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Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Published online by Cambridge University Press:  20 August 2015

Franz Georg Fuchs ,
Andrew D. McMurry ,
Siddhartha Mishra ,
Nils Henrik Risebro  and
Knut Waagan
Franz Georg Fuchs*
Affiliation:
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern N-0316 Oslo, Norway
Andrew D. McMurry*
Affiliation:
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern N-0316 Oslo, Norway
Siddhartha Mishra*
Affiliation:
Seminar for Applied Mathematics, D-Math, ETH Zürich, HG G. 57.2, Rämistrasse 101, Zürich-8092, Switzerland
Nils Henrik Risebro*
Affiliation:
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern N-0316 Oslo, Norway
Knut Waagan*
Affiliation:
Center for Scientific Computation and Mathematical Modeling, The University of Maryland, CSCAMM 4146, CSIC Building #406, Paint Branch Drive College Park, MD 20742-3289, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

Keywords

35L6574S1065M12Conservation lawsMHDdivergence constraintGodunov-Powell source termsupwinded source termshigh-order schemes
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 2 , February 2011 , pp. 324 - 362
Copyright
Copyright © Global Science Press Limited 2011

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