We propose a second-order accurate semi-implicit and well-balanced finite volume scheme for the equations of ideal magnetohydrodynamics (MHD) including gravitational source terms. The scheme treats all terms associated with the acoustic pressure implicitly while keeping the remaining terms part of the explicit sub-system. This semi-implicit approach makes the method particularly well suited for problems in the low Mach regime. We combine the semi-implicit scheme with the deviation well-balancing technique and prove that it maintains equilibrium solutions for the magnetohydrostatic case up to rounding errors. In order to preserve the divergence-free property of the magnetic field enforced by the solenoidal constraint, we incorporate a constrained transport method in the semi-implicit framework. Second order of accuracy is achieved by means of a standard spatial reconstruction technique with total variation diminishing (TVD) property, and by an asymptotic preserving (AP) time stepping algorithm built upon the implicit-explicit (IMEX) Runge-Kutta time integrators. Numerical tests in the low Mach regime and near magnetohydrostatic equilibria support the low Mach and well-balanced properties of the numerical method.

翻译：我们提出了一种针对理想磁流体力学（MHD）方程（包含引力源项）的二阶精确半隐式及平衡保持型有限体积格式。该格式对所有与声压相关的项进行隐式处理，同时将剩余项保留为显式子系统的组成部分。这种半隐式方法使得该格式特别适用于低马赫数条件下的问题。我们将半隐式格式与偏差平衡保持技术相结合，并证明其能够维持磁流体静力学平衡解（精度可达舍入误差水平）。为满足螺线管约束所强加的磁场无散特性，我们在半隐式框架中融入了约束输运方法。通过采用具有总变差递减（TVD）特性的标准空间重构技术，以及基于隐式-显式（IMEX）龙格-库塔时间积分器的渐近保持（AP）时间推进算法，实现了二阶精度。低马赫数区域及磁流体静力学平衡附近的数值实验验证了该数值方法在低马赫数特性与平衡保持特性方面的有效性。