BDF3 method Stability region: The stability regions of BDFk methods for k = 1,..., 4. Stable regions with R(z) 1 are indicated white while unstable regions are red Solution strategy with error estimator Artificial Viscosity: Approximation of the ramp function (left) and sonic speed (right) Rotating Gaussian: The initial distribution (left) and final distribution at t = π 4 (right) of the Gaussian Rotating Gaussian: Convergence of w w h L 2 under uniform spatial and temporal refinement with p = 2 (left) and p = 3 (right) Rotating Gaussian: Error in dependence of time step size t (left) and tolerance tol (right) on a fixed mesh with N e = 512 elements and cubic ansatz functions Rotating Gaussian: time step evolution for different tolerances tol on a fixed mesh with N e = 512 elements and cubic ansatz functions Rotating Gaussian: Error evolution on different grids with cubic ansatz functions (left). The tolerance is divided by a factor of eight or sixteen depending on the order of the method (right) Radial Expansion Wave: Density ρ on the domain at t = t 0 (left) and t = T (right) Radial Expansion Wave: Quadratic ansatz functions and N e = 2048 elements Radial Expansion Wave: Cubic ansatz functions and N e = 8192 elements Kármán vortex street: Section of the mesh Kármán vortex street: Mach number distribution at two instances approximately a half period apart showing the periodic behavior of the flow Von Kármán vortex street: Results obtained for tol = 0.05 and 10 2 t Sod s shock tube: Initial data Sod s shock tube: Exact solution at T = Sod s shock tube: Sketch of the mesh Sod s shock tube: Errors in the density and velocity for constant viscosity ε Sod s shock tube: Exact and approximated solution of density and velocity for different bulk viscosities ε 0 at the discontinuity and shock. A mesh with N e = 400 elements and polynomials of degree p = 2 have been used Sod s shock tube: Error in density ρ and velocity u for different polynomial degrees p and values for bulk viscosity ε Sod s shock tube: Exact and approximated solution of density and velocity for different bulk viscosities ε 0 at the discontinuity and shock. A mesh with N e = 400 elements and polynomials of degree p = 2 have been used A.1. Radial Expansion Wave: Quadratic ansatz functions and N e = 8192 elements A.2. Radial Expansion Wave: Cubic ansatz functions and N e = 2048 elements

10 List of Tables 4.1. Coefficients of Cash s SDIRK method Coefficients of Al-Rabeh s SDIRK method Coefficients of Hairer s and Wanner s SDIRK method Range of values for γ to retain an A-stable method Range of values for γ to retain an L-stable method Coefficients of BDFk-methods up to order Angle α of A(α)-stable BDFk-methods up to order Kármán vortex street: Mean drag coefficients c D and Strouhal numbers Sr from experiments and previous publications Kármán vortex street: Mean drag coefficients c D and Strouhal numbers Sr determined by Cash s SDIRK method using different tolerances and bounds for the time step Kármán vortex street: Mean drag coefficients c D and Strouhal numbers Sr determined with Al-Rabeh s SDIRK method using different tolerances and bounds for the time step Kármán vortex street: Mean drag coefficients c D and Strouhal numbers Sr determined with Hairer s and Wanner s SDIRK method using different tolerances and bounds for the time step A.1. Rotating Gaussian: Convergence of BDF methods on fixed mesh N e = 512, cubic ansatz functions and fixed time step A.2. Rotating Gaussian: Convergence of SDIRK methods on fixed mesh N e = 512, cubic ansatz functions and fixed time step A.3. Rotating Gaussian: Convergence of BDF2 for quadratic and cubic ansatz functions. 61 A.4. Rotating Gaussian: Convergence of BDF3 method for quadratic and cubic ansatz functions A.5. Rotating Gaussian: Convergence of Cash s method for quadratic and cubic ansatz functions A.6. Rotating Gaussian: Convergence of Al-Rabeh s method for quadratic and cubic ansatz functions A.7. Rotating Gaussian: Convergence of Hairer s and Wanner s method for quadratic and cubic ansatz functions A.8. Rotating Gaussian: Convergence of SDIRK methods on fixed mesh N e = 512, cubic ansatz functions and variable time step A.9. Rotating Gaussian: Convergence of SDIRK methods when mesh and tolerance are refined using cubic ansatz A.10.Radial Expansion Wave: Entropy error and total number of Newton steps on the coarse mesh with quadratic ansatz functions A.11.Radial Expansion Wave: Entropy error and total number of Newton steps on the fine mesh with quadratic ansatz functions A.12.Radial Expansion Wave: Entropy error and total number of Newton steps on the coarse mesh with cubic ansatz functions A.13.Radial Expansion Wave: Entropy error and total number of Newton steps on the fine mesh with cubic ansatz functions A.14.Sod s Shock Tube: Density errors ρ(t ) ρ h (T ) L 2 for constant viscosity A.15.Sod s Shock Tube: Velocity errors u(t ) u h (T ) L 2 for constant viscosity A.16.Sod s Shock Tube: Density errors ρ(t ) ρ h (T ) L 2 for adaptive viscosity A.17.Sod s Shock Tube: Velocity errors u(t ) u h (T ) L 2 for adaptive viscosity

11 1. Introduction For a long time experiments have been one of the most important tools in science and industry. However, experiments are usually expensive, time consuming and sometimes hardly executable. With the occurrence of computers and their performance increasing, simulations become a more and more interesting alternative to experiments. They offer a cheaper environment where certain experiments can be modeled. It is possible to easily change key quantities to study their influence on the experiment. In this work we focus on computational fluid dynamics (CFD), i.e., the modeling of fluid flows such as the flow of air around an airfoil, for example. Such flows are usually described by partial differential equations (PDEs). These are differential equations of multivariate functions in which their partial derivatives occur. In many cases it is only possible to solve these equations approximately using simulations. A common simulation technique for PDEs arising in CFD are finite-volume methods (FV methods) [41 43, 70 73]. Current codes employing finite-volume schemes are usually second order accurate. With order of accuracy q we denote that the error e is proportional to h q with h being the mesh size. One often speaks of high-order methods when the order of the method is higher than two. By now, finite-volume methods are well-accepted in engineering. They show stable behavior in particular for steady-state problems due to strong numerical dissipation. However, for unsteady problems that need to preserve complex structures of a flow over a longer time, e.g. vortices, the dissipation becomes disadvantageous because the structures are smoothened out. Moreover, for increased accuracy the mesh size needs to be decreased much more than for higher order methods. In general, this leads to a much higher number of unknowns such that also the time needed for a simulation is larger. A promising method for high-order CFD are the discontinuous Galerkin methods (DG) [5, 11, 19, 28, 29, 34, 35, 37, 38, 53, 63]. The computational domain is triangulated and then the solution is represented by a polynomial of degree p on each of the elements. A drawback is the higher number of degrees of freedom compared to continuous Galerkin (CG) methods [37]. Similar to DG methods the solution is approximated using polynomials, but the approximated solution is assumed to be continuous over the whole triangulation. For the DG method continuity is only enforced on each element of the triangulation such that jumps in the solution between elements are possible. Problems in CFD are usually described by hyperbolic or nearly hyperbolic partial differential equations for which classical CG methods are usually not stable without stabilization [14, 39]. However, DG methods are stable for such equations and therefore of greater interest in CFD. The degrees of freedom that are coupled globally for a DG, but also other methods, can be reduced using hybridization [26]. A new characterization for elliptic problems has been given by Cockburn and Gopalakrishnan [20] and later unified to be applied to different numerical methods by Cockburn et al. [21]. Based on this the hybridized discontinuous Galerkin method (HDG) has been developed for different partial differential equations including the equations of fluid dynamics, especially the compressible Euler and Navier-Stokes equations [45 47, 49, 59, 61]. The method introduces additional unknowns on the edges between elements increasing the total number of degrees of freedom. However, this allows the reformulation of the problem such that the number of globally coupled unknowns is much lower than for the initial problem at the cost of a more complicated assembling of the system of equations. Since the computational costs and the memory requirements are mainly dependent on the system size of globally coupled unknowns the HDG method still reduces the overall computational work. We focus on the HDG method for time dependent problems in this work. An implementation based on Netgen, NGSolve [57] and PETSc [7 9] has been done in an earlier work [59, 77]. It has been extended by time integration methods based on backward differentiation formulas (BDF) by Schütz et al. [62]. Other experiments with BDF and diagonally implicit Runge-Kutta (DIRK)

12 1. Introduction 2 methods have been executed by Nguyen and Peraire [45, 47]. We examine time integration based on singly diagonally implicit Runge-Kutta (SDIRK) methods with an embedded error estimator [1, 15, 18, 33]. These methods are quite popular for solving stiff ordinary differential equations (ODEs) that arise from many different real world applications as chemical reactions, for example. With the error estimator it is possible to adapt the time step during the simulation in order to control the temporal error. It is investigated how well the error estimator works to obtain simulation results with a desired accuracy. Furthermore, these methods are compared to the BDF methods for which it is much more complicated to have a time step adaptation because of the nature of the method and the absence of an integrated error estimator. In order to apply the HDG method to a wide range range of problems the solver has to be able to handle discontinuities. These often occur in supersonic fluid flow in the shape of shocks or contact discontinuities. One way to handle these discontinuities is to use of an artificial viscosity model. In the work we study an artificial viscosity model that has been introduced by Nguyen and Peraire [44] for HDG methods. It has been originally used for steady state problems only. We investigate whether it can be applied to time-dependent problems as well. This work is structured in the following way. First, the governing equations used in this work are introduced, namely the convection-diffusion equation, the Euler equations and the Navier-Stokes equations. In the following chapter we introduce the spatial discretization using the hybridized discontinuous Galerkin method by applying it to the convection-diffusion equation. Then, in Chapter 4, different time integration methods for ordinary differential equations are presented. Especially their stability properties and the possibility of time step adaptation for these methods are examined. The application of these methods to the hybridized DG method is shown afterwards in Chapter 5. Following, we describe the artificial viscosity model in Chapter 6. The implementation is validated in Chapter 7. Moreover, some results for different test cases are discussed to quantify how good the time integration methods work. Finally, a short conclusion and an outlook of future work is given in Chapter 8.

13 2. Governing Equations The problems discussed in this work are mainly motivated by computational fluid dynamics (CFD). This means in most cases the flow field of a certain fluid, mostly air, has to be determined. Additionally to the flow field itself and occuring phenomena as vortices or shocks e.g., the transport of some scalars within the flow field may be of interest. For a simulation a model of the problem in form of a set of equations must exist. These equations are presented in this section and are all partial differential equations (PDE). First we introduce the general notation. Then we present the (nonlinear) convection-diffusion equation. It is often used for developing numerical methods in computational fluid dynamics as it describes the two main types of transport that occur, namely convection and diffusion. Real-world flows are often modeled using the Euler equations or the Navier-Stokes equations that are introduced afterwards. The Euler equations describe inviscid flows which can be seen as a special case of the Navier-Stokes equations under the assumption of vanishing viscosity Notation To apply the discretization method to a wide range of problems in a satisfactory manner, it is beneficial to introduce generalized notation for problems. For the hybridized DG method we write problems in the following form: w t + (f(w) f v (w, w)) = h(w, w). (2.1) Here, w is unknown and may be vector valued. With w t we denote the time derivative w t := w t. Moreover, there are spatial derivatives, denoted by the gradient operator and divergence operator, of the convective flux f(w) and the viscous flux f v (w, w). On the right hand side h(w, w) is a source or sink term. Depending on the problem certain terms may vanish. In this work we focus on two dimensional (d = 2) problems. Therefore two spatial directions x 1 and x 2 exist. This is no restriction of the generality of the method. It can also be applied to one (d = 1) or three dimensional (d = 3) problems. For the sake of completeness we define the used operators for spatial derivatives of vectors and matrices. We introduce these notations under the assumption of exactly two spatial dimensions (d = 2). The gradient operator applied to some vector v R m is defined as v = v 1 v 1 x 1 x 2. v m x 1. v m x 2 (2.2) and is a matrix of size m 2. The divergence of a vector v R 2 and of a matrix A R m 2 are defined as 2 v i div v = v = = v 1 + v 2 (2.3) x i x 1 x 2 and a 11 a 12 A :=.. a m1 a m2 = i=1 (a 11, a 12 ) T. (a m1, a m2 ) T = a 11 x 1 a m1 x 1 + a12 x 2. + am2 x 2. (2.4)

14 2. Governing Equations Scalar Convection-Diffusion Equation Convective transport and diffusion processes can be described using the convection-diffusion equation. In general, the equation can be written as w t + (f(w) f v (w, w)) = 0, (2.5) In our case w is an unknown scalar. Nevertheless, the convection-diffusion equation can also be formulated in case w is a vector. The fluxes are given by f(w) = uw, f v (w, w) = ε w. (2.6) The term convective flux f(w) describes the convective transport of quantity w by a flow field with velocity u = (u 1, u 2 ) T. In case w contains components of the velocity the problem becomes nonlinear. The viscous flux f v (w, w) accounts for diffusive transport with ε > 0 being the diffusivity. We require positive values for ε because otherwise the problem is ill-conditioned. The right hand side is zero, because no sources and sinks h(w, w) are present. All terms may be functions of space (x 1, x 2 ) and time t. As already mentioned in the introduction to this chapter, equation (2.5) is often used as a model equation when developing numerical methods. The reason for this is that flow problems are described by coupled convective and diffusive transport processes. While yielding a simpler equation than the full equations of fluid dynamics the main features are still covered by the convection-diffusion equation. It also covers different types of partial differential equations. For u = (0, 0) T and w t = 0 one has a so-called elliptic problem. A common property is that this equation leads to smooth solutions even if the initial conditions are non-smooth. On the other side one has a hyperbolic problem for vanishing diffusivity ε = 0. Then in the nonlinear case non-smooth solutions may develop even if the initial data is smooth. In the setting of fluid dynamics the discontinuities are often referred to as shocks. Since the equation is a partial differential equation of mixed type many different effects may occur depending on the initial conditions and diffusivity ε. This makes it challenging to develop numerical methods because the discretizetion method must be able to handle a large number of effects that may occur. On the other hand, this makes the convection-diffusion a very powerful test equation Euler Equations A very important set of equations in fluid dynamics are the Euler equations. They describe the special case of an inviscid flow and are often used in gas dynamics to model high speed flows of air. Using our notation the equations can be written as w t + f(w) = 0. (2.7) Neither viscous fluxes f v (w, w) nor source terms h(w, w) appear in the Euler equations. The convective flux is split into two parts f(w) := (f 1 (w), f 2 (w)) with The vector of unknowns f 1 (w) = ( ρu 1, P + ρu 2 1, ρu 1 u 2, u 1 (E + P ) ) T (2.8) f 2 (w) = ( ρu 2, ρu 1 u 2, P + ρu 2 2, u 2 (E + P ) ) T. (2.9) w = (ρ, ρu 1, ρu 2, E) T (2.10) contains the density ρ, momentum ρu 1 and ρu 2 in the two spatial directions and the energy E. Therefore, this is a system of four equations. The first equation is called continuity equation and it ensures mass conservation. The second and third equation describe the momentum conservation in each spatial direction. There occurs a term containing the gradient of the pressure P. It describes the change of momentum due to forces acting on the surface of the fluid. The last equation describes the energy conservation with E being the energy. In general, the gravitational acceleration g may

15 2. Governing Equations 5 have an influence on the fluid, but as we focus on flows of gases in this work, the density has rather small values. Therefore, the influence of the gravitational acceleration on the fluid is neglected such that the terms containing g are assumed to be zero. In two dimensions, there are five unknowns (density ρ, velocities u 1 and u 2, energy E and the pressure P ), but only four equations, leading to an under-determined system of equation. To resolve this issue a closure is needed. A common choice is the equation of state ( P = (γ 1) E 1 ) 2 ρ u 2 2 (2.11) where γ is the ratio of specific heats. It depends on the fluid and is approximately γ = 1.4 for air. The Euler equations are a nonlinear, hyperbolic system of PDEs. As already mentioned, problems of this type may lead to non-smooth solutions containing shocks. This coincides with observations of so-called transonic and supersonic flows in gas dynamics where shocks appear Navier-Stokes Equations A wider set of flows than described by the Euler equations are covered by the Navier-Stokes equations. In contrast to the Euler equations the fluid is assumed to be viscous. Therefore, also a viscous flux appears in the equations w t + (f(w) f v (w, w)) = 0. (2.12) The convective flux f(w) = (f 1 (w), f 2 (w)) equals the convective flux of the Euler equations (cf. eq. (2.8) and (2.9)). The viscous flux f v (w, w) is split in two parts similar to the convective flux with components f v,1 (w, w) = f v,2 (w, w) = ( 0, τ 11, τ 21, τ 11 u 1 + τ 12 u 2 + κ θ ) T, (2.13) x 1 ( 0, τ 21, τ 22, τ 21 u 1 + τ 22 u 2 + κ θ ) T. (2.14) x 2 The Navier-Stokes equations describe conservation of mass, momentum and energy as the Euler equations. Therefore, the vector of unknowns w is the same as for the Euler equations (cf. eq. (2.10)). However, additional terms occur for the momentum and energy equations because of viscous effects. The entries τ ij in the viscous flux refer to the entries of the stress tensor S. For a Newtonian fluid it can be written as ( ) τ11 τ S = 12 = µ ( u + ( u) T 23 ) τ 21 τ ( u)i. (2.15) 22 with the dynamic viscosity µ and the identity matrix I. It causes change of momentum and energy due to friction. The second additional term is the heat flux q which is approximated using Fourier s law [30] q = κ θ (2.16) with the temperature θ and heat conductivity κ. In the case of an inviscid flow µ = 0 also the heat flux q = 0 vanishes and one obtains the Euler equations again. The viscosity is approximated using Sutherland s law [68] µ = µ(θ) = C 1θ 3 2 θ + C 2 (2.17) with C 1 and C 2 being constants depending on the fluid. Yet another closure is needed for the temperature θ. It can be computed using the ideal gas law θ = µγ ( E κ Pr ρ 1 ( u u 2 ) ) 1 P 2 = (γ 1)c v ρ. (2.18)

16 2. Governing Equations 6 Here, the Prandtl number Pr = µcp κ and the specific heats of constant pressure c p and of constant volume c v occur. Compared to the Euler equations the Navier-Stokes equations are more complex especially due to the stress tensor. This allows the description of phenomena as turbulence [51] and boundary layers which can not be described by the Euler equations because these are viscous effects Dimensionless Numbers An important tool in fluid dynamics are dimensionless numbers. They are used in similitude and can be determined by doing a dimensional analysis of the equations describing a problem. We briefly introduce the dimensionless numbers we are using in this work. Reynolds Number The Reynolds number Re quantifies the ratio of inertial to viscous forces. It is defined as Re = ρ 0UL µ 0 = UL ν 0. (2.19) For the density ρ 0 and viscosity µ 0 a specific value at a reference point is taken. This could be in the free stream, for example. The viscosity ν is the kinematic viscosity ν = µ ρ that allows a shorter representation since the density cancels out. The velocity U is some representative velocity of the fluid and L is a characteristic length influencing the fluid. This may be the height or diameter of a channel or the the diameter of an obstacle. The Reynolds number is useful when determining if a flow contains instationary effects as vortex shedding or turbulence. However, it is usually not possible to specify sharp values of Re for which these effects occur because they also depend on quantities not covered by the Reynolds number [51, 58]. Prandtl Number The Prandtl number Pr describes the ratio of kinematic viscosity to the thermal diffusivity: Pr = ν a = µc p κ. (2.20) It only depends on the fluid properties, such as the thermal diffusivity a, but not on the length scale as the Reynolds number does. We assume the Prandtl number to be constant and set Pr = 0.72 which is a common choice for air at moderate temperatures. It is needed to compute the temperature θ from the ideal gas law (cf. equation (2.18)). Mach Number For flows described by the Euler or Navier-Stokes equations the Mach number Ma is important to measure the influence of compressible effects. It is defined as the ratio of the characteristic velocity U and the speed of sound c: Ma = U c. (2.21) In case of an ideal gas the speed of sound can be determined from c = γ P ρ. The speed of sound actually defines the speed with which information is transported within the fluid. For Ma < 0.3 the density variations are less than 5% such that simplifications as incompressibility, i.e., ρ = const, are often applied. For Mach numbers greater than one, i.e., Ma > 1, information can only be transported downstream.

17 2. Governing Equations 7 Strouhal Number In the special case of a flow around an obstacle, vortices shed for certain Reynolds numbers. Using the frequency f of the vortex shedding the Strouhal number Sr is defined as: Sr = fl U. (2.22) The characteristic length L is the diameter of the obstacle and U is the flow velocity.

18 3. Spatial Discretization In this section the hybridized DG method (HDG) is introduced. It is explained using the steady-state convection-diffusion equation. First we present some general notes on the hybridized DG method. Then we illustrate how to hybridize the convection-diffusion. Finally, the actual discretization of the convection-diffusion equations is described. For this, some function spaces are needed. The space of square integrable functions on a domain Ω is { } L 2 (Ω) := f : f 2 dx =: f L 2 (Ω) <. (3.1) Ω With this the space of functions whose derivatives are up to order m square integrable is defined as where the derivatives are denoted by H m (Ω) := { f : D α f L 2 (Ω), α m } (3.2) D α f = x α1 α 1... f, α = α α n. (3.3) xαn n So m is the order up to which the derivatives have to exist in the sense of distribution The Hybridized Discontinuous Galerkin Method It has been already mentioned in the introduction that discontinuous Galerkin methods are of interest in computational fluid dynamics. This is especially the case for high-order methods because the order can be easily varied by the polynomial degree of the used ansatz functions. However, these methods have a large number of globally coupled degrees of freedom. From this arises a large, linearized system of equations that mainly determines the computational costs and memory requirements of the method. A way to reduce the number of globally coupled unknowns is the so-called hybridization of a method. One introduces new unknowns on the edges between elements such that the arising system of equations allows to reduce the number of globally coupled unknowns. Although the total number of unknowns is increased, this is often worthwhile because the total computational costs mainly depend on the number of globally coupled unknowns. Hybridization itself is a technique already introduced in 1965 by Fraeijs de Veubeke [26] for problems arising from linear elasticity. It was later shown by Arnold and Brezzi [4] that hybridization can be used to improve the approximation of a method. At this point it was mostly applied to so-called mixed methods [3], but not yet to DG methods. For the approximate solution of hybridized mixed methods a new characterization has been introduced by Cockburn and Gopalakrishnan [20]. Later, Cockburn et al. developed a way to apply hybridization to finite element methods for elliptic problems using a unifying framework [21]. With this framework different finite element methods, including DG methods, can be coupled. The finite element method can even vary from one element to another and the coupling is achieved by hybridization. Such methods combining a DG method for hyperbolic parts of a problem and a mixed method for elliptic parts with hybridization have been developed, amongst others, by Egger and Schöberl [27] and Schütz and May [62]. Furthermore, DG methods have been hybridized and applied to a wide range of problems in CFD [45, 46, 49, 61, 62]. These methods have shown an optimal convergence rate q = p + 1 in the vector of unknowns w and its gradient w with p being the degree of the polynomial used to represent the solution. At the same time, the number of globally coupled of unknowns can be reduced. For a standard DG method the number of globally coupled is proportional to O(p d ) while for the hybridized DG methods it is proportional to O(p d 1 ) with d being the number of spatial

19 3. Spatial Discretization 9 n 2 n 1 Ω 1 Ω 2 Γ Figure 3.1.: Numerical method: Domain Ω split into two parts by an edge Γ. dimensions. Therefore, the hybridized DG method becomes more beneficial as the polynomial degree is increased [59]. All in all, the hybridized DG method is a promising high-order method. Besides the reduced number of globally coupled unknowns, it allows many local computations which can be exploited by parallel computations. Furthermore, spatial adaptation is not only possible by refining the mesh, but also by choosing ansatz functions of different polynomial degree within a mesh Hybridization of the Nonlinear Convection-Diffusion Equation In Section 2.2 the general case of the convection-diffusion equation was introduced. Here, we focus on the problem for a given open bounded domain Ω with homogeneous Dirichlet boundary conditions. Other boundary conditions can also be applied as von Neumann boundary conditions, for example. A more detailed discussion on the application of boundary conditions to the hybridized methods is given in [45]. Here, we investigate the stationary case, i.e., w t = 0; the time-dependent case is treated in Section 5. The nonlinear convection-diffusion equation with boundary conditions is given by: f(w) ε w = h, x Ω (3.4) w = 0, x Ω. We assume that the solution w has certain regularity such that it is at least in H 2 (Ω). Then, the weak formulation of the problem is: N(w, ϕ) := f(w) ϕdx + ε w ϕdx =! ϕhdx, ϕ H0 1 (Ω). (3.5) Ω Ω with a test function ϕ. Further, we assume that the domain Ω can be split into two parts Ω = Ω 1 Ω 2 by an edge Γ := Ω 1 Ω 2 with normal vectors n 1,2 as in Figure 3.1. Then, the initial problem (3.4) can be formulated separately on each domain Ω 1 and Ω 2 : f(w 1 ) ε w 1 = h x Ω 1, w 1 = 0 x Ω Ω 1, w 1 = λ x Γ f(w 2 ) ε w 2 = h x Ω 2, w 2 = 0 x Ω Ω 2, w 2 = λ x Γ. It has the same structure as the convection-diffusion equation on the initial domain Ω, but with an additional boundary condition on the edge Γ. This condition λ is introduced to retain continuity of the solution over the edge. Moreover, the solutions w 1 and w 2 of Problem (3.6) inherit the regularity assumption of the global solution w such that they have to be at least in H 2 (Ω i ), i = {1, 2}. Now, we introduce a new global solution ŵ on Ω that is constructed by the partial solutions w 1 and w 2 : { w 1, x Ω 1 ŵ =, w i H 2 (Ω i ), i = {1, 2}. (3.7) w 2, x Ω 2 The initial problem (3.4) and the split problem (3.6) must have the same solution ŵ =! w since the problems shall be equivalent. To ensure that ŵ coincides with w the variable λ has to be chosen such that continuity is retained and that the initial problem is solved. This means we require N(w, ϕ) =! N(ŵ, ϕ) = ϕhdx. (3.8) Ω Ω (3.6)

20 3. Spatial Discretization 10 From this, an additional equation, that λ has to fulfill, can be derived by partially integrating N(ŵ, ϕ) such that N(ŵ, ϕ)= f(ŵ) ϕdx + ε ŵ ϕdx (3.9) Ω Ω = f(w 1 )ϕdx + f(w 2 )ϕdx + ϕε( ŵ 1 ŵ 2 ) n 2 dσ (3.10) Ω 1 Ω 2 Γ ϕ(f(ŵ 1 ) n 2 + f(ŵ 2 ) n 1 )dσ ε w 1 ϕdx ε w 2 ϕdx Γ Ω 1 Ω 2 = ϕhdx ϕ(f(ŵ 1 ) n 2 + f(ŵ 2 ) n 1 )dσ + ε ϕ( w 1 w 2 ) n 2 dσ (3.11) Ω Γ Γ with ŵ i = lim w(x + τn i). In comparison to the weak formulation of the initial problem (3.5) two τ 0+ additional integrals over the edge Γ occur. To ensure the equivalence of the problems the additional terms must vanish such that equation (3.8) holds true. Therefore we require: ϕ (ε( ŵ 1 ŵ 2 ) n 2 f(ŵ 1 ) n 2 f(ŵ 2 ) n 1 ) dσ = 0. (3.12) Γ In our case ŵ 1,2 are actually ŵ 2 = w 1 and ŵ 1 = w 2 on the edge Γ. The variable λ is used to enforce that the integral (3.12) vanishes. With this additional equation the split problem (3.6) can be formulated as f(w 1 ) ε w 1 = h x Ω 1, w 1 = 0 x Ω Ω 1, w 1 = λ x Γ (3.13) f(w 2 ) ε w 2 = h x Ω 2, w 2 = 0 x Ω Ω 2, w 2 = λ x Γ (3.14) ε w f(w) = 0, x Γ (3.15) such that it is finally equivalent to the original problem (3.4). Here denotes the jump-operator that is defined as w := w 1 n 1 w 2 n 1. (3.16) It can be seen that it is possible to divide the initial problem into two problems. These subproblems only depend on given boundary conditions and the unknown λ on the edge Γ. From this it follows that the problems can be solved independently once λ is known. The idea of HDG methods is to reformulate the problem in a way such that a problem arises which depends only on λ. This represents all globally coupled unknowns and then allows to solve locally for the solution within each subdomain Ω i. For more than two subdomains, more edges respectively, the problem can be rewritten accordingly Discretization of the Nonlinear Convection-Diffusion Equation Now, we rewrite the steady-state convection-diffusion equation (3.4) into the mixed form. This means we introduce an additional unknown σ = w such that only first derivatives occur in the problem: σ = w x Ω (3.17) (f(w) εσ) = h x Ω (3.18) w = g x Ω. (3.19) The mixed form is often used for DG methods because second derivatives have strong continuity requirements. These requirements cannot be guaranteed due to the discontinuities that are allowed in the solution. Furthermore, the domain Ω is partitioned into N disjoint partitions such that Ω = N i=1 Ω k with Ω k being the set of edges of the domain Ω k. The set Γ denotes all edges e k that occur in the partition. This means edges that intersect with the domain boundary Ω and edges of intersecting sub-domains Ω i Ω j, i j. Edges e k between intersecting sub-domains occur only once in the

21 3. Spatial Discretization 11 set Γ. The total number of edges e k is given by N := Γ. Additionally we define the set Γ 0 Γ that contains all edges e k which do not intersect with the domain boundary Ω. All elements and edges have a normal vector n pointing outwards the corresponding domain Ω k. Based on this w ± is defined as w ± = lim w(x ± τn). τ 0+ For the actual discretization, we need ansatz spaces for the unknowns. We choose H h := {f L 2 (Ω) f Ωk Π p (Ω k ) k = 1,..., N} d m (3.20) V h := {f L 2 (Ω) f Ωk Π p (Ω k ) k = 1,..., N} m (3.21) M h := {f L 2 (Γ) f ek Π p (e k ) k = 1,..., N, e k Γ} m. (3.22) with d = 2 being the spatial dimension. The space of polynomials up to a degree p on a domain Ω is Π p (Ω) while m is the number of unknowns. In case of the scalar convection-diffusion equation the number of unknowns is m = 1. The Euler and Navier-Stokes equations in two dimensions have m = 4 unknowns (density, momentum in both spatial directions and energy). For a more convenient notation we introduce a set of abbreviations for integration and summation: (g, h) := N k=1 Ω k g h dx, g, h Ωk := N k=1 Then the weak formulation of the hybridized method is: Ω k g h dσ, g, h Γ := N k=1 e k g h dσ (3.23) Method 1 (Hybridized DG-Method) Find a solution (σ h, w h, λ h ) H h V h M h such that for all test functions (τ h, ϕ h, µ h ) H h V h M h the following equations hold true: (σ h, τ h ) + (w h, τ h ) g, τ h n λ Ω k Ω h, τ h n = 0 (3.24) Ω k \ Ω (f(w h ), ϕ h ) (ε σ h, ϕ h ) + f(g) n, ϕ h Ω k Ω + f(λ h ) n α(λ h w h ), ϕ h = (h, ϕ Ω k \ Ω h) (3.25) σ h n σ+ h n + α(2λ h w h w+ h ), µ h + λ Γ h g, µ h 0 Γ\Γ0 = 0. (3.26) The analytical flux f(w) has been replaced on edges by a numerical flux, a modified Lax-Friedrichs flux [43] f(u) n f num (λ, v, n) := f(λ) n α(λ v) (3.27) with a parameter α that is problem dependent. Since the solution is only required to be continuous on an element Ω k, there may occur two different solutions on the edge between two neighboring elements Ω k and Ω k. To retain conservation the numerical flux is introduced. It ensures that the in- or outgoing flux is the same for both elements. It is a common technique borrowed from finite volume methods. This method does not yet look too interesting because a lot of new unknowns for σ and the function λ on the edges have been introduced. However, this formulation can be rewritten in a more convenient way. As already indicated in a previous section, the unknowns (w h, σ h ) can be expressed as a function of λ such that (w h (λ), σ h (λ)). Then w h (λ) and σ h (λ) can be computed once λ is known. One way to interpret this, is given by the system of equations. After linearization of the possibly nonlinear problem, one retains a linear system of equations that can be written in the following form: A B R Σ F C D S W = G. (3.28) L M N Λ H This is a very general way to express the problem with block-matrices. The vectors Σ, W and Λ correspond to the unknown basis coefficients of (σ h, w h, λ h ) and the vectors F,G and H correspond

22 3. Spatial Discretization 12 to the residual, i.e., equations (3.24)-(3.26). The system of equations can be divided into two parts that are given as ( ( ) ( ( A B Σ F R = Λ (3.29) C D) W G) S) ( ) ( 1 (( ) ( ) Σ A B F R = Λ (3.30) W C D) G S) and ( ) ( ) Σ L M + NΛ = H. (3.31) W Then equation (3.30) can be inserted into equation (3.31) yielding a linear system of equations that only depends on the hybrid variable λ h : ( N ( L M ) ( A B C D ) 1 ( R S ) ) Λ = H ( L M ) ( A B C D ) 1 ( ) F. (3.32) G The block structure of the matrices allows to invert the matrices locally in an efficient manner. It has to be noted that the construction of the matrices before the system of equations is solved for Λ is more involved than for other methods. It requires solving a problem on each element. These local solves are needed to construct the system of equations and reconstruct the solution once Λ is known. Depending on the memory available for a simulation, these local solves can be stored for a faster reconstruction. More specific information about the implementation of the used framework can be found in the Master s thesis of Woopen [77]. After obtaining the vector Λ the remaining unknowns can be computed by solving equation (3.30) in an elementwise fashion. It leads to a larger number of total computations, but the required peak memory is lowered due to the smaller system when solving for the number of globally coupled unknowns. In some cases the hybridized method actually increases the number of globally coupled unknowns compared to a standard DG method. This happens if the polynomial degree p is small. Then, there are only few degrees of freedom on each element Ω k. Since the degrees of freedom of the hybrid variable λ h exist on the edges and there are usually more edges than elements, this may lead to an increased number of unknowns in total [59]. Nevertheless, hybridization is usually beneficial. The condition of the matrix resulting from the system of equations is often better than for standard DG methods leading to faster convergence. At the 2nd International Workshop on High-Order CFD Methods in Cologne, the results have shown that even for ansatz functions with low polynomial order p the hybridized method can be faster than standard DG methods.

23 4. Time Integration Methods In this section the integration methods are explained and a way of time step adaptation is discussed. These methods have been developed for ordinary differential equations (ODE) and we also stay with these type of problems in this chapter. How they can be applied to the HDG method is described afterwards in Chapter 5. First, a short introduction to the general problem setting is given. After that, the key properties convergence, consistency and stability of time integration methods are introduced briefly. Then two types of schemes, diagonally implicit Runge-Kutta methods (DIRK methods) and backward differentiation formulas (BDF), are presented and especially their stability properties are discussed. For DIRK methods a strategy for time step adaptation is shown. The information given in this section are taken from the books of Dahmen and Reusken [25], Hairer and Wanner [33], Hairer et al. [32] and Strehmel et al. [67] if nothing else is stated General Setting The methods that are used in this work were actually developed to solve a first order ordinary differential equation with given initial conditions. Such a problem can be formulated as: Problem 1 Find a function y = y(t), with t being the time variable, satisfying y (t) = g(t, y(t)), t [t 0, T ], y R n (4.1) with given initial condition y(t 0 ) = y 0. (4.2) The idea of time integration methods is to use the slope of the solution y(t) at certain points that is given by g(t, y(t)), that has to be Lipschitz continuous, to determine a solution. Although the methods have been developed for ODEs, they are also useful in the case of partial differential equations. They are often applicable to the system of equations that occurs after the spatial discretization of a PDE, not only in the case of hybridized DG methods. First we consider so-called one step methods. A very famous and also simple method is the explicit Euler method: Method 2 (Explicit Euler Method) Given a step size t = T t0 N j = 0,..., N 1 t j+1 = t j + t y j+1 = y j + tg(t j, y j ). with N N. Compute for The idea behind the method is to use the slope of the solution given by g(t, y) and construct the tangent at point (t j, y j ). Another important scheme is the implicit or backward Euler method. Method 3 (Implicit Euler Method) Given a step size t = T t0 N j = 0,..., N 1 t j+1 = t j + t y j+1 = y j + tg(t j+1, y j+1 ). (4.3) with N N. Compute for In this case the slope at point (t j+1, y j+1 ) is used. However, this point is not known beforehand, as it is actually the point that shall be computed. Therefore, the method is implicit and involves solving a, possibly nonlinear, system of equations. Although this renders the method much more expensive than the explicit method, where the solution can be directly determined from given data, implicit methods are used for the HDG method. The reason for this is on the one hand side (4.4)

24 4. Time Integration Methods 14 the stability properties of implicit methods (cf. Section 4.1.2) which allow larger time steps than explicit methods. On the other hand side the HDG method itself is implicit and leads to a set of differential algebraic equations, as already mentioned, to which explicit time integration methods cannot be applied Consistency and Convergence For the analysis of one step methods we introduce an alternative notation, i.e., y j+1 := y j + tφ g (t j, y j, t). (4.5) The function Φ g is called increment function and gives a convenient way to encode the whole method. All methods, no matter if explicit or implicit, can be written this way. We assume that the time step size t is constant with t = T t0 N, N N. It is possible to use a varying time step, what is actually our aim in the end. However, a constant time step size makes the notation much easier without loss of generality. We come back to variable time steps when we discuss the time step adaptation (cf. Section 4.2.2). For the explicit Euler method the increment function is while for the implicit Euler method it is Φ g (t j, y j, t) = g(t j, y j ) Φ g (t j, y j, t) = g(t j + t, y j + tφ g (t j, y j, t)). The time interval [t 0, T ] is discretized into a set of discrete points in time t 0 < t 1 <... < t N = T defining the temporal grid G h = {t 0,..., t N }. We define the grid function y h as the solution that is computed for a given point in time t j that means y h (t j ) = y j, j = 0,..., N. From this we can introduce the local error in two different ways as in Dahmen and Reusken [25]. It quantifies the error that occurs when computing a single time step. First, we define the general form of the local error, also called local error per unit time step, δ(t a, y a, t) = y(t a + t; t a, y a ) y h (t a + t; t a, y a ) (4.6) with (t a, y a ) being initial values in a neighborhood of the global solution and y(t; t a, y a ) is the solution at time t for the initial value problem y (t) = g(t, y), t [t a, t a + t] y(t a ) = y a. It is assumed that there exists a unique solution to this problem. Since we determine a solution for a certain set of times t j we take certain points from our grid G h. This leads to a local error of (t a, y a ) = (t j, y(t j )). δ j, t := y(t j+1 ; t j, y(t j )) y h (t j+1 ; t j, y(t j )) = y(t j+1 ) y(t j ) tφ g (t j, y(t j ), t) (4.7) which is actually the difference between the true solution y(t j+1 ) and the solution determined with the one step method assuming one has the exact value of y(t j ) at time t j. It is also possible to define the local error when doing one time step based on the approximated solution (t a, y a ) = (t j, y j ). This leads to a second definition of the local error δ j, t := y(t j+1 ; t j, y j ) y h (t j+1 ; t j, y j ) = y(t j+1 ; t j, y j ) y j tφ g (t j, y j, t). (4.8)

25 4. Time Integration Methods 15 Both definitions can be used for analysis of the time integration methods. However, the first definition is used in this section since it allows easier notation. The second one is important when introducing error estimators for time step control. With the local error we can establish the notion of consistency. A one step method is consistent iff δ(t a, y a, t) lim = 0, t a [t 0, T ] t 0 t holds true for all initial value problems (4.1) with a vector norm on R n. This can additionally be used to quantify the quality of the approximation. Definition 1 (Consistency of one step methods) A one step method is consistent of order q N with the initial value problem (4.1), if the relation δ(t a, y a, t) C t q+1, t 0 (4.9) holds true for all points (t a, y a ) in a neighborhood of the solution {(t, y(t)) t [t 0, T ]} with a constant C independent of t. Besides the local error there exists the global discretization error e h (t j ) = y(t j ) y h (t j ), j = 0,..., N. It takes into account that the local errors sum up from t 0,..., t N. Convergence is defined as e h (t j ) := max e h (t j ) = 0, t 0. (4.10) j The order of convergence can also be determined and is an important quantity that describes how fast the solution converges. Definition 2 (Convergence of one step methods) A one step method is of order q N convergent if the relation e h C t q, t 0 (4.11) holds true with a constant C independent of t. In the end the global error is of most interest because it determines the temporal error added to the solution. Nevertheless, the knowledge of the local errors is important for the time step size control and there is a useful connection between the consistency and convergence of a one step method. Theorem 1 Let g(t, y) and Φ g (t, y, t) be Lipschitz continuous in y. Then from consistency of order q follows convergence of order q. This theorem is very useful as it is sufficient to do an analysis of the error in one time step to get a conclusion about the global error Stability As already mentioned only implicit methods are used in this work. This comes from the fact that the discretized equations impose restrictions on the solution technique. One constraint is the CFL-condition introduced by Courant, Friedrichs and Lewy [23]. For DG methods a modified fashion has been introduced by Cockburn et al. [22] e l t min ( Ω k, Ω k ) f (w) n L CFL, l = 1,..., N. Here e l denotes an edge of the triangulation. It restricts the largest possible time step size for hyperbolic equations. This restriction can be loosened by using of implicit methods. On the other side, the discretized equations usually form a stiff problem. It is hard to give an exact definition what stiff actually means. Most textbooks refer to Curtiss and Hirschfelder [24] who observed that explicit methods had trouble solving a system of ordinary differential equations describing chemical reactions. The reaction rates of this system differed in several orders of magnitude. Implicit methods, in their case backward differentiation formulas, where much more successful. Therefore,

26 4. Time Integration Methods 16 they identified stiff equations as equations where certain implicit methods [...] perform better [...] than explicit ones [24]. Further analysis of stiff problems has shown that often the Jacobian J = g y of a problem and especially its eigenvalues λ determine the stiffness of a problem. To show this, we linearize the initial value problem (4.1) around a smooth solution h(t) such that it reads By substituting y(t) h(t) =: ȳ(t) we obtain y (t) = g(t, h(t)) + g (t, h(t))(y(t) h(t)) (4.12) y ȳ (t) = g (t, h(t))ȳ(t) +... = J(t)ȳ(t) (4.13) y We assume that the Jacobian is constant J(t) = J = const and truncate terms of higher order. For improved readability we additionally drop the bars. Then the problem becomes y (t) = Jy(t), y(0) = y 0. (4.14) It is obvious that the solution of this problem is given by the exponential function as y(t) = y 0 e Jt. (4.15) Now, we can apply one step methods to the problem. In the case of the explicit Euler method (4.3) the solution can be determined by y j+1 = (1 + hj)y j =: R(hJ)y j. (4.16) With R(hJ) we denote the so-called stability function. It plays an important role when analyzing the time integration methods. Under the assumption that the Jacobian is diagonalizable with eigenvectors v 1,..., v n the initial data can be written in the eigenvector basis as y 0 = n c i v i. (4.17) i=1 By using (4.16) recursively the solution y j+1 can be written in terms of the initial data and therefore can also be expressed in an eigenvector basis y j = R(hJ) j y 0 = n R( tλ i ) j c i v i (4.18) with the eigenvalues λ i of J. The problem itself now is called stiff in the case that all components decay while the speed of decay varies strongly. In the linear case this means the problem is stiff if Re(λ i ) < 0, max i,j i=1 λ i λ j 1, λ i C, i = 1,..., n (4.19) It can be seen that y j stays bounded if z = tλ i lies in the stability region S := {z C R(z) 1}. (4.20) In this case, stiff components of the solution are damped by the stability function R(z). This is hardly possible for the explicit Euler method. From equation (4.18) follows that the stability function is R(z) = 1 + z. Therefore extremely small time steps are needed such that R(z) is at least near to one. This is a problem for all explicit methods making it very hard to apply them to stiff problems. In this case implicit methods are needed. The stability function of the implicit Euler method is R(z) = 1 1+z and therefore there exist z C for which the method is stable.

27 4. Time Integration Methods 17 From the fact that we assumed the Jacobian to be diagonalizable it is also possible to decouple the linearized problem (4.14) into a set of n scalar problems. Based on this results we define a scalar test equation. It is called the Dahlquist test equation given by y (t) = λy, y 0 = 1. (4.21) We use this problem for the stability analysis of the methods. The solution to this problem is y = e λt. The eigenvalues of the linearized problem (4.14) are an important tool when investigating the stiffness of a problem. However, this is not always a sufficient condition. A lot of assumptions have been made to get to the linearized problem. Furthermore, smoothness of the solution, the integration interval and the size of the problem have also shown to influence on the stiffness of a problem [33] A- and A(α)-Stability A very important kind of stability, when dealing with stiff problems, is A-stability. A method is called A-stable if the whole negative half-plane C = {z C Re(z) 0} is part of the stability region S. This is the region where also the solution of Dahlquist s problem (4.21) is stable, meaning e λt decays. Using the definition of the stability function from the previous section a method is A-stable iff R(z) 1, z C (4.22) holds true. The explicit Euler method with stability function R(z) = 1 + z is therefore not A-stable. In fact, there does not exist any explicit one step methodwhich is A-stable as already mentioned before. Only implicit methods can be A-stable. The implicit Euler method, for example, with the stability function R(z) = 1 1+z is obviously A-stable. In some cases A-stability is a too strong requirement for a solution technique. It may be sufficient if the method is stable for most, but not all, z of the negative half plane C. Therefore, also a weaker form of stability is introduced called A(α)-stability. A method is A(α)-stable if its stability region contains a symmetric domain on the left half-plane S {z C arg(z) π < α}. (4.23) The domain is limited by two axis-symmetric lines with an inner angle of 2α. In the case of α = π 2 the stability region is the whole negative half-plane z C, meaning it is the same as A-stability. This kind of stability is especially interesting for linear multistep methods as is explained in Section 4.3. The stability regions are often displayed using a two dimensional plot. In Figure 4.1 we show such plots with the regions that have to be at least in the stability region S for A-stability (left) and A(α)-stability with α = π 4 (right) L-Stability It has been mentioned that the damping of stiff components is a desired property. A method is damping if the norm of the stability function is strictly smaller than one, R(z) < 1. It may appear that a method loses its damping property. For example, the trapezoidal rule y j+1 = y j + t 2 is an implicit one step method with stability function ( g(tj, y j ) + g(t j+1, y j+1 ) ) (4.24) R(z) = z 1 0.5z. (4.25) It is A-stable since it fulfills R(z) 1, z C, but it looses its damping property for large amplitudes because the stability function reaches the stability bound for large arguments z, i.e., lim R(z) = 1. (4.26) z

28 4. Time Integration Methods Im(z) 0 Im(z) 0 α α Re(z) (a) Stability region: A-stable Re(z) (b) Stability region: A(α)-stable Figure 4.1.: Stability region: Examples of the stability regions of an A-stable method (left) and an A(α)-stable method with α = π 4 (right) are shown. This may become a problem depending on the problem to be solved because stiff components may be damped only very slowly. Therefore an additional kind of stability is introduced. Definition 3 (L-Stability) A one step method is L-stable if it is A-stable and for the stability function the following holds true: lim R(z) = 0. (4.27) z L-stability is not a requirement as important as A-stability for solving stiff problems because A-stable methods that are not L-stable can solve many stiff problems successfully. Nevertheless, it may be useful because our solver should be applicable to a wide range of different problems. Therefore, it is possible to have problems for which L-stability is beneficial. In contrast to one step methods, multistep methods may be L-stable even if they are only A(α)-stable. A more detailed discussion on the stability of multistep methods, especially BDF methods, is given in Section Diagonally Implicit Runge-Kutta Methods A famous class of one step methods are the so-called Runge-Kutta methods (RK methods). The idea is that the initial value problem (4.1) can be solved by integrating g(t, y(t)) over time from a known point (t j, y j ) to a new time t j+1 The integral is approximated using a quadrature rule tj+1 y j+1 = y j + g(x, y(x))dx. (4.28) t j tj+1 with some weights b i and certain nodes t j g(x, y(x))dx t All Runge-Kutta methods can be written in the following form: s b i k i (4.29) i=1 k i g(x i, y(x i )), i = 1,..., q. (4.30)

29 4. Time Integration Methods 19 Method 4 (Runge-Kutta method with s stages) Given weights c i, b i, 1 i s and a i,l, 1 i, l s with step size t = T t0 N. Compute for j = 0,..., N 1: t j+1 = t j + t (4.31) s k i = g(t j + c i t, y j + t a i,l k l ), i = 1,..., s (4.32) y j+1 = y j + t l=1 s b l k l (4.33) l=1 We call s the number of stages. The weights have to be chosen such that a desired stability and consistency is achieved. A common way to write the weights is the Butcher-Tableau c 1 a 1,1 a 1,2... a 1,s c 2 a 2, c s a s,1 a s,2... a s,s b 1 b 2... b s (4.34) where all coefficients are displayed. This can also be viewed as an array of two vectors b, c and a matrix A with b = (b 1,..., b s ) T, c = (c 1,..., c s ) T, A = (a i,l ) s i,l=1 (4.35) such that the Butcher-Tableau can also be displayed as c A b T. (4.36) For explicit methods the matrix A has a lower triangular structure with all diagonal elements being zero while implicit methods have at least one non-zero element on the diagonal or upper triangular part of the matrix. In this work we consider of the special case where A has a lower triangular structure with diagonal entries being non-zero. We call these methods diagonally implicit Runge-Kutta (DIRK) methods. In fact, we focus on methods where all diagonal elements are alike, i.e., a 11 = a ii = a ss = γ 0. Then the Butcher-Tableau becomes c 1 γ c 2 a 2,1 γ c s a s,1 a s,2... γ b 1 b 2... b s (4.37) and the method is called singly diagonally implicit Runge-Kutta method (SDIRK method). An important work about these methods is from Alexander [2]. The actual naming of these methods varies depending on the sources and authors. Alexander has called his methods DIRK methods although they had the same element on the diagonal. Other authors have called these methods semi-explicit [16, 48] or semi-implicit [15, 17] Runge-Kutta methods because these methods are somewhere between the full-implicit and the explicit case. One advantage of (S)DIRK methods compared to the fully implicit methods is that they require solving one system of equations of size n n in each stage instead of solving one system of equations of size sn sn where n is the number of unknowns. Therefore, the required memory to solve the system of equations is much lower for (S)DIRK methods compared to full-implicit Runge-Kutta methods. This especially becomes important for growing problem sizes n and number of stages s where the memory needed for solving such a system of equations becomes a bottleneck. Moreover, there is the possibility to freeze the Jacobian arising in the linearized system of equations for SDIRK methods to lower the computational effort because of the constant diagonal term γ. However, we do

30 4. Time Integration Methods b i ˆbi Table 4.1.: Coefficients of Cash s SDIRK method. not exploit on this feature in this thesis. This is left open for future work. The actual application of the SDIRK methods to the hybridized DG method is discussed in detail in Chapter 5. A drawback compared to the fully implicit schemes is the lower order of the SDIRK methods. The Runge-Kutta-Gauss methods have order q = 2s while being A-stable, for example, and are fully implicit. An A-stable SDIRK method, however, has maximum order q s + 1 and for an L-stable method the order is limited to q s [33]. A huge advantage of all one step methods, that already has been mentioned, is the possibility to adapt the time step size. For this an error estimator is needed. In the case of (S)DIRK methods it is possible to embed an additional (S)DIRK method of different order into an existing method for error estimation. This way it is possible to get an error estimate with only few extra work. To refer to the stages and the order of the SDIRK methods we use the notation (s, q) which refers to a method of order q with s stages. For the case with an embedded method we write (s 1 /s 2, q 1 /q 2 ) meaning the method with s 1 stages has order q 1 and second scheme has s 2 stages and order q 2. For example, (2/3, 4/5) would refer to a SDIRK scheme with 3 stages of order 5 where an SDIRK scheme with 2 stages of order 4 is embedded. The Butcher-Tableau of such a scheme can be displayed as c 1 γ c 2 a 2,1 γ (4.38) c s a s,1 a s,2... γ b 1 b 2... b s ˆb1 ˆb2... ˆbs. We denote the coefficients of the higher order method with b i and the coefficients of the embedded method with ˆb i. From this, two different solutions can be determined y j+1 = y j + t ŷ j+1 = y j + t s b l k l, (4.39) l=1 s ˆbl k l. (4.40) The local error then can be estimated using the two solutions to adjust the time step size. Further remarks regarding the estimated error and time step size adaptation are given later in Section In this work we restrict ourselves to three different SDIRK methods. Cash has derived different formulas for embedding an error estimating method into a DIRK method [18]. We use Cash s (2/3, 2/3) SDIRK method. It is the method of lowest order in our analysis, but also has the lowest number of stages meaning it is possibly less costly for a time step than the other methods. Its coefficients are given in Table 4.1. Another approach has been used by Al-Rabeh [1]. He has constructed a (3,3) method very similar to the one of Cash and then developed a (4,4) SDIRK method enveloping the (3,3) formula. The coefficients are given in Table 4.2. The last method is a (5,4) method of Hairer and Wanner [33]. It has a (4,3) method embedded. In contrast to the other methods it has one additional stage compared to its order q meaning q = s 1. The other methods have the same order as number of stages q = s. This may lead to a method with higher amount of computational work due to the additional stage, but also may be covered if the error estimator works better than for the other methods. l=1

31 4. Time Integration Methods b i ˆbi Table 4.2.: Coefficients of Al-Rabeh s SDIRK method b i ˆbi Table 4.3.: Coefficients of Hairer s and Wanner s SDIRK method Stability As already mentioned in Section 4.1.2, stability of the time integration method is a crucial property to solve stiff problems. In general, the stability function of a Runge-Kutta method can be computed using the Butcher-Tableau in matrix vector notation (4.35). The stability function then becomes R(z) = det ( I za + zob ) T (4.41) det (I za) with I being the identity matrix and o := (1,..., 1) T is a vector of ones [33]. For SDIRK-methods the stability function only depends on the diagonal element a ii = γ. Using this knowledge, Hairer and Wanner [33] computed valid values for γ to retain a certain order and stability depending on the number of stages s. The values for γ to have an A-stable method are given in Table 4.4 and for L-stable methods are given in Table 4.5. The method of Cash has three stages (s = 3) and γ = As it is third-order consistent, it is A- and L-stable according to the Tables. The embedded formula with two stages s = 2 is only A-stable. Al-Rabeh s embedded method with three stages is A- and L-stable similar to the method of Cash. However, the higher-order method with four stages is only A-stable. Whether this may be a disadvantage for stiff problems is examined for different problems in section 7. The (5,4) SDIRK method of Hairer and Wanner with γ = 0.25 is A- and L-stable. The embedded (4,3) method, however, is not A-stable since γ < 1 3. For a better understanding, the stability regions are plotted in Figure 4.2. The x-axis represents the real part of z while the y-axis represents the imaginary part of z. White regions in the plots show where R(z) 1 holds true, meaning the SDIRK method is stable. One can see that the (3,3) method of Cash and Al-Rabeh coincide due to the same γ. The (4,3) method of Hairer and Wanner has only a small stability region. This, however, is acceptable if this method is only used as an error estimator. For the other methods it is possible to choose between the higher and lower order method as the error estimator. It depends on how important L-stability is for the problem. In this work we always use the lower order method for error estimating Time Step Adaption A key feature of one step methods we utilize in this work, is the possibility to adapt the time step. The structures of flows that have to be resolved during a simulation may change over time. In this

32 4. Time Integration Methods Im(z) 0 Im(z) Re(z) Re(z) (a) Stability region: Cash s (2,2) method (b) Stability region: Cash s (3,3) method Im(z) 0 Im(z) Re(z) Re(z) (c) Stability region: Al-Rabeh s (3,3) method (d) Stability region: Al-Rabeh s (4,4) method Im(z) 0 Im(z) Re(z) Re(z) (e) Stability region: Hairer s and Wanner s (4,3) method (f) Stability region: Hairer s and Wanner s (5,4) method Figure 4.2.: Stability region: The stability regions of all SDIRK methods. Stable regions with R(z) 1 are indicated white while unstable regions are red.

33 4. Time Integration Methods 23 q s q = s s = 1 2 γ < γ = s = 2 4 γ < γ = s = 3 3 γ γ = s = γ s = γ γ γ = Source: Textbook of Hairer and Wanner [33] Table 4.4.: Range of values for γ to retain an A-stable method. q s 1 q = s 2 s = γ γ = ± s = γ γ = s = γ γ = s = γ γ = Source: Textbook of Hairer and Wanner [33] Table 4.5.: Range of values for γ to retain an L-stable method. case also the time step can be adapted such that as few time steps as possible are computed while retaining a sufficient temporal resolution. When the number of time steps is minimized also the total simulation time can be minimized. An example is the shedding of vortices behind an obstacle. While a vortex is developing a small time step is needed. Once the vortex has separated and is transported by the flow a higher time step can be used until a new vortex develops. This will be shown as one of the test cases in Section 7.3. Now, we allow a varying time step size t j := t j+1 t j in this section. The index j is used to identify the time step size associated with the jth time step. One usually aims at solving a problem with a given accuracy while using the largest possible time step. For this the accuracy tol is specified and one requires that the error fulfills y(t ) y h (T ) tol (T t 0 ). (4.42) We have already introduced the error of one time step as the local error δ j, tj (cf. eq. (4.7)) respectively δ j, tj (cf. eq. (4.8)). As the error at t = T shall be limited, it is actually necessary to limit the accumulated local error. A reasonable estimator for the local error is given by [25] N 1 y(t ) y h (T ) δ j, tj. (4.43) If it can be ensured that the error in each time step is bounded by j=0 δ j, tj tol (t j+1 t j ) = tol t j (4.44) then it follows that the accumulated error stays in the specified bound N 1 δ j, tj j=0 N 1 j=0 tol (t j+1 t j ) = tol (T t 0 ). (4.45) Therefore, we can use the local discretization error for time step adaptation. In each time step the local error should be approximately equal to the tolerance tol t j to ensure that the error bound is satisfied and the time step is as large as possible.

34 4. Time Integration Methods 24 For the analysis, we define an error estimator s( t j ) for δ j, tj such that δ j, tj s( t j ) c t q+1 j. (4.46) Here q is the order of accuracy of the estimated error. To test if a time step is valid or not, the ratio r( t j ) := s( t) tol t j (4.47) is very important. It describes the ratio of the estimated error to the error that is allowed for a time step. In the case r( t j ) > 1 the error in the current time step is too large and it has to be recomputed with a smaller time step size t j. If r( t j ) 1 the time step is accepted. Depending on the actual value of r( t j ) the time step size t j+1 may be increased for the next time step as it is preferable to have r( t j ) close to one. Therefore, we require for a time step 1! c tq+1 new tol t new. (4.48) When we cancel t new from the denominator and multiply with the old time step size t we obtain c t q+1 ( ) q new = c tq+1 tnew. (4.49) tol t new tol t t Since our estimator shall be s( t j ) c t q+1 j s( t) tol t ( tnew t (cf. eq. (4.46)), we finally get ) q = r( t) ( tnew t ) q. (4.50) This expression still is required to be close to one. Therefore, the new time step size t new is computed as ( ) q tnew r( t) 1 t new r( t) 1 q t. (4.51) t In our case we use the embedded formula to compute two solutions y j+1 (cf. eq. (4.39)) and ŷ j+1 (cf. eq. (4.40)). The local error is estimated as δ j, tj s( t j ) := ŷ j y j = y j + t j = t j ( s ˆbl k l y j + t j l=1 s (ˆb l b l )k l. l=1 s b l k l ) l=1 (4.52) Additionally, one can use information from the solution process of the nonlinear system of equations. The number of iterations n it needed to solve the system of equations using Newton s method decreases, the better the starting value and the condition of the system of equations is. If the solution changes only slowly, then Newton s method is more likely to converge after a few steps. The opposite happens in the case of too large time steps or a rapidly changing solution. Therefore, we introduce information about the solution process into the time step adaptation. For this we define an additional safety factor α that depends on the highest number of iterations of all stages n it and the maximum allowed number of iterations for the Newton solver n it,max as: α(n it ) = 0.9 2n it,max + 1 2n it,max + n it. (4.53) If nothing else is stated we set n it,max := 10 such that 0.63 α(n it ) 0.9. Combining equation (4.51) and the safety factor (4.53) we set the new time step size using the following formula: t new := α(n it )r( t j ) 1 q tj. (4.54)

35 4. Time Integration Methods 25 a 0 a 1 a 2 a 3 a 4 BDF BDF BDF BDF Source: Textbook of Dahmen and Reusken [25] Table 4.6.: Coefficients of BDFk-methods up to order 4. Even when setting the tolerance tol carefully, it may occur that the time step size grows too large to resolve all interesting features of a solution. On the other hand it may happen that the time step size gets ridiculously small, e.g. when a boundary layer evolves. To maintain some certainty about the solution and the time needed to run a simulation we specify an upper t max and lower bound t min for the time step. In each time step it is checked if the time step size stays within these bounds t min t j t max. If this is not the case the time step size t new is rejected and set to t min or t max corresponding to the bound reached. The full time step adaptation process is also displayed in a flow chart, see Fig Backward Differentiation Formulas Another popular class of time integration methods are linear multistep methods. As the name suggests these methods approximate a solution of (4.1) based on the solution of one or more previous time steps. It is possible to use an adaptive time step for BDF methods like for one step methods [32]. However, BDF methods have no integrated error estimator such that one cannot decide whether a time step is adequate. Therefore, we only use multistep methods with a fixed time step size t = T t0 N. The general form of a k-step method can be written as Then the method is given as y j+k = Φ h (t j+k 1, y j, y j+1,..., y j+k ), j = 0,..., m k. (4.55) k a l y j+l = t l=0 k b l g(t j+l, y j+l ) (4.56) l=0 where y j+k is the value that has to determined. From this follows that a k 0 is required. The coefficients a l and b l are fixed and depend on the method. Multistep methods can be implicit (b k 0) or explicit (b k = 0). Similar to the one step methods, explicit methods have poor stability properties. Therefore, we focus on a class of implicit methods called backward differentiation formula. It is a very popular class of multistep methods for stiff problems. A short form to refer to these methods is BDFk-method where k is the number of previous steps needed for method. At the same time k is the order of consistency and convergence of the method. Method 5 (BDFk-method) Given a step size t = T t0 N with N N, coefficients a l (0 l k) and initial values y 0,..., y k 1. Compute for j = 0,..., N k t j+1 = t j + t k a l y j+l = t g(t j+k, y j+k ) l=0 (4.57) The coefficients of the BDFk-methods up to fourth order are given in Table 4.6. They are determined by polynomial interpolation using the Lagrange polynomials with the data points (t j, y j ), (t j+1, y j+1 ),..., (t j+k, y j+k ). (4.58)

36 4. Time Integration Methods 26 impl. Euler BDF2 BDF3 y 0 y 1 y 2 t 0 t 0 + t t t t t t t t Figure 4.3.: Initialization of the time integration for the BDF3 method. The resulting polynomial p k (t) then is differentiated and required to equal the right hand side of the initial value problem (4.1) such that p k(t j+k ) = g(t j+k, y j+k ) (4.59) holds true. A problem occurs for k > 1 since only initial data y 0 are given, but for the computations data at the points y j, j = 0,..., k 1 are needed. The idea is to compute the missing values using alternative methods. They have to be computed with sufficient accuracy because the order of consistency and convergence misbehaves if the error in the initial data is too large. For the BDF2 method one implicit Euler step is computed with time step size t to compute y 1. The implicit Euler method is of second order consistent for one step and therefore preserves the needed accuracy. For higher order methods another initialization procedure is needed. We sketch it for the BDF3 method (cf. Figure 4.3): 1. Determine a new time step size t that fulfills t 2 < t 3 and t = l t for an l N. 2. Compute one implicit Euler step using t. 3. Compute BDF2 steps until t 1 = t 0 + t and t 2 = t t to obtain y 1 and y 2. After the starting values y 0, y 1 and y 2 are computed the BDF3 method is used. An alternative approach is to use one step methods to compute starting values. Depending on the order of the one step method the time step size has to be adjusted, but it may yield less computational costs than initializing with lower order BDF methods. This is not investigated in this work as it is not directly obvious whether the computational work is actually reduced. The amount of computational work depends on the time step size that has to be used for initialization and error constants of the methods, for example Stability In contrast to one step methods, an additional kind of stability is needed for multistep methods such that stability and consistency lead to a convergent method. For this the test equation y (t) = 0, y(0) = 1 (4.60) is defined. Obviously, the solution is y(t) = 1. Applying a multistep method to this problem leads to the equation k a l y j+l = 0, j = 0, 1,.... (4.61) l=0 meaning that the solution y j fulfills a homogeneous difference equation. In order to determine the solution of this equation we use the ansatz y j = ξ j with j being the jth power of ξ C, ξ 0. Then, the so-called characteristic polynomial ρ(ξ) is ρ(ξ) = k a l ξ l = 0 (4.62) l=0

37 4. Time Integration Methods 27 where we have already divided the equation by ξ j. The equation is fulfilled if ξ is a root of the equation. Using all roots ξ 1,..., ξ k of ρ(ξ), the general solution of equation 4.61 can be written as y j = p 1 (j)ξ j p k(j)ξ j k (4.63) where p 1 (j),..., p k (j) are polynomials of degree m 1 with m being the multiplicity of the root. In order to obtain a stable solution we require y j to be bounded for j. This is fulfilled for ξ l 1 if all ξ l are single roots. For a multiple root ξ l the series j i ξ j l with i = 1,..., m is a solution, too. Since j i grows unbounded for i > 1 the root must fulfill ξ l < 0 in this case to obtain a bounded solution. This requirement leads to zero-stability. Definition 4 (Zero-Stability) A method is zero-stable if the root condition holds true: In case ρ(ξ l ) = 0 for a ξ l C, then ξ l 1 (4.64) and if ξ l is a multiple root then is fulfilled. ξ l < 1 (4.65) Linear one step methods are always zero-stable. Applying a one step problem to equation 4.60 and using the ansatz y j = ξ j again gives the characteristic polynomial ρ(ξ) = 1.0ξ 0 1.0ξ 1 = 0 (4.66) with only one root ξ 1 = 1. Now, with consistency and zero-stability it is possible to formulate the following theorem: Theorem 2 If a multistep method is zero-stable and consistent of order q then it is convergent. The order of convergence is also q. For the stability concerning stiff problems we use again Dahlquist s test equation (4.21). Applying a multistep method on this problem gives ( k k ) a l y j+l = tλ b l y j+l. (4.67) l=0 In contrast to zero-stability the right hand side does not vanish. The problem can be rewritten as l=0 k ( (al b l z)y j+l) = 0, z = λ t. (4.68) l=0 Now, we use the ansatz y j = ξ j again, where ξ j is still the jth power of ξ, and divide by ξ j such that we obtain the characteristic equation k ( (al b l z)ξ l) = ρ(ξ) zµ(ξ) = 0 (4.69) l=0 with ρ(ξ) as defined in equation 4.62 and µ(ξ) = k b l ξ l. (4.70) l=0 The polynomials ρ(ξ) and µ(ξ) are also called the generating polynomials of the method. As for the zero-stability, the solution can be expressed as a linear combination (4.63) of the roots ξ l, l = 1,..., k of equation (4.69). In this case the roots also depend on the complex number z such that ξ l = ξ l (z). So, the solution is, again, stable if the solution y j stays bounded for j. This leads to the same conditions for a stable method as for the zero stability. The solution is stable if the norm of all roots are less or equal than one ( ξ l (z) 1) or strictly less than one ( ξ l (z) < 1) in the case of multiple roots. From this a more general stability concept is derived than includes zero-stability:

38 4. Time Integration Methods Im(z) 0 Im(z) Re(z) Re(z) (a) Stability region: BDF1 method (b) Stability region: BDF2 method Im(z) 0 Im(z) Re(z) Re(z) (c) Stability region: BDF3 method (d) Stability region: BDF4 method Figure 4.4.: Stability region: The stability regions of BDFk methods for k = 1,..., 4. Stable regions with R(z) 1 are indicated white while unstable regions are red. BDF1 BDF2 BDF3 BDF4 α Source: Textbook of Hairer and Wanner [33] Table 4.7.: Angle α of A(α)-stable BDFk-methods up to order 4. Definition 5 (Stability of Multistep Methods) The characteristic equation of a linear multistep method ρ(ξ) zµ(ξ) = 0 (4.71) has roots ξ l depending on z C. The stability domain of the method is defined as S = {z C; for all roots ξ l 1 holds true, for all multiple roots ξ l < 1 holds true }. (4.72) For z = 0 characteristic equation of the method becomes ρ(ξ) = 0 which is actually the condition for zero-stability. Therefore, a multistep method is zero-stable if 0 S. Using this definition of stability, one can transfer the A- and A(α)-stability conditions that have been already introduced for one step methods (cf. eq. (4.22) and (4.23)) to multistep methods. A linear multistep method is A-stable if the negative complex half plane is part of the stability region C S. (4.73)

39 4. Time Integration Methods 29 The method is A(α)-stable if {z C arg(z) π α} S (4.74) holds true. The stability regions for multistep methods can be displayed in the same way as for the one step methods. In Figure 4.4 the stability regions of BDFk methods are shown for k = 1,..., 4 where red regions denote unstable values for z. Therefore, the BDF methods are only A-stable up to order q = 2 and A(α)-stable for higher orders with angles α given in Table 4.7. The angle α decreases with increasing order of the method. Therefore, BDF methods of order larger than q = 3 are rarely used. Actually, for k 7 BDF methods are not zero-stable and therefore are not convergent. The observation that multistep methods of order q > 2 are not A-stable holds true for all multistep methods, not only BDF methods. It is also known as the second Dahlquist barrier. Nevertheless, it is possible to construct multistep methods of higher order with α close to π. Therefore, A(α)-stability is an important condition for these methods. In contrast to one step methods, BDF methods are L-stable even if it is only A(α)-stable. For one step method A-stability was necessary in order to obtain an L-stable method. The demand of vanishing roots for z is fulfilled through the construction of the BDF methods [67].

40 4. Time Integration Methods 30 Initialize solver: y 0, t 0, tol, n it,max j 0 t j min { t new, T t j } t j+1 t j + t j Compute y j+1, ŷ j+1 r( t j ) ŷj y j 2 tol t j t j t new Compute new time step size α(n it ) 0.9 2n it,max + 1 2n it,max + n it t new α(n it )r( t j ) 1 q t j t new max( t new, t min ) t new min( t new, t max) no Check if r( t j ) 1 or t j = t min yes Accept time step j j + 1 Finished! yes Reached final time t j = T? no t j t new Figure 4.5.: Solution strategy with error estimator.

41 5. Time Integration for the Hybridized Discontinuous Galerkin Method In this section we discuss how the time integration methods from the previous section can be applied to the hybridized discontinuous Galerkin method. For this we study the discretized nonlinear convection-diffusion equation. In contrast to Section 3 we investigate the time-dependent case. Therefore, the time derivative of w h explicitly appears in the equations (σ h, τ h ) + (w h, τ h ) g, τ h n Ω k Ω λ h, τ h n Ω k \ Ω = 0 (5.1) ((w h ) t, ϕ h ) (f(w h ), ϕ h ) (ε σ h, ϕ h ) + f(g) n, ϕ h Ω k Ω + f(λ h ) n α(λ h w h ), ϕ h = (h, ϕ) (5.2) Ω k \ Ω σ h n σ+ h n + α(2λ h w h w+ h ), µ h + λ Γ h g, µ h 0 Γ\Γ0 = 0 (5.3) where the same ansatz spaces for the unknowns and test functions are used as in Section 3.1. In addition to boundary conditions, we need initial conditions w(x, t = t 0 ) = w 0 (x) in the timedependent case. The equations (5.1)-(5.3) define only a semi-discrete set of equations because the time derivative (w h ) t still has to be discretized. For an easier notation we rewrite the problem set in the following way: T ((w h ) t, ϕ h ) + N (σ h, w h, λ h ; τ h, ϕ h, µ h ) = 0. (5.4) The term N concentrates all terms of the equations in which no time-derivative occurs. Then T is a vector T ((w h ) t, ϕ h ) = (0, ((w h ) t, ϕ h ), 0) T (5.5) containing zeros from the first and third equation. We introduce the time index j to distinguish the solution of different times as in the previous chapter. With j + k for BDF methods and j + 1 for SDIRK methods we denote quantities at a new time for which the solution has to be computed Notes on Differential-Algebraic Equations The discretized problem (cf. equations (5.1)-(5.3)) forms a special problem set. Because the time-derivative occurs only in one of the equations it is not an ordinary differential equation, but a differential algebraic equation (DAE) [33, 67]. In our case we have the differential variable w h and algebraic variables σ h and λ h. The term algebraic emphasizes that there are no time-derivatives on these variables. Differential algebraic equations can be written in the following general form G(t, y(t), y (t)) = 0. (5.6) The underlying problem often looks similar to our problem which can be written as y (t) = g(t, y(t), z(t)) 0 = h(t, y(t), z(t)) where y denotes the differential variables and z the algebraic variables. Solving these equations incorporates (implicit) differentiation. However, numerical differentiation is unstable making these problems hard to solve. The problems become stiff. In the case of equation (5.7) we can rewrite the problem into an ordinary differential equation if the derivative (5.7) z h(t, y(t), z(t)) = h z(t, y(t), z(t)) (5.8)

42 5. Time Integration for the Hybridized Discontinuous Galerkin Method 32 exists and is invertible. In this case there exists a locally unique solution z = H(t, y(t)) following from the implicit function theorem. Then we can rewrite the problem to y (t) = g(t, y(t), H(t, y(t))) = φ(t, y(t)). (5.9) The number of differentiations to do so can be used as a measurement how complicated it is to solve the system of equations. It is called the differentiation index di. We want to make this more clear using an example from the book of Strehmel et al. [67]. The differential algebraic equation e y 1 (t) + y 2(t) + y 3 (t) = g 1 (t) y 1(t) + 2y 2 (t) + y 3(t) = g 2 (t) y 1 (t) y 2 (t) = g 3 (t) (5.10) shall be solved. We assume that all derivatives with respect to t and y exist. Rewriting this into the general form (cf. eq. (5.6)) gives G(t, y(t), y (t)) = ey 1 (t) + y 2(t) + y 3 (t) g 1 (t) y 1(t) + 2y 2 (t) + y 3(t) g 2 (t) = 0 (5.11) y 1 (t) y 2 (t) g 3 (t) with y(t) = (y 1 (t), y 2 (t), y 3 (t)) T being the vector of unknowns. In order to transform this equation into an explicit ODE (cf. eq. (5.9)) the Jacobian G y has to be regular. This is not the case for this problems because the Jacobian is given by G y = ey 1 (t) (5.12) which is not regular. However, it is possible to rewrite the problem such that the Jacobian is invertible. After one differentiation of G with respect to t the equations become G t = ey 1 (t) y 1 (t) + y 2 (t) + y 3(t) g 1(t) y 1 (t) + 2y 2(t) + y 3 (t) g 2(t) = 0. (5.13) y 1(t) y 2(t) g 3(t) From this system of equations we take the last equation in order to obtain a relationship for the derivative of y 2(t). Then combining the equation with the initial DAE (5.10) it can be written in the general form (5.6). The problem is then given by G(t, y(t), y (t)) = ey 1 (t) + y 1(t) + y 3 (t) g 1 (t) g 3(t) y 1(t) + 2y 2 (t) + y 3(t) g 2 (t) = 0 (5.14) y 1(t) y 2(t) g 3(t) where the last equation has been replaced and we have used the relation y 2(t) = y 1(t) g 3(t) in the first equation to eliminate y 2(t). This is now an implicit ODE with the Jacobian G y = ey 1 (t) (5.15) being regular such that locally a unique solutions exists. Therefore, the problem can be written as an explicit ODE (cf. eq. (5.9)). Since one differentiation of G is necessary to rewrite the DAE it has differentiation index di = 1. For the HDG method the index is di = 1. In general, one cannot expect numerical methods constructed to solve ordinary differential equations to keep their consistency and convergence properties. However, in the case of BDF methods and the used SDIRK methods the properties are retained. This mainly follows from the strong stability properties of the methods [67].

43 5. Time Integration for the Hybridized Discontinuous Galerkin Method Backward Differentiation Formulas We first show the time discretization using the implicit Euler method which is actually the BDF1 method, but could also be interpreted as an one stage DIRK method with γ = 1. Applying the implicit Euler method onto equation (5.4) leads to T (w j+1 h, ϕ h ) T (w j h, ϕ h) t j + N (σ j+1 h, w j+1 h, λ j+1 h ; τ h, ϕ h, µ h ) = 0. (5.16) We can split terms such that unknown terms appear on the left hand side and known terms on the right hand side such that the problem becomes: T (w j+1 h, ϕ h ) + t j N (σ j+1 h, w j+1 h, λ j+1 h ; τ h, ϕ h, µ h ) = T (w j h, ϕ h). (5.17) This system of equations then can be solved using Newton s method. Now, it is easy to write the formula for general BDFk methods because they only depend on one evaluation of the nonlinear term N. So, the BDFk method (4.57) applied to the hybridized discontinuous Galerkin method leads to the following system of equations: a k T (w j+k h, ϕ h ) + t j N (σ j+k h k 1, w j+k h, λ j+k h ; τ h, ϕ h, µ h ) = l=0 a l T (w j+l h, ϕ h). (5.18) It can be viewed as a modified implicit Euler step with an additional factor a k and a modified right hand side (see also [45, 62]) Singly Diagonally Implicit Runge-Kutta Methods For the s-stage DIRK methods (4.33) the time integration is a bit more involved, because there is usually more than one system of equations that has to be assembled and saved in each time step. A DIRK method applied to equation (5.4) yields T (w j+1 h, ϕ h ) = T (w j h, ϕ h) + t j s b i k i i=1 = T (w j h, ϕ h) t j s i=1 b i N (σ j,i h, wj,i h, λj,i h ; τ h, ϕ h, µ h ) (5.19) where the index j, i are the unknown intermediate solutions. They are determined by s systems of equations that have to be solved consecutively. These equations are given as T (w j,i h, ϕ h) = T (w j h, ϕ h) t j i l=1 a il N (σ j,i h, wj,i h, λj,i h ; τ h, ϕ h, µ h ) (5.20) for i = 1,..., s. When we move unknown terms to the left and known terms to the right hand side the problem T (w j,i h, ϕ h)+ t j a ii N (σ j,i h, wj,i h, λj,i h ; τ h, ϕ h, µ h ) = T (w j h, ϕ i 1 h) t j a il N (σ j,i h, wj,i h, λj,i h ; τ h, ϕ h, µ h ) l=1 (5.21) can be viewed as a Euler step with modified right hand side again. Therefore, BDF and SDIRK methods fit into the same solver framework. Only the given coefficients need to be taken into account when assembling the matrices and vectors. We note that the coefficients c i from the Butcher-Tableau do not appear explicitly in the formula. Nevertheless, they are important for the consistency of the method.

44 5. Time Integration for the Hybridized Discontinuous Galerkin Method 34 Once all intermediate solutions w j,i h are known, we we can compute the solution wj+1 h from equation (5.19). For this, no further system of equations has to be solved. Furthermore, we estimate the temporal error as ) s( j ) = t j ( s i=1 (ˆb i b i )N (σ j,i h, wj,i h, λj, h ; τ h, ϕ h, µ h ) L 2. (5.22) This can be done more efficiently than it may seem because N needs to be computed anyway to assemble the solution in equation (5.19) and for assembling the system of equations (5.20). The norm used in the estimator is defined as v L 2 = n vi 2 (5.23) for a vector v R n. i=1

45 6. Shock-Capturing In Section 2 we have already mentioned that non-smooth solutions may occur. This is an effect of the nonlinearity of the governing equations. It actually reflects processes also occurring in real world experiments where for flows with local Mach numbers larger than one, i.e., Ma > 1, shocks develop and contact discontinuities may form. Over shocks most quantities, as the density, pressure and velocity, for example, are discontinuous. Nevertheless, at shocks the Rankine-Hugoniot conditions are satisfied [42, 43]. At contact discontinuities only some of the quantities, as the density and energy, are discontinuous. This may lead to stability issues of the method, but at least leads to oscillations in the solution (Gibb s phenomenon). The approximation of non-smooth problems using numerical methods is a severe challenge. Especially for finite element methods this is problematic because the solution is approximated using polynomials. Therefore, even for discontinuous Galerkin methods, the solution is continuous on each element. Different shock capturing methods have been proposed, for example, flux limiter [42, 43, 55, 69 71] and artificial viscosity models [10, 12, 50, 52, 74]. These two methods are very popular for high-order methods. However, flux limiter have not been of great interest for hybridized DG methods by now. This mainly results from the fact that current limiters do not preserve the locality of the scheme such that it is not longer possible to use static condensation to reduce the number of unknowns. In this case the HDG method is not beneficial anymore compared to standard DG methods. Therefore, we use an artificial viscosity model in this work An Artificial Viscosity Model for the Hybridized Discontinuous Galerkin Method For many different kinds of methods, artificial viscosity models have been introduced. The idea of these models is to introduce an additional diffusive term with viscosity to smear out shocks, but there are many ways how to apply the diffusive term and how to determine the size of artificial viscosity. In this work we use a model motivated by an artificial viscosity model introduced by Nguyen and Peraire [44] especially for HDG. The Euler equations have been introduced in Section 2 in the following form: w t + f(w) = 0. (6.1) Artificial viscosity models rely on the idea that viscous (elliptic) processes lead to a smooth solution. Therefore, one introduces an additional term modeling these viscous effects such that one obtains always a smooth solution even if a shock occurs. This can be formulated as w t + (f(w) f v ( ε, w, w)) = 0 (6.2) where f v is the artificial viscous term with artificial viscosity ε. We set the artificial viscous flux to f v ( ε, w, w) := ε w. (6.3) In contrast to Nguyen and Peraire [44] we use w instead of a modified vector in which the total energy is substituted by enthalpy. It is desirable to add viscosity only at regions where actual discontinuities occur. Nguyen and Peraire suggest to use the dilatation u = u 1 x 1 + u 2 x 2 (6.4)

46 6. Shock-Capturing x ra(x) v(x) c c (ln) c (log 10 ) γp ρ (a) Smooth approximation of the ramp function. (b) Function supposed to smoothly approximate the speed sound c. Figure 6.1.: Artificial Viscosity: Approximation of the ramp function (left) and sonic speed (right). as shock sensor. This originates from physical observations. In presence of a shock the dilatation is strongly negative. Since its magnitude is a measure for the strength of the shock the artificial viscosity shall be scaled with the dilatation. In case the dilatation is non-negative no shocks are present and therefore no or at least only few viscosity shall be added. Nguyen and Peraire propose the following relation for the viscosity ( ) l u ε = ε 0 v (6.5) c where ε 0 is the bulk viscosity, l is a characteristic length and c is the speed of sound. The characteristic length is determined as the minimum of the size of an element d e and ten times the shortest distance to a wall d w l = min(d e, 10d w ). (6.6) This prevents the model from adding viscosity near walls. The function v is a smooth approximation of the ramp function { β x, x β ra(x) := (6.7) 0, x > β defined as v(x) := α ln(1 + e β x α ) (6.8) with two parameters α and β (cf. Figure 6.1(a)). In their work the authors suggest that α = 0.1 and β = 0.5 are a good choice and the parameters to adjust are ε 0 and d e. However, this means one needs to refine or coarsen the mesh in order to modify d e. Nevertheless, one may also consider to modify α and β. In the original work, the authors did not specify which logarithm they have used, but because of the exponential function occurs in the formula we suggest that the natural logarithm is the correct choice. Remarks Flows with large Mach numbers lead to stronger shocks. In presence of strong shocks the pressure P is usually very small. This leads to a very stiff sonic speed c and may affect the stability of the shock capturing scheme. Therefore, Nguyen and Peraire suggest to replace the sound speed by a smooth approximation. For this, they define the function [ ( ( c c(c) := c { log exp 2.0 c 2 1.0))]}. (6.9)

47 6. Shock-Capturing 37 Here, c is the speed of sound at free stream. Besides not specifying which logarithm is used in this formula, it turns out that this formula has to be wrong. In Figure 6.1 we plot the speed of sound and the approximated value for c = 1 and using the natural logarithm and logarithm to base 10. Instead of approximating c it is almost constant at a value near 1. Therefore, it is not feasible to use the approximation. Further, one may question the choice of the shock sensor together with the scaling function v(x). While the dilatation becomes negative for shocks it does not become negative at contact discontinuities at which the velocity is constant. Nevertheless, it may be beneficial to add artificial viscosity on purpose to improve the approximation of the discontinuity of the density. Moreover, the method may add artificial viscosity to regions where solution is smooth. In case of constant velocity or only slight acceleration of the flow the dilatation is very small u 0. Then, the scaling function becomes v(x 0) α ln(1 + e β α ) , for α = 0.1 and β = 0.5, such that artificial viscosity is added at this point. The effective amount of viscosity still depends on the bulk viscosity ε 0.

48 7. Numerical Results In this chapter we present numerical results to prove the correctness of the implementation and to test the performance of the different time integration methods. First we consider a rotating Gaussian. It is transported due to diffusion and linear convection. For this test case there exists an exact solution such that the convergence and the behavior of the error estimator regarding the tolerance tol can be observed. Next, a radial expansion wave, a test case of the International Workshop on High-Order CFD Methods [75], is simulated. Then, viscous flow around a cylinder is examined. The flow develops a periodic behavior that is compared to experimental data. Finally, we analyze the shock capturing with Sod s shock tube problem [64]. All simulations run Newton s method until n it,max = 10 iterations or the residual drops below In case the number of iterations is exceeded the time step is neglected for SDIRK methods and recomputed with a smaller time step as long t j t min. The time step sizes may vary in different orders of magnitude between each test case since we use non-dimensional governing equations and therefore non-dimensional times. Parts of these results are presented in a paper by Jaust and Schütz that has been accepted for publication in the Journal of Computers and Fluids under the title A Temporally Adaptive Hybridized Discontinuous Galerkin Method for Time-Dependent Compressible Flow [40]. Most of the results are presented using graphs of the results. In case the reader is interested in the tabulated results, these are given in appendix A Rotating Gaussian This test case describes a scalar w that is initialized according to a two dimensional Gaussian distribution on the domain Ω = [ 0.5, 0.5] [ 0.5, 0.5]. It is transported by a velocity field and at the same time damped by diffusion. The velocity field is fixed to u = ( 4x 2, 4x 1 ) T. Therefore the linear convection-diffusion equation (2.5) has to be solved. Then the convective flux is f(w) = ( 4x 2 w, 4x 1 w) T and the diffusive flux is f v (w) = ε w. The diffusivity is set to ε = 10 3 and the standard deviation σ of the Gaussian distribution is set to σ = 0.1. So, the initial condition reads w(x 1, x 2, 0) = w 0 (x 1, x 2 ) = exp ( (x 1 x 1,c ) 2 + (x 2 x 2,c ) 2 ) 2σ 2 (7.1) with (x 1,c, x 2,c ) being the center of the distribution. In our case we shift the pulse in the negative x 1 -direction such that (x 1,c, x 2,c ) = ( 0.2, 0). The simulation is run from t 0 = 0 to T = π 4. This is exactly the time needed to transport the pulse a half rotation in counter-clockwise direction. As there is diffusion, the pulse slowly damps out over time. The solution at every point in time is given by 2σ 2 w(x 1, x 2 ) = ( 2σ 2 + 4εt exp ( x 1 x 1,c ) 2 + ( x 2 x 2,c ) 2 ) 2σ 2 + 4εt (7.2) x 1 = x 1 cos(4t) + x 2 sin(4t) x 2 = x 1 sin(4t) + x 2 cos(4t) such that exact Dirichlet boundary conditions can be prescribed. This test case also has been used by Nguyen and Peraire [46] as a time-dependent case to analyze the hybridized discontinuous Galerkin method. In Figure 7.1 the Gaussian pulse for t = t 0 and t = T are displayed. One sees the radial symmetric Gaussian pulse and that it is slightly flattened at the final time.

49 7. Numerical Results 39 w w Figure 7.1.: Rotating Gaussian: The initial distribution (left) and final distribution at t = π 4 (right) of the Gaussian. Quadratic Ansatz Functions Cubic Ansatz Functions w(t ) wh(t ) L Hairer and Wanner Al-Rabeh Cash BDF2 BDF3 w(t ) wh(t ) L Hairer and Wanner Al-Rabeh Cash BDF2 BDF Time Step t Time Step t Figure 7.2.: Rotating Gaussian: Convergence of w w h L 2 refinement with p = 2 (left) and p = 3 (right). under uniform spatial and temporal Convergence Study We study the convergence of the methods on different grids. The coarsest grid has N e = = 32 elements. By uniformly refining the mesh in both spatial directions we create finer grids up to N e = = elements. The time step is set to t = π 32 on the coarsest mesh and divided by a factor of two for each mesh refinement. Additionally we use two different orders of polynomial ansatz functions, namely quadratic p = 2 and cubic p = 3 polynomials. Then the best order of convergence we can expect is q = 3 and q = 4 respectively. Therefore, the convergence of the fourth order time integration methods of Al-Rabeh and Hairer and Wanner may be limited due to the spatial error introduced when using quadratic ansatz functions. The results given in Figure 7.2 show exactly this behavior. In case of the quadratic ansatz functions the time integration methods show at most convergence up to third order. The BDF2 method shows second order convergence. Cash s SDIRK method and the BDF3 show third order convergence. The other SDIRK methods show only third order convergence due to the spatial error dominating. Actually, the fourth order methods produce almost exactly the same errors such that the lines lie almost perfectly on top of each other.

50 7. Numerical Results 40 w(t ) wh(t ) L Hairer and Wanner Al-Rabeh Cash BDF2 BDF3 Grid Error w(t ) wh(t ) L Hairer and Wanner Al-Rabeh Cash Grid Error t (a) Uniform temporal refinement tol (b) Adaptive temporal refinement Figure 7.3.: Rotating Gaussian: Error in dependence of time step size t (left) and tolerance tol (right) on a fixed mesh with N e = 512 elements and cubic ansatz functions. With cubic ansatz functions the spatial error is small enough such that the expected order of convergence of the fourth order methods can be observed as well. Nevertheless, the errors of the fourth order methods are again very similar to each other. The BDF methods and Cash s method do not benefit at all from the reduced spatial order. Their total error was dominated by the temporal error already for the quadratic ansatz functions. It is interesting to see that the BDFk methods reach their final convergence rate only from N e = 512 or more elements. The SDIRK methods seem to be much less mesh dependent in this case by reaching their rate of convergent also on coarser meshes. Furthermore, the total error of Cash s method is lower than the error of the BDF3 although both methods are of third order Time Step Adaptation The time step adaptation is analyzed by running the simulation on a grid with N e = 512 elements and cubic ansatz functions. The minimum error, the grid error, has been evaluated to be approximately w(t ) w h (T ) L2,min by using Hairer s and Wanner s SDIRK method with t = The lower and upper bounds of the time step size are set to t min = 0 and t max =. This is done to purely analyze the error estimator and the adaptation strategy. First, we perform a time study to see how the error w(t ) w h (T ) L 2 develops on this grid for the methods using the time step sizes t from the previous section. As it can be expected it depends on the order of convergence of the methods how fast the error reaches the grid error, see Figure 7.3(a). Hairer and Wanner s methods converges fastest with Al-Rabeh s method being slightly slower. The second order BDF method does not reach the grid error at all with the given time step size t. At this point it suffers from its lower rate of convergence. We run the time adaptive simulation with three different tolerances tol {10 1, 10 2, 10 3 }. It can be seen in Figure 7.3(b) that even for the loosest tolerance tol = 10 1 the errors are already close to the grid error. Furthermore, it can be seen that the temporal error is controlled by the time step adaptation. Although the methods have different orders of accuracy and error constants the errors in time of all methods are very close to each other. Even though Cash s method has a lower order than Al-Rabeh s method, the errors of both methods almost perfectly coincide. Hairer s and Wanner s SDIRK method produces a slightly lower error. It shows how well the time step adaptation works in this case. For smaller tolerances the grid error is reached for all methods. The effects of the time step adaptation can be clearly seen in the time step evolution for the different tolerances in Figure 7.4. The more restrictive the tolerance is the smaller the time steps size gets. It may even occur that the time step t 0 at the beginning is too large such that time steps are discarded and recomputed with a smaller time step size. This especially happens during

51 7. Numerical Results Hairer and Wanner Al-Rabeh Cash tj Time t (a) tol = Hairer and Wanner Al-Rabeh Cash tj Time t (b) tol = Hairer and Wanner Al-Rabeh Cash tj Time t (c) tol = Figure 7.4.: Rotating Gaussian: time step evolution for different tolerances tol on a fixed mesh with N e = 512 elements and cubic ansatz functions.

52 7. Numerical Results 42 w(t ) wh(t ) L Hairer and Wanner Al-Rabeh Cash Number of Elements Tolerance tol N e q = 3 q = Figure 7.5.: Rotating Gaussian: Error evolution on different grids with cubic ansatz functions (left). The tolerance is divided by a factor of eight or sixteen depending on the order of the method (right). the first time step with tolerances smaller than tol = This is the only point at which the solution has been recomputed. It can be avoided by choosing a smaller initial time step t 0. The safest initial time step size is t 0 = t min. After the first step the time step grows up to an almost constant value depending on the order of the method and the chosen tolerance. Cash s method always has the smallest time step size as one could expect. The method with the lowest order needs the smallest time step size to achieve the same error level as a method of higher order. The fourth order methods converge to alike time step sizes. The difference between their time step sizes results most probably from a different error constant. In some cases, the time step size of the last time step is much smaller than the previous time step. It is caused by the fact that the time step has been too large such that it has been limited to t j = T t j in order to reach the final time T exactly. As final test within this test case we study the error behavior when refining the mesh beginning from N e = 32 as in the beginning of this test case. We use cubic ansatz functions and activate the time step control. The tolerance tol for the coarsest grid is set to tol = 10 1 and adapted to the refined meshes based on the order of the SDIRK method. For Cash s method the tolerance is divided by a factor of eight for each refinement and by a factor of sixteen for the other methods. The results and actual values for the tolerance are given in Figure 7.5. In all cases the tolerance tol is strict enough such that all methods reach the grid error of the corresponding grid. Therefore, the time step adaptation performs very well. The effect of temporal errors is minimized no matter which method is used. However, it is not possible to obtain a solution on the finest mesh with Al-Rabeh s method. The time step size t 0 is successively decreased because the ratio r( t 0 ) (cf. eq. (4.47)) is always larger than one. For time step sizes t 0 < 10 6 it converges to r( t 0 ) such that the time step is never accepted.

53 7. Numerical Results 43 ρ : ρ : Figure 7.6.: Radial Expansion Wave: Density ρ on the domain at t = t 0 (left) and t = T (right) Radial Expansion Wave This test case is part of the problems that have been proposed for the first and second International Workshop on High-Order CFD Methods [75]. It describes a radial expansion wave in a quadratic domain Ω = [ 4, 4] [ 4, 4]. The flow is inviscid meaning that it is described by the Euler equations (cf. Section 2.3). Since the problem is radialsymmetric the initial conditions are expressed in terms of the radius r(x 1, x 2 ) = x x2 2. Fir this, the functions q(r, t) and a(r, t) are defined as 0, ( ( )) 0 r < r 1 1 q(r, 0) = γ 1 + tanh 0.25 (r 1), 2 2 r < 3 2 (7.3) 2 γ, r 3 2 a(r, 0) = 1 γ 1 q(r, 0) (7.4) 2 which are used to determine the initial values for the density, velocity and pressure as: ρ(r, 0) = γa(r, 0) 2 γ 1 (7.5) u 1 (x 1, x 2, 0) = x 1 r u 2 (x 1, x 2, 0) = x 2 P = r q(r, 0) (7.6) q(r, 0) (7.7) ρa(r, 0)2. (7.8) γ The fluid is at rest in the center of the domain for r(x 1, x 2 ) < 0.5 at the beginning and accelerates because of the expansion such that it gets supersonic. At the outflow the Mach number is Ma = 2 meaning that no information goes into the domain from the boundaries. Therefore the problem is solely dependent on the initial conditions. The initial density distribution and the distribution at the end of the simulation are given in Figure 7.6. The simulation runs from t 0 = 0 to T = 2 and the ratio of specific heats is set to γ = 3. It was initially supposed to be done with γ = 1.4 and T = 3; here, however, a discontinuity evolved at some time t > 2 which could be overcome by increasing γ and/or reducing the final time T. With a discontinuity in the solution it is not possible to measure the order of accuracy of the method. Similar to many real life applications this test case lacks an analytical solution. Therefore the error is measured in terms of the global entropy error. The local entropy error is defined as: ε s (x 1, x 2, t) = ln P (x 1, x 2, t) γ(ln ρ(x 1, x 2, t) ln γ). (7.9)

54 7. Numerical Results 44 The averaged local error of one element is denoted as ε s,k (t) with k being the index of the element. It is computed on each element using Gaussian quadrature. Then the global error is measured as ε s (t) L 2 = 1 N e ( ε s,k (t) N 2 ). (7.10) e Again, we use meshes that are uniformly partitioned into triangles as for the rotating Gaussian with N e = 2048 and N e = 8192 elements. The ansatz functions are quadratic to compare the BDF2 method with the SDIRK methods and cubic when comparing with the BDF3 method such that the temporal error should be dominating for the BDF methods. The time step is set to t = 0.01 on the coarse mesh and t = on the fine mesh for both BDF methods. These time steps have been identified to be a bound were the entropy error of the BDF methods hardly changes when decreasing the time step further. The SDIRK methods use the same time step for the first step t 0 as BDF methods, but may reject it due to the estimated error. On the coarsest grid with quadratic ansatz functions and N e = 2048 elements the tolerance is set to tol = 0.5 and in all other cases tol = 0.1. We emphasize that we always estimate the error of the variables in w for time step adaptation. For this test case this is the error in density, momentum and energy. However, for the convergence study the error is measured in terms of the entropy error that is derived from the quantities in w. This means we are not directly controlling the entropy error with our error estimator. The results on the coarsest mesh using quadratic ansatz functions and on the finest mesh using cubic ansatz functions are displayed in Figures 7.7 and 7.8. Results of the two remaining cases can be found in the appendix A.2. All SDIRK methods are able to follow the entropy error evolution of the BDF methods with the chosen tolerances. Only on the finest mesh with cubic ansatz functions the error of Al-Rabeh s and Cash s method are slightly higher. The total amount of work is quantified using the cumulated number of Newton steps that are needed during the simulation. Although the computational work associated with one time step is much higher for SDIRK methods than for BDFk methods it turns out that the time step adaptation in this case can overcome this drawback. Right from the beginning of the simulation the time step is increased such that the total amount of Newton steps needed by the SDIRK methods is always lower than for the BDF methods. Especially on the fine mesh with N e = 8192 the SDIRK methods perform much better. Moreover, for a constant tolerance tol = 0.1 these methods tend to converge to a time step of t j 0.15 for the fourth order methods and to t j 0.08 for Cash s method. This seems to be independent of the mesh and the polynomial order of the ansatz functions. Nevertheless, the time step evolution at the start looks different. For finer meshes and higher polynomial order the SDIRK methods increase the time step size much slower. A possible reason for this is that the solution at this point of time changes more rapidly than for later times with t > 0.2 and this is resolved better the finer the mesh is. There are some pulses in the entropy error especially for small times, but also some smoother pulses for times t > 0.2 when using cubic ansatz functions. It supports the assumption that certain effects can only be resolved when the spatial resolution is high enough. Comparing the SDIRK methods, it shows that fourth order methods are a good choice even for lower order ansatz functions. Their total amount of Newton steps is in most cases lower than for Cash s method although the methods need to solve one respectively two more (nonlinear) systems of equations in each time step. Furthermore, the embedded method for error estimating works notably well in Hairer s and Wanner s method. This is the only method that can follow the entropy error of the BDF3 method on the finest mesh (cf. Fig. 7.8(a)). Therefore, the additional stage compared to Al-Rabeh s method is beneficial at this point. Moreover, the cumulated number of Newton steps is only slightly larger than for the Al-Rabeh s method. All in all, the SDIRK methods work very well for the test case. The time step control chooses time steps such that the results are comparable between all methods. Moreover, the time step mainly depends on the chosen tolerance and the order of the method. Furthermore, the time steps are increased when possible without overshooting. In all cases the newly set time step size has been accepted. k=1

55 7. Numerical Results Quadratic Ansatz Functions, N = 2048 εs(t) L Hairer and Wanner Al-Rabeh Cash BDF Time t (a) Entropy error over time tj Hairer and Wanner Al-Rabeh Cash BDF Time t (b) Time step evolution. Cumulated Newton Steps Hairer and Wanner Al-Rabeh Cash BDF Time t (c) Cumulated Newton steps. Figure 7.7.: Radial Expansion Wave: Quadratic ansatz functions and N e = 2048 elements

56 7. Numerical Results 46 Cubic Ansatz Functions, N = 8192 εs(t) L Hairer and Wanner Al-Rabeh Cash BDF Time t (a) Entropy error over time tj Hairer and Wanner Al-Rabeh Cash BDF Time t (b) Time step evolution. Cumulated Newton Steps 1,500 1, Hairer and Wanner Al-Rabeh Cash BDF Time t (c) Cumulated Newton steps. Figure 7.8.: Radial Expansion Wave: Cubic ansatz functions and N e = 8192 elements

57 7. Numerical Results 47 Figure 7.9.: Kármán vortex street: Section of the mesh. Figure 7.10.: Kármán vortex street: Mach number distribution at two instances approximately a half period apart showing the periodic behavior of the flow Kármán Vortex Street As last a test case without shocks we investigate viscous flow around a cylinder. It is described by the Navier-Stokes equations (cf. Section 2.4). The free stream Mach number and Reynolds number are set to Ma = 0.2 and Re = 180. Under these conditions vortices shed periodically behind the cylinder. This phenomena is called Kármán vortex street. The simulations are done on a mesh with N e = 2916 elements that extends to 20 diameters away from the cylinder. A section of the mesh where the vortices can be observed is presented in Figure 7.9. The fluid flows from the left to the right according to the free stream conditions. On the cylinder surface no-slip boundary conditions are applied. On the outer boundaries characteristic far field conditions are applied such that the vortices can leave the computational domain. The flow evolves from free stream initial conditions. In this case the startup phase, when for example the boundary layer evolves, is crucial. Especially too large time steps lead to unsatisfactory results or even a stationary flow. Such behavior has been observed when investigating the same test case using BDF methods. We compare our results with the one from previous work of Schütz et al. [62], but also with the experiments of Gopinath and Jameson [31], Henderson [36] and Williamson [76]. Therefore we determine the drag coefficient c D and the Strouhal number Sr. The Strouhal number is determined from the periodic behavior of the lift coefficient c L of the cylinder. We run the simulation for all three SDIRK methods using different tolerances tol = {0.1, 0.05, 0.01}. Additionally, we set a lower and an upper bound for the time step such that t min t t max because otherwise the first time steps get extremely small while the boundary layer evolves and may grow too large after this leading to a stationary flow. To be more precise we use two different bound configuration t min = 10 3 and t max = 5 as well as t min = 10 2 and t max = 8. We refer to these configurations as (1) and (2). On each element cubic basis functions are used. In Figure 7.10 one can see the Mach number distribution of the fully developed flow computed with Hairer s and Wanner s method. The snapshots are taken such that between both images approximately half a period has elapsed. It can be seen that the images are almost symmetric. Due to the variable time step size it is hardly possible to extract two snapshot with exactly half a period between them. An excerpt of the evolution of the lift and drag coefficients for 1000 t 1200 as well as the time step evolution for 0 t 1200 of configuration (2) with tol = 0.05 are given in Figure As it has been mentioned already the flow field is very complex at the beginning because of the