High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers

TL;DR

This paper introduces a high-order, asymptotic preserving finite difference WENO scheme with divergence-free constraints for MHD equations, effective across all sonic Mach numbers, including the incompressible limit.

Contribution

It develops a novel high-order semi-implicit WENO scheme with divergence-free constraints that is asymptotic preserving and accurate in both compressible and incompressible MHD regimes.

Findings

The scheme is formally proven to be asymptotic preserving.

Numerical experiments confirm the scheme's accuracy and divergence-free properties.

The method effectively captures shocks and is efficient in the incompressible limit.

Abstract

In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergence-free finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number $\varepsilon$ ranging from $0$ to $\mathcal{O}(1)$. High-order accuracy in time is obtained by SI implicit-explicit Runge-Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially non-oscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit.