Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.

翻译：针对不同底地形条件下存在磁场的一维浅水方程，构建了不变性有限差分格式。基于作者近期完成的群分类结果，获得了原始微分模型守恒律的有限差分模拟量。数值计算了若干典型问题，并对存在磁场与无磁场（标准浅水模型）的情形进行了对比。同时讨论了拉格朗日坐标系下差分格式的不变性及数值解的能量保持特性。