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We design ecient 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 (31, 32). 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 eciency and robustness of the GMD schemes.

Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations