  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.
