Geodesic
Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations
ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 661-680

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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.

DOI : 10.1051/m2an/2011059
Classification : 65M06, 35L65
Keywords: multidimensional evolution equations, magnetohydrodynamics, constraint transport, central difference schemes, potential-based fluxes 
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     title = {Constraint preserving schemes using potential-based fluxes. {III.} {Genuinely} multi-dimensional schemes for {MHD} equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {661--680},
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Mishra, Siddhartha; Tadmor, Eitan. Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations. ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 661-680. doi: 10.1051/m2an/2011059

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