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

最近在科学和工程领域有相当多的兴趣围绕着能够以高精确度模拟流动。有限体积格式（fv格式）是此类流动模拟中最常用的格式之一。对于复杂几何图形，特别是基于非结构网格的标准fv格式具有二阶精度。然而，当需要长时间或长距离模拟高度不稳定现象时，二阶精度可能是非常不足的。目前的提议是由美国和德国的研究小组合作提出的，旨在通过采用基于空间和时间任意精度的不连续伽辽金（DG）算法来改善这种情况。这些方案即使在扭曲的非结构化网格上也具有较高的阶精度。第一个资助期的一个重要方面是提高高阶DG方案的冲击捕获性能。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。这里，额外的要求必须通过数值方法来满足——磁场的无散度特性和电荷守恒。本方案采用了两种策略：美国组的特殊空间离散化方法和德国组的一般后验散度清理策略。所开发方法的测试问题和应用涵盖了空气动力学以及天体物理学和空间物理学中的典型冲击问题。发散校正也将用于脉冲等离子体动力推力器的模拟。

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.