Commun. Comput. Phys., 9 (2011), pp. 324-362.


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

Franz Georg Fuchs 1*, Andrew D. McMurry 1, Siddhartha Mishra 2, Nils Henrik Risebro 1, Knut Waagan 3

1 Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern N-0316 Oslo, Norway.
2 Seminar for Applied Mathematics, D-Math, ETH Zurich, HG G. 57.2, Ramistrasse 101, Zurich-8092, Switzerland.
3 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.

Received 17 November 2009; Accepted (in revised version) 7 May 2010
Available online 27 August 2010
doi:10.4208/cicp.171109.070510a

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.

AMS subject classifications: 35L65, 74S10, 65M12
Key words: Conservation laws, MHD, divergence constraint, Godunov-Powell source terms, upwinded source terms, high-order schemes.

*Corresponding author.
Email: franzf@math.uio.no (F. G. Fuchs), a.d.mcmurry@ifi.uio.no (A. D. Mcmurry), smishra@sam.math.ethz.ch (S. Mishra), nilshr@math.uio.no (N. H. Risebro), knutwa@ucar.edu (K. Waagan)
 

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