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.

Comment: 22 pages, 6 figures. This version (v7) has been specially prepared for publication in the Open Commun. Nonlinear Math. Phys., Special Issue in Memory of Professor Decio Levi. Notice that in the previous version (v5) of the paper an appendix was added with an informal discussion on stability issues. By the decision of the authors, this appendix was not included in v6 and v7

• 76B10 - Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
• 76M20 - Finite difference methods applied to problems in fluid mechanics
• 76M60 - Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
• 76W05 - Magnetohydrodynamics and electrohydrodynamics