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Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws

Published online by Cambridge University Press:  03 June 2015

Andreas Hiltebrand  and
Siddhartha Mishra
Andreas Hiltebrand*
Affiliation:
Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG J 49, Zürich -8092, Switzerland
Siddhartha Mishra
Affiliation:
Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG G 57.2, Zürich -8092, Switzerland; Center of Mathematics for Applications (CMA), University of Oslo, P. O. Box -1053, Blindern, Oslo-0316, Norway
*
*Corresponding author. Email addresses: andreas.hiltebrand@sam.math.ethz.ch (A. Hiltebrand), smishra@sam.math.ethz.ch (S. Mishra)
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Abstract

An entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper to approximate hyperbolic systems of conservation laws. This numerical scheme involves the solution of a very large nonlinear system of algebraic equations, by a Newton-Krylov method, at every time step. In this paper, we design efficient preconditioners for the large, non-symmetric linear system, that needs to be solved at every Newton step. Two sets of preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are designed. Fourier analysis of the preconditioners reveals their robustness and a large number of numerical experiments are presented to illustrate the gain in efficiency that results from preconditioning. The resulting method is employed to compute approximate solutions of the compressible Euler equations, even for very high CFL numbers.

Keywords

65M6035L65Preconditioningspace-time DGentropy stablesystems of conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 5 , May 2015 , pp. 1320 - 1359
Copyright
Copyright © Global-Science Press 2015 

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