Arbitrary High-Order Discontinuous Galerkin Schemes for the Magnetohydrodynamic Equations

Authors:
Arne Taube;Michael Dumbser;Dinshaw S. Balsara;Claus-Dieter Munz
Affiliations:
Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Stuttgart, Germany 70569;Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Stuttgart, Germany 70569;Department of Physics, University of Notre Dame, Notre Dame, USA 46556;Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Stuttgart, Germany 70569
Venue:
Journal of Scientific Computing
Year:
2007

Citing 8
Cited 4

Extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics

Journal of Computational Physics
Notes on the eigensystem of magnetohydrodynamics

SIAM Journal on Applied Mathematics
Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws

Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems

Journal of Computational Physics
ADER: Arbitrary High Order Godunov Approach

Journal of Scientific Computing
Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case

Journal of Computational Physics
ADER schemes for three-dimensional non-linear hyperbolic systems

Journal of Computational Physics
Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes

Journal of Scientific Computing

A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

Journal of Computational Physics
Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics

Journal of Computational Physics
WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions

Journal of Computational Physics
Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - Speed comparisons with Runge-Kutta methods

Journal of Computational Physics

Quantified Score

Hi-index	0.02

Visualization

Abstract

In this paper, we propose a discontinuous Galerkin scheme with arbitrary order of accuracy in space and time for the magnetohydrodynamic equations. It is based on the Arbitrary order using DERivatives (ADER) methodology: the high order time approximation is obtained by a Taylor expansion in time. In this expansion all the time derivatives are replaced by space derivatives via the Cauchy-Kovalevskaya procedure. We propose an efficient algorithm of the Cauchy-Kovalevskaya procedure in the case of the three-dimensional magneto-hydrodynamic (MHD) equations. Parallel to the time derivatives of the conservative variables the time derivatives of the fluxes are calculated. This enables the analytic time integration of the volume integral as well as that of the surface integral of the fluxes through the grid cell interfaces which occur in the discrete equations. At the cell interfaces the fluxes and all their derivatives may jump. Following the finite volume ADER approach the break up of all these jumps into the different waves are taken into account to get proper values of the fluxes at the grid cell interfaces. The approach under considerations is directly based on the expansion of the flux in time in which the leading order term may be any numerical flux calculation for the MHD-equation. Numerical convergence results for these equations up to 7th order of accuracy in space and time are shown.