求解多维欧拉方程的二阶旋转输运格式

A second-order rotational upwind transport scheme for solving multi-dimensional compressible Euler equations

提出了一个基于旋转近似Riemann求解器的二阶精度迎风型有限体积方法。不同于“网格相关”(Grid-aligned)的有限体积方法或者维数分裂的有限差分方法,本格式在求解Riemann问题时不依赖网格的方向,而是沿具有一定的物理意义的两个方向(称为迎风方向)。我们发现当迎风方向取为控制体界面两侧速度差矢量方向(及与之正交的方向)时,该格式能够完全消除基于Riemann求解器的通量差分裂格式存在的激波不稳定或者所谓“红斑”(carbuncle)现象。为了提高格式的时间和空间精度,我们通过在控制体界面处求解线化的广义Riemann问题的方法体现输运过程对通量计算的影响。这种方案,使得我们有可能以此为基础,构造真正多维的有限体积型迎风格式。

A second-order rotational upwind transport scheme for solving multi-dimensional compressible Euler equations is presented in this paper. In this scheme, the numerical fluxes are evaluated by solving two Generalized Riemarm Problems （GRP） in two upwind directions which include the direction of velocity-difference vector and the direction which is perpendicular to it. The GRPs are solved based on the Roe＇ linearization technique which takes the transportation effects into consideration. It is found that the scheme can eliminate the shock instabilities or carbuncle phenomena of the flux-difference splitting type schemes completely.