Effect of Geometric Conservation Law on Improving Spatial Accuracy for Finite Difference Schemes on Two-Dimensional Nonsmooth Grids (Communications in Computational Physics, 2018)
It is well known that grid discontinuities have significant impact on the performanceof finite difference schemes (FDSs). The geometric conservation law (GCL) isvery important for FDSs on reducing numerical oscillations and ensuring free-streampreservation in curvilinear coordinate system. It is not quite clear how GCL works infinite difference method and how GCL errors affect spatial discretization errors especiallyin nonsmooth grids. In this paper, a method is developed to analyze the impactof grid discontinuities on the GCL errors and spatial discretization errors. A violationof GCL causes GCL errors which depends on grid smoothness, grid metrics methodand finite difference operators. As a result, there are more source terms in spatial discretizationerrors. The analysis shows that the spatial discretization accuracy on non-sufficientlysmooth grids is determined by the discontinuity order of grids and canapproach one higher order by following GCL. For sufficiently smooth grids, the spatialdiscretization accuracy is determined by the order of FDSs and FDSs satisfying theGCL can obtain smaller spatial discretization errors. Numerical tests have been doneby the second-order and fourth-order FDSs to verify the theoretical results.