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

by V. Wheatley and P. Huguenot and H. Kumar

(Report number 2009-39)

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 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.

Keywords:

@Techreport{WHK09_420,
  author = {V. Wheatley and P. Huguenot and H. Kumar},
  title = {On the role of Riemann solvers in discontinuous Galerkin methods for magnetohydrodynamics},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-39.pdf },
  year = {2009}
}

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