A structure-preserving compact high-order method for multi-dimensional hyperbolic conservation laws

多维守恒律比一维守恒律具有更多的现象，例如具有涡和非平凡定态的湍流。同时，计算资源是有限的，并且对于多维模拟而言，细化特别昂贵。目前可用的数值方法只能捕捉多维现象过度网格细化。这是因为他们增加了根植于一维思维的数值扩散，也因为通量是使用黎曼问题计算的。对于亚音速流，后者破坏了解.我们提出了一种新的混合有限元-有限体积方法，通过局部演化连续数据来实现迎风，而不是解决黎曼问题。其自由度是单元平均值/力矩和位于单元界面处的点值，后者在相邻单元之间共享。平均值的演化是保守的，这意味着该方法能够收敛到弱解。最低（3）阶方法可以在文献中找到，名称为主动通量，并已被证明在粗网格上实现上级结果，因为它是结构保持。在我们的新方法中，高阶矩的演化使其具有任意高阶精度。目前，主动通量法可用于线性或一维问题.本项目旨在发展我们的新方法，用于多维守恒律系统，特别是全多维Euler方程，并分析其结构保持性。该项目将与法国波尔多大学Wasilij Barsukow密切合作完成。

Multi-dimensional conservation laws possess many more phenomena than their one-dimensional counterparts, for example turbulent flow with vortices and non-trivial stationary states. At the same time, computational resources are limited, and refinement is particularly costly for multi-dimensional simulations. Currently available numerical methods are only able to capture multi-dimensional phenomena upon excessive grid refinement. This is because they add numerical diffusion that is rooted in one- dimensional thinking, and also because the fluxes are computed using Riemann problems. For subsonic flow, the latter spoil the solution. We propose a new hybrid finite element – finite volume method that achieves upwinding by locally evolving continuous data, instead of solving a Riemann problem. Its degrees of freedom are cell averages/moments and point values located at cell interfaces, the lat- ter being shared between adjacent cells. The evolution of the averages is conservative, which means that the method is able to converge to the weak solution. The lowest (3rd) order method can be found in the literature under the name of Active Flux and has been shown to achieve superior results on coarse grids, because it is structure preserving. In our new method, the evolution of higher moments makes it arbitrarily high-order accurate. Currently, Active Flux methods are available for linear, or one- dimensional problems. This project aims at developing our new method for multi-dimensional systems of conservation laws, in particular the full multi-dimensional Euler equations and analyzing its structure preservation properties. This project will be done in close cooperation with Wasilij Barsukow / University of Bordeaux, France.