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

多维守恒定律比一维守恒定律拥有更多的现象，例如具有涡旋的湍流和非平凡的定态。同时，计算资源是有限的，并且对于多维模拟来说，改进是非常昂贵的。目前可用的数值方法只能通过过度的网格细化来捕捉多维现象。这是因为它们添加了植根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，后者破坏了解决方案。本文提出了一种新的混合有限元-有限体积方法，该方法通过局部演化的连续数据来实现上绕，而不是解决黎曼问题。它的自由度是单元平均值/矩和位于单元界面的点值，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，以主动通量的名义，并已被证明在粗网格上取得较好的结果，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，有源通量法可用于线性或一维问题。本项目旨在发展我们的多维守恒律系统的新方法，特别是全多维欧拉方程，并分析其结构保存特性。该项目将与Wasilij Barsukow /法国波尔多大学密切合作。

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.