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An Efficient Implementation of the Divergence Free Constraint in a Discontinuous Galerkin Method for Magnetohydrodynamics on Unstructured Meshes

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  07 February 2017

Christian Klingenberg ,
Frank Pörner  and
Yinhua Xia
Christian Klingenberg*
Affiliation:
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
Frank Pörner*
Affiliation:
Department of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
Yinhua Xia*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
*
*Corresponding author.Email addresses:klingen@mathematik.uni-wuerzburg.de (C. Klingenberg), frank.poerner@mathematik.uni-wuerzburg.de (F. Pörner), yhxia@ustc.edu.cn (Y. Xia)
*Corresponding author.Email addresses:klingen@mathematik.uni-wuerzburg.de (C. Klingenberg), frank.poerner@mathematik.uni-wuerzburg.de (F. Pörner), yhxia@ustc.edu.cn (Y. Xia)
*Corresponding author.Email addresses:klingen@mathematik.uni-wuerzburg.de (C. Klingenberg), frank.poerner@mathematik.uni-wuerzburg.de (F. Pörner), yhxia@ustc.edu.cn (Y. Xia)
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Abstract

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint (∇·B=0) of its magnetic field B. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements [19], and another approach of the divergence cleaning technique given by Dedner et al. [15]. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

Keywords

Ideal magnetohydrodynamics equationsdiscontinuous Galerkin methoddivergence free constraintlocally divergence free projectiondivergence free cleaning technique

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M20: Method of lines 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35L65: Conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 423 - 442
Copyright
Copyright © Global-Science Press 2017 

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