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The Discontinuous Galerkin Method with Explicit Runge-Kutta Time Integration for Hyperbolic and Parabolic Systems with Source Terms

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The Discontinuous Galerkin Method with Explicit Runge-Kutta Time Integration for Hyperbolic and Parabolic Systems with Source Terms The discontinuous Galerkin method with explicit Runge-Kutta time integration for hyperbolic and parabolic systems with source terms Martien A. Hulsen MEMT 19 Delft University of Technology Laboratory for Aero and Hydrodynamics Rotterdamseweg 145 2628 AL Delft The Netherlands November 19, 1991 Copyright c 1991 Fakulteit der Werktuigbouwkunde en Maritieme Techniek, Technische Universiteit Delft. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted by any means, electronic, mechanical, pho- tocopying, recording, or otherwise, without the prior written permission of the author. ISBN 90-370-0061-4 ii Summary In this report numerical methods are described that are capable of accurately solving hyperbolic and parabolic problems. The straightforward extension of these methods to include non-conservative systems is a very attractive feature, which is of high importance for the simulation of time-dependent viscoelastic fluid flows. The methods are based on the discontinuous Galerkin method and a mixed method for the space discretization and on Runge-Kutta methods for the discretization in time. In chapter 1 some background on the problem is given with respect to vis- coelastic fluid flow. However, the equations are given in a formal form, which makes the described methods applicable to a very wide range of physical prob- lems. In chapter 2 the discontinuous Galerkin method is introduced for an ordinary differential equation in order to get some feeling for this non-standard finite element method. In chapter 3 the method is extended to multidimensional unsteady hyper- bolic systems. Conservative and non-conservative systems are treated indepen- dently. A basic concept in the method is the characteristic decomposition which also determines the initial and boundary conditions that should be specified. The method is also compared to other finite element formulations. In chapter 4 some examples of hyperbolic problems are solved by the method, in particular a one- and two-dimensional scalar transport problem, one-di- mensional scalar transport with relaxation and a wave problem with two vari- ables and reflecting boundary conditions. In chapter 5 the method is extended to include diffusion (parabolic systems) by using a mixed formulation that introduces gradients as independent variables. Examples are given for the one-dimensional convection-diffusion equation. In chapter 6 some conclusions are given. iii Contents List of symbols vii 1 Introduction 1 2 The discontinuous Galerkin method for an ordinary differential equation 4 2.1 Introduction .............................. 4 2.2 The DG method ........................... 4 3 The DG method for hyperbolic systems 10 3.1 Preliminaries ............................. 10 3.2 Conservative systems ......................... 10 3.3 Non-conservative systems ...................... 11 3.4 Characteristic decomposition and splitting ............. 13 3.5 Inversion of the mass matrix ..................... 17 3.6 Initial and boundary conditions ................... 18 3.7 Time integration ........................... 19 3.8 Comparison with other finite element formulations ........ 20 4 Examples of hyperbolic problems 24 4.1 One dimensional scalar transport .................. 24 4.1.1 Equations ........................... 24 4.1.2 Stability ............................ 26 4.1.3 Practical results ....................... 28 4.2 Two-dimensional scalar transport .................. 33 4.2.1 Equations ........................... 35 4.2.2 Stability ............................ 36 4.2.3 Practical results ....................... 37 4.3 One-dimensional transport of a scalar with relaxation ...... 44 4.3.1 Equations ........................... 44 4.3.2 Stability ............................ 44 4.3.3 Practical results ....................... 50 4.4 A hyperbolic system ......................... 50 4.4.1 Equations ........................... 50 4.4.2 Stability ............................ 52 4.4.3 Practical results ....................... 53 5 The DG method combined with a mixed method for parabolic systems 57 5.1 A mixed formulation ......................... 57 5.2 One-dimensional convection-diffusion equation .......... 59 5.2.1 Equations ........................... 59 5.2.2 Stability ............................ 61 5.3 Practical results ............................ 63 v 6 Conclusions 67 References 68 vi List of symbols Only symbols that are used in more than one chapter are listed. The first appearance of a symbol is indicated by the equation number in parenthesis or by the page number. Roman symbols First appearance Ai coefficient matrix (1.1) C¯ (z) amplification factor (2.27) c wave speed (4.1) F (n) flux vector in the direction of ~n (3.2) ˜ F i flux vector in the xi-direction (1.2) f˜(u) source vector (1.1) ˜ f˜1 stability factor for convection terms (4.18) f2 stability factor for relaxation terms (4.44) G right-hand side (3.35) ˜ h, hj element size (4.13) Ij subinterval j 5 ` degree of the polynomial interpolation 5 m number of components in u 1 M (km), M mass matrix˜ (3.38) N ¯ number of subintervals 5 ne number of elements 10 nv number of variables 29 P`(Ij ) space of polynomials up to the degree ` 5 Sj coefficient matrix (4.10) t¯ time (1.1) [u]j jump of u at point xj (2.23) U discretized vector of unknowns (3.35) ˜ U j element unknowns of interval Ij (4.6) ˜ ∗ U j vector of unknowns (4.10) u˜ vector of unknown variables (1.1) ˜ ua, ub boundary values of u (2.2) uh, uh approximate functions of u and u 5, 10 V ˜ function space˜ (3.5) (e) Vh approximation space 10 wi characteristic variables 13 ~x position vector 1 x, xi co-ordinates (2.1), (1.1) Greek symbols α upwind parameter (2.17) ∆t time step (3.57) νij , ν diffusion coefficient matrices (1.4), (1.5) θ¯ ¯ relaxation time (4.40) φ(k) base function (3.36) Ω domain (1.1) vii Ωe element domain 10 ∂Ωe boundary of an element (3.2) viii Chapter 1 Introduction The simulation of viscoelastic fluid flow has become an active research area during the last fifteen years (Crochet 1989). The majority of articles contain simulations of creeping steady flow, motivated by the importance of industrial processing of polymer melts. For polymer solutions however, it has become clear that inertia can have major influence on the flow behaviour (Joseph 1990). Simulations that include inertia to show these major effects have only recently beginning to appear (Hulsen 1990; Delvaux & Crochet 1990a, 1990b; Hu & Joseph 1990). The problems are still assumed to be steady. At higher flow rates the flow becomes unsteady or even chaotic. In order to study this behaviour time-dependent equations have to be solved. Phelan, Malone & Winter (1989) showed that allowing a weak compressibil- ity into the equations of viscoelastic flow makes these equations hyperbolic (see also Joseph 1990). In that case the equations can formally be written in the following form d ∂u X ∂u ˜ + Ai(u) ˜ + f(u)=0 in (0,T) × Ω, (1.1) ∂t i=1 ¯ ˜ ∂xi ˜ ˜ ˜ d T where Ω is a bounded subdomain of R , d =1, 2or3,u =(u1,u2,...,um) , T ˜ f(u)=(f1,f2,...,fm) is a source term and the matrices Ai(u), i =1,...,d ˜ ˜ Pd ¯ ˜ are such that any linear combination i=1 ξiAi(u)hasm real eigenvalues and a complete set of eigenvectors. A system having¯ ˜ the latter property is called hyperbolic (Whitham 1974). An important application of hyperbolic systems is in gas dynamics. The equations (conservation of mass, momentum and energy), are usually written in the so-called conservation form d ∂u X ∂ ˜ + F i(u) + f(u)=0, (1.2) ∂t i=1 ∂xi ˜ ˜ ˜ ˜ ˜ where u is the vector of conserved variables (specific mass, momentum and ˜ energy) and F i, i =1,...,d are the flux vectors with respect to all co-ordinate directions. Comparing˜ (1.2)with(1.1)weseethat1 ∂Fi Ai(u)= ˜ ,i=1,...,d. (1.3) ¯ ˜ ∂u ˜ The majority of numerical methods for hyperbolic systems have been developed with equation (1.2) in mind, including the possibility of shocks, i.e. discontinu- ous solutions satisfying the conservation laws in integral form (Whitham 1974). 1 We have assumed that Ai(u), F i(u), and f(u) are functions of u only. An explicit depen- dence on ~x presents no difficulty¯ ˜ in˜ the˜ following˜ ˜ as long as the dependence˜ is smooth. 1 Our main application will be visco-elastic fluid flow. Although of course mass, momentum and energy are still conserved, the full system including the stresses cannot be written in conservation form (Hulsen 1986). Whether discontinuous solutions are a physical reality for viscoelastic flows is far from a resolved prob- lem and we will assume that for the general non-conservative system (1.1)the solutions are smooth. To solve the equations numerically, Phelan, Malone & Winter (1989) apply a finite difference method that was originally developed for the Euler equa- tions but does not require a conservative form. Unfortunately their method is only a first-order upwinding method, which is known to be much too diffusive. Higher-order methods have been developed (see for example, Hirsch 1990), but typically within the frame of conservative systems. These schemes achieve to be of higher order by considering the variables in multiple neighbouring cells. A different approach towards higher-order methods is followed by Cockburn & Shu (1989). They use the discontinuous Galerkin method with higher-order polynomials. Although their work is still within the framework of conservative hyperbolic systems, we will show in this report that it is easily adaptable to non- conservative systems if we assume that the solutions are smooth. Advantages of this finite element approach are the flexibility in discretization of domains with complex boundaries and the easy implementation of boundary conditions. The method seems well suited for viscoelastic flow simulations. We should note, however, that the equations for viscoelastic flow in the usual form are not fully hyperbolic because a. The flow is assumed to be incompressible, i.e. the compression wave speed is assumed to be ∞. Only the introduction of a weak (artificial) compressibility resolves this.
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