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含磁场一维浅水方程，构建了不变有限差分格式。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。通过数值求解若干典型问题，对比了存在磁场与无磁场（标准浅水模型）的情形。同时讨论了拉格朗日坐标系下差分格式的不变性与所得数值解的能量守恒性。