Efficient High Accuracy Simulation for Euler and MHD Flow in Complex Geometries

最近对科学和工程的相当大的兴趣围绕着能够以高水平的精度模拟流动。有限体积格式(FV格式)是此类流动模拟最常用的格式之一。对于复杂几何，特别是基于非结构网格的标准FV格式是二阶精度的。然而，当需要长时间或远距离地模拟高度非定常现象时，二阶精度可能被证明是非常不足的。本提案是美国和德国研究小组之间的合作，旨在通过使用基于不连续Galerkin(DG)的空间和时间任意精度的算法来改善这种情况。即使在扭曲的非结构网格上，这些格式也具有很高的精度。第一个资助期的一个重要方面是改善高阶DG格式的激波捕捉性能。除了基于欧拉方程的流体流动模拟外，我们的兴趣还扩展到与电磁现象相互作用的流动模拟上。这里，额外的要求必须通过数值方法来满足--磁场的无发散特性和电荷守恒。在这项提议中，遵循两种策略：美国小组的特殊空间离散化方法和德国小组的一般后验散度清理策略。所开发的测试问题和方法的应用涵盖了空气动力学以及天体物理和空间物理中的典型激波问题。发散修正也将用于脉冲等离子体动态推进器的模拟。

A considerable amount of recent interest in science and engineering revolves around being able to simulate flows with a high level of accuracy. Finite volume schemes (FV-schemes) are amongst the most well-used schemes for such flow simulations. The standard FV-schemes for complex geometries especially based on unstructured grids are second order accurate. However, second order accuracy can be shown to be very inadequate, when the simulation of highly unsteady phenomena is required for long times or over long distances. The present proposal, which is a collaboration between research groups in the USA and Germany, seeks to improve this situation by employing Discontinuous Galerkin (DG) based algorithms of arbitrary accuracy in space and time. These schemes give the high order accuracy even on distorted unstructured grids. An important aspect of the first funding period is to improve the shock-capturing properties for high order DG schemes.Beside the simulation of fluid flow based on the Euler equation our interest also extends to the simulation of flows interacting with electro-magnetic phenomena. Here, additional requirements have to be fulfilled by the numerical method - the divergence-free property for the magnetic field and the charge conservation. In this proposal two strategies are being followed: a special space discretization method by the US group and a general a posteriori divergence cleaning strategy by the German group. The test problems and the applications of the methods developed cover typical shock problems in aerodynamics as well as in astrophysics and space physics. The divergence corrections will also be used in the simulation of pulsed plasma dynamic thrusters.