A multi well-balanced scheme for the shallow water MHD system with topography

The shallow water magnetohydrodynamic system involves different families of physically relevant steady states. In this paper, we design a well-balanced numerical scheme for the shallow water magnetohydrodynamic system with topography, that resolves exactly a large family of steady states. It is obtained by a generalized hydrostatic reconstruction algorithm involving the magnetic field. It is positive in height and semidiscrete entropy satisfying, which ensures the robustness of the scheme.