High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

In this paper, we develop a high-order semi-implicit (SI) structure-preserving finite difference weighted essentially nonoscillatory (WENO) scheme for magnetohydrodynamic (MHD) equations with a gravitational source. The proposed scheme is well-balanced for magnetic steady states, divergence-free for the magnetic field, conservative in the high Mach regime, and exhibits asymptotic preserving (AP) and asymptotically accurate (AA) properties in the incompressible low sonic Mach regime. The constrained transport method is applied to maintain a discrete divergence-free magnetic field. The sonic Mach number \(\varepsilon \) $\epsilon$ ranging from 0 to \(\mathcal {O}(1)\) $\mathcal{O}(1)$ is taken into account for all Mach flows. One of the crucial and novel ingredients is the addition of an evolution equation for the perturbation of potential temperature as an auxiliary equation to the conservative MHD system. This addition ensures a correct asymptotic low sonic Mach limit and helps to effectively capture shocks in the compressible high Mach regime. A well-balanced finite difference WENO scheme is designed for conservative variables of the resulting system. With stiffly accurate SI implicit-explicit Runge–Kutta time discretizations, the AP and AA properties are formally proven. Numerical experiments are provided to validate the effectiveness and structure-preserving properties of the proposed scheme.