有关有限差分高精度格式两个应用问题的讨论

Discussions on two problems in applications of high-order finite difference schemes

激波装配法结合有限差分高精度格式是发展全场一致高精度算法的一种途径。在对流场内的间断全部进行装配后,对高精度格式的应用研究成为发展本算法的主要研究问题。本文将有限差分的高精度格式应用于贴体坐标系时发现,对均匀流场,高精度格式因不满足几何守恒律而产生的数值误差比一阶迎风格式大,初步分析认为是由于高精度格式所用的模板比一阶格式更宽,涉及的网格点数更多,从而引入了更多的误差。而作者提出的基于离散等价方程的相容性算法可消除这一误差。此外,本文在利用激波捕捉法求解正方形均匀网格上的正激波运动问题时发现因通量分裂格式的使用,在激波处会产生随着特征线传播的非物理波动,这一波动在激波与网格不完全匹配时表现为多维波动相互干扰的虚假"数值湍流"现象,高精度格式的高分辨率特性使得这一现象更加明显。这是因为激波捕捉法假设激波为空间连续函数,用于包含激波的流场时必然得到数值解表示的过渡区,导致可信度评估困难,使用激波装配法可以避免这一问题。

One way to achieve high degree fidelity in the whole flow field is by developing shock-fitting methods combined with high-order finite difference schemes.After all the discontinuities in shock-embedded compressible fluid flow are assembled,the application of the high-order finite difference schemes becomes the main problem which we need to consider.It has been found that the finite difference schemes always make numerical errors under curvilinear coordinate system due to the unfulfilled geometric conservation law.Moreover,we found that the errors of high-order schemes are greater than that of first-order upwind scheme for uniform flow.The reason given in this paper is that the stencils of high-order schemes involve more mesh points than that of first-order scheme.The source-coupled algorithm based on discrete equivalence equations proposed by the authors can eliminate the errors.Besides,the problem of normal shock’s motion on uniform grid is also discussed in this paper.Because of the flux splitting,two waves can be created after the moving normal shock.These two waves travel at two different characteristic velocities,and can lead to numerical turbulence when the shock does not exactly match to grid points.Since the shock-capture methods assume that shock is spatial continuous,transition regions represented by the numerical solution leads to difficulties in the credibility evaluation.Shock-fitting methods can avoid this problem.