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的主要目的是建立一个数值方案，该方案不仅能产生关于精度、稳定性和一致性的数学上可靠的结果，而且还能产生物理上可靠的解决方案。关键思想是利用Cartan的运动框架方法（[29,30,31,32]），最终目标是增强线性化二维磁流体动力浅水微分方程的现有数值格式，作为迈向更复杂应用的第一步。与其他必须从一开始就构造不变数值格式的方法相比，移动框架的方法创造了使现有数值格式不变的可能性，从而继承了基本微分方程所要求的相同对称性。后者的离散化模型将使用移动框架的方法开发，最后，数值格式的新模型将在现有的开源数值代码中实现，并与标准离散化格式在稳定性和守恒性质方面进行比较。

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.