On the role of Riemann solvers in Discontinuous Galerkin methods for magnetohydrodynamics

Wheatley, V., Kumar, H. and Huguenot, P. (2010) On the role of Riemann solvers in Discontinuous Galerkin methods for magnetohydrodynamics. Journal of Computational Physics, 229 3: 660-680. doi:10.1016/j.jcp.2009.10.003

Author Wheatley, V.
Kumar, H.
Huguenot, P.
Title On the role of Riemann solvers in Discontinuous Galerkin methods for magnetohydrodynamics
Journal name Journal of Computational Physics   Check publisher's open access policy
ISSN 0021-9991
Publication date 2010-02-01
Year available 2009
Sub-type Article (original research)
DOI 10.1016/j.jcp.2009.10.003
Open Access Status Not yet assessed
Volume 229
Issue 3
Start page 660
End page 680
Total pages 21
Editor Tryggvason, G
Place of publication Maryland Heights, United States
Publisher Academic Press
Language eng
Subject C1
970109 Expanding Knowledge in Engineering
0299 Other Physical Sciences
Abstract It has been claimed that the particular numerical flux used in Runge–Kutta Discontinuous Galerkin (RKDG) methods does not have a significant effect on the results of high-order simulations. We investigate this claim for the case of compressible ideal magnetohydrodynamics (MHD). We also address the role of limiting in RKDG methods. For smooth nonlinear solutions, we find that the use of a more accurate Riemann solver in third-order simulations results in lower errors and more rapid convergence. However, in the corresponding fourth-order simulations we find that varying the Riemann solver has a negligible effect on the solutions. In the vicinity of discontinuities, we find that high-order RKDG methods behave in a similar manner to the second-order method due to the use of a piecewise linear limiter. Thus, for solutions dominated by discontinuities, the choice of Riemann solver in a high-order method has similar significance to that in a second-order method. Our analysis of second-order methods indicates that the choice of Riemann solver is highly significant, with the more accurate Riemann solvers having the lowest computational effort required to obtain a given accuracy. This allows the error in fourth-order simulations of a discontinuous solution to be mitigated through the use of a more accurate Riemann solver. We demonstrate the minmod limiter is unsuitable for use in a high-order RKDG method. It tends to restrict the polynomial order of the trial space, and hence the order of accuracy of the method, even when this is not needed to maintain the TVD property of the scheme. © 2009 Elsevier Inc. All rights reserved.
Keyword Riemann solvers
Discontinuous Galerkin methods
Hyperbolic conservations-laws
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ
Additional Notes Available online 13 October 2009.

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Created: Sun, 10 Jan 2010, 10:04:56 EST