Communications in Applied Mathematics and Computational Science

On the origin of divergence errors in MHD simulations and consequences for numerical schemes

Friedemann Kemm

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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.

Article information

Commun. Appl. Math. Comput. Sci., Volume 8, Number 1 (2013), 1-38.

Received: 20 October 2010
Revised: 14 May 2012
Accepted: 10 December 2012
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 76W05: Magnetohydrodynamics and electrohydrodynamics 39A12: Discrete version of topics in analysis 35L45: Initial value problems for first-order hyperbolic systems 35L65: Conservation laws 35L80: Degenerate hyperbolic equations
Secondary: 35N10: Overdetermined systems with variable coefficients 65M06: Finite difference methods 39A70: Difference operators [See also 47B39] 65Z05: Applications to physics

involutions constraint magnetohydrodynamics plasma physics Maxwell equations divergence curl operator scheme finite differences finite volume method resonance hyperbolic PDE compressible flow


Kemm, Friedemann. On the origin of divergence errors in MHD simulations and consequences for numerical schemes. Commun. Appl. Math. Comput. Sci. 8 (2013), no. 1, 1--38. doi:10.2140/camcos.2013.8.1.

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