Evaluation of generalized curvilinear coordinate transformations applied to high-accuracy finite-difference schemes

1998 ◽  
Author(s):  
R. Hixon ◽  
S.-H. Shih ◽  
T. Dong ◽  
R. Mankbadi
Keyword(s):  
Finite Difference ◽  
High Accuracy ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Coordinate Transformations ◽  
Curvilinear Coordinate

Effect of Geometric Conservation Law on Improving Spatial Accuracy for Finite Difference Schemes on Two-Dimensional Nonsmooth Grids

2015 ◽  
Vol 18 (3) ◽  
pp. 673-706
Author(s):  
Meiliang Mao ◽  
Huajun Zhu ◽  
Xiaogang Deng ◽  
Yaobing Min ◽  
Huayong Liu
Keyword(s):  
Finite Difference ◽  
Conservation Law ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Spatial Discretization ◽  
Curvilinear Coordinate ◽  
Discretization Errors ◽  
The Impact ◽  
Geometric Conservation Law

AbstractIt 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.

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