High order structure-preserving finite difference WENO schemes for MHD equations with gravitation in all sonic Mach numbers. (English) Zbl 1535.65129
Summary: 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 \(\varepsilon\) ranging from 0 to \(\mathcal{O}(1)\) 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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L04 | Numerical methods for stiff equations |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 76H05 | Transonic flows |
| 80A19 | Diffusive and convective heat and mass transfer, heat flow |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
MHD with gravitation; all Mach; asymptotic preserving; asymptotically accurate; well-balanced; divergence-freeReferences:
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