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

这项为期三个月的联合项目提案的主要目的是建立一个数值方案，不仅在准确性、稳定性和一致性方面产生数学上可靠的结果，而且在物理上也产生可靠的解决方案。关键思想是利用嘉当的移动标架方法（[29，30，31，32]），最终目标是增强线性化二维磁流体动力学浅水微分方程的现有数值方案，作为迈向更复杂应用的第一步。与其他方法相比，不变的数值方案必须从一开始就构建，移动框架的方法创造了使现有的数值方案不变的可能性，继承了基础微分方程所需的相同对称性。后者的离散化模型将使用移动标架的方法来定义，最后，新的数值方案模型将在现有的开源数值代码中实现，并与标准离散化方案在稳定性和守恒性方面进行比较。

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.