Application of the "Method of Moving Frames" to the magnetohydrodynamic shallow water equations - Conservation Properties and Robustness

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The main aim of the proposal for a three-month joint project is to construct a numerical scheme which not only produces mathematically reliable results regarding accuracy, stability and consistency, but also physically reliable solutions. The key idea is to make use of Cartan's method of moving frames ([29, 30, 31, 32]) with the final goal of enhancing existing numerical schemes for the linearized two-dimensional magnetohydrodynamic shallow water differential equations, as a first step towards more complex applications. In contrast to other methods where invariant numerical schemes have to be constructed from the outset, the method of moving frames creates the possibility to invariantize already existing numerical schemes, to inherit the same symmetries as required by the underlying differential equations. A discretization model for the latter will be de- veloped using the method of moving frames, and finally, the new model of numerical scheme will be implemented in an existing open-source numerical code and compared to a standard discretization schemes with respect to stability and conservation properties.