Vol. 8, No. 1, 2013

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
On the origin of divergence errors in MHD simulations and consequences for numerical schemes

Friedemann Kemm

Vol. 8 (2013), No. 1, 1–38
Abstract

This paper investigates the origin of divergence errors in MHD simulations. For that purpose, we introduce the concept of discrete involutions for discretized conservation laws. This is done in analogue to the concept of involutions for hyperbolic conservation laws, introduced by Dafermos. By exploring the connection between discrete involutions and resonance, especially for constrained transport like MHD, we identify the lack of positive central viscosity and the assumption of one-dimensional physics in the calculation of intercell fluxes as the main sources of divergence errors. As an example of the consequences for numerical schemes, we give a hint how to modify Roe-type schemes in order to decrease the divergence errors considerably and, thus, stabilize the scheme.

Keywords
involutions, constraint, magnetohydrodynamics, plasma physics, Maxwell equations, divergence, curl, operator scheme, finite differences, finite volume method, resonance, hyperbolic PDE, compressible flow
Mathematical Subject Classification 2010
Primary: 76W05, 39A12, 35L45, 35L65, 35L80
Secondary: 35N10, 65M06, 39A70, 65Z05
Milestones
Received: 20 October 2010
Revised: 14 May 2012
Accepted: 10 December 2012
Published: 13 March 2013
Authors
Friedemann Kemm
Institute for Applied Mathematics and Scientific Computing
Brandenburg University of Technology Cottbus
Platz der Deutschen Einheit 1
D-03046 Cottbus
Germany