This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations. They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity for the given convex entropy function and preserving the steady states of the lake at rest (with zero magnetic field). The key is to match both discretizations for the fluxes and the non-flat river bed bottom and Janhunen source terms, and to find the affordable EC fluxes of the second-order EC schemes. Next, by using the second-order EC schemes as building block, high-order accurate well-balanced semi-discrete EC schemes are proposed. Then, the high-order accurate well-balanced semi-discrete ES schemes are derived by adding a suitable dissipation term to the EC scheme with the WENO reconstruction of the scaled entropy variables in order to suppress the numerical oscillations of the EC schemes. After that, the semi-discrete schemes are integrated in time by using the high-order strong stability preserving explicit Runge-Kutta schemes to obtain the fully-discrete high-order well-balanced schemes. The ES property of the Lax-Friedrichs flux is also proved and then the positivity-preserving ES schemes are studied by using the positivity-preserving flux limiters. Finally, extensive numerical tests are conducted to validate the accuracy, the well-balanced, ES and positivity-preserving properties, and the ability to capture discontinuities of our schemes.

本文为浅水磁流体动力学（SWMHD）方程开发了高阶精确熵稳定（ES）有限差分方案。它们建立在用Janhunen源项修正SWMHD方程的数值近似的基础上。首先，构建了二阶精确平衡良好的半离散熵保守（EC）方案，该方案满足给定凸熵函数的熵标识并保留静止（零磁场）下的湖泊稳态。关键是要使通量的离散化与非平坦河床底部和Janhunen源项相匹配，并找到二阶EC方案可承受的EC通量。接下来，通过使用二阶EC方案作为构建块，提出了高阶精确平衡良好的半离散EC方案。然后，通过用比例熵变量的WENO重构对EC方案添加合适的耗散项，可以推导高阶精确均衡的半离散ES方案，从而抑制EC方案的数值振荡。之后，通过使用高阶强稳定性，保留显式Runge-Kutta方案，将半离散方案及时集成，以获得全离散高阶均衡方案。还证明了Lax-Friedrichs磁通的ES性质，然后使用保正通量限制器研究了保正ES方案。最后，进行了广泛的数值测试，以验证准确性，均衡性，ES和阳性保留特性，以及捕获我们方案不连续性的能力。