Effect of Geometric Conservation Law on Improving Spatial Accuracy for Finite Difference Schemes on Two-Dimensional Nonsmooth Grids
Effect of Geometric Conservation Law on Improving Spatial Accuracy for Finite Difference Schemes on Two-Dimensional Nonsmooth Grids
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DOI：
10.4208/cicp.250614.060215a
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发表时间：
2015-09
影响因子：
3.7
通讯作者：
M. Mao;Huajun Zhu;Xiaogang Deng;Min Yaobing;Huayong Liu
中科院分区：
文献类型：
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作者：
M. Mao;Huajun Zhu;Xiaogang Deng;Min Yaobing;Huayong Liu
众所周知，网格不连续性对有限差分方案（FDS）的性能有显着影响。几何守恒定律（GCL）对于 FDS 在减少数值振荡和确保曲线坐标系中的自由流保持方面非常重要。目前尚不清楚 GCL 在有限差分法中的工作原理以及 GCL 误差如何影响空间离散化误差，特别是在非光滑网格中。本文提出了一种方法来分析网格不连续性对 GCL 误差和空间离散化误差的影响。违反 GCL 会导致 GCL 错误，该错误取决于网格平滑度、网格度量方法和有限差分算子。因此，空间离散误差中有更多的源项。分析表明，非充分光滑网格的空间离散精度由网格的不连续阶数决定，通过GCL可以逼近高一阶。对于足够光滑的网格，空间离散精度由FDS的阶数决定，满足GCL的FDS可以获得较小的空间离散误差。采用二阶和四阶FDS进行了数值试验，验证了理论结果。
It is well known that grid discontinuities have significant impact on the performance of finite difference schemes (FDSs). The geometric conservation law (GCL) is very important for FDSs on reducing numerical oscillations and ensuring free-stream preservation in curvilinear coordinate system. It is not quite clear how GCL works in finite difference method and how GCL errors affect spatial discretization errors especially in nonsmooth grids. In this paper, a method is developed to analyze the impact of grid discontinuities on the GCL errors and spatial discretization errors. A violation of GCL cause GCL errors which depend on grid smoothness, grid metrics method and finite difference operators. As a result there are more source terms in spatial discretization errors. The analysis shows that the spatial discretization accuracy on non-sufficiently smooth grids is determined by the discontinuity order of grids and can approach one higher order by following GCL. For sufficiently smooth grids, the spatial discretization accuracy is determined by the order of FDSs and FDSs satisfying the GCL can obtain smaller spatial discretization errors. Numerical tests have been done by the second-order and fourth-order FDSs to verify the theoretical results.