Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations∗,∗∗
Centre of Mathematics for Applications (CMA), University of
Oslo, P.O. Box
2 Department of Mathematics, Center of Scientific Computation and Mathematical Modeling (CSCAMM), Institute for Physical sciences and Technology (IPST), University of Maryland, 20741-4015 MD, Maryland, USA
Received: 27 September 2009
Revised: 23 May 2010
We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.
Mathematics Subject Classification: 65M06 / 35L65
Key words: Multidimensional evolution equations / magnetohydrodynamics / constraint transport / central difference schemes / potential-based fluxes
The work on this paper was started when S.M. visited the Center of Scientific Computation and Mathematical Modeling (CSCAMM) and he thanks CSCAMM and all its members for the excellent hospitality and facilities. E. T. Research was supported in part by NSF grants DMS07-07949, DMS10-08397 and ONR grant N00014-091-0385. He thanks the Centre for Advanced Study at the Norwegian Academy of Science and Letters, for hosting him as part of its international research program on Nonlinear PDEs during the academic year 2008-09.
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