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

Part of:Hyperbolic equations and systemsNumerical approximation and computational geometryPartial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  14 September 2015

Meiliang Mao,
Huajun Zhu,
Xiaogang Deng,
Yaobing Min and
Huayong Liu
Meiliang Mao*
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. ChinaComputational Aerodynamics Institute, CARDC, Mianyang, 621000, P.R. China
Huajun Zhu
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. China
Xiaogang Deng
Affiliation:
National University of Defense Technology, Changsha, Hunan, 410073, P.R. China
Yaobing Min
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. China
Huayong Liu
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. China
*
*Corresponding author. Email addresses: mlmao@cardc.cn (M. Mao), hjzhu@skla.cardc.cn (H. Zhu), xgdeng2000@vip.sina.com (X. Deng), ybmin@skla.cardc.cn (Y. Min), hyliu@skla.carde.cn (H. Liu)

Abstract

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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.

Keywords

Geometric conservation lawfinite difference schemespatial discretization errornonsmooth gridshigh-order accuracy

MSC classification

Secondary:35L04: Initial-boundary value problems for first-order hyperbolic equations65D25: Numerical differentiation65D18: Computer graphics, image analysis, and computational geometry65M06: Finite difference methods65M15: Error bounds

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015, pp. 673 - 706
Copyright
Copyright © Global-Science Press 2015 

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