Volume 17, Issue 5
Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws

Andreas Hiltebrand & Siddhartha Mishra

Commun. Comput. Phys., 17 (2015), pp. 1320-1359.

Published online: 2018-04

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  • Abstract

An entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper [20] 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, nonsymmetric 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.

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@Article{CiCP-17-1320, author = {}, title = {Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws}, journal = {Communications in Computational Physics}, year = {2018}, volume = {17}, number = {5}, pages = {1320--1359}, abstract = {

An entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper [20] 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, nonsymmetric 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.140214.271114a}, url = {http://global-sci.org/intro/article_detail/cicp/11014.html} }
TY - JOUR T1 - Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws JO - Communications in Computational Physics VL - 5 SP - 1320 EP - 1359 PY - 2018 DA - 2018/04 SN - 17 DO - http://doi.org/10.4208/cicp.140214.271114a UR - https://global-sci.org/intro/article_detail/cicp/11014.html KW - AB -

An entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper [20] 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, nonsymmetric 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.

Andreas Hiltebrand & Siddhartha Mishra. (2020). Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws. Communications in Computational Physics. 17 (5). 1320-1359. doi:10.4208/cicp.140214.271114a
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