Volume 12, Issue 2
High-Order Finite Difference Schemes Based on Symmetric Conservative Metric Method: Decomposition, Geometric Meaning and Connection with Finite Volume Schemes

Adv. Appl. Math. Mech., 12 (2020), pp. 436-479.

Published online: 2020-01

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• Abstract

High-order finite difference schemes (FDSs) based on symmetric conservative metric method (SCMM) are investigated. Firstly, the decomposition and geometric meaning of the discrete metrics and Jacobian based on SCMM are proposed. Then, high-order central FDS based on SCMM is proved to be a weighted summation of second-order finite difference schemes (FDSs). Each second-order FDS has the same vectorized surfaces and cell volume as a second-order finite volume scheme (FVS), and the cell volume is uniquely determined by the vectorized surfaces. Moreover, the decomposition and connection with FVSs are also discussed for general high-order FDSs. SCMM can be applied for high-order weighted compact nonlinear scheme (WCNS). Numerical experiments show superiority of high-order WCNS based on SCMM in stability, accuracy and ability to compute flows around complex geometries. The results in this paper may to some extent explain why high-order FDSs based on SCMM can solve problems with complex geometries and may give some guidance in constructing high-order FDSs on curvilinear coordinates.

65D18, 68U05, 68U07

hjzhu@skla.cardc.cn (Huajun Zhu)

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@Article{AAMM-12-436, author = {Deng , XiaogangZhu , HuajunMin , YaobingLiu , HuayongMao , Meiliang and Wang , Guangxue}, title = {High-Order Finite Difference Schemes Based on Symmetric Conservative Metric Method: Decomposition, Geometric Meaning and Connection with Finite Volume Schemes}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {2}, pages = {436--479}, abstract = {

High-order finite difference schemes (FDSs) based on symmetric conservative metric method (SCMM) are investigated. Firstly, the decomposition and geometric meaning of the discrete metrics and Jacobian based on SCMM are proposed. Then, high-order central FDS based on SCMM is proved to be a weighted summation of second-order finite difference schemes (FDSs). Each second-order FDS has the same vectorized surfaces and cell volume as a second-order finite volume scheme (FVS), and the cell volume is uniquely determined by the vectorized surfaces. Moreover, the decomposition and connection with FVSs are also discussed for general high-order FDSs. SCMM can be applied for high-order weighted compact nonlinear scheme (WCNS). Numerical experiments show superiority of high-order WCNS based on SCMM in stability, accuracy and ability to compute flows around complex geometries. The results in this paper may to some extent explain why high-order FDSs based on SCMM can solve problems with complex geometries and may give some guidance in constructing high-order FDSs on curvilinear coordinates.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0243}, url = {http://global-sci.org/intro/article_detail/aamm/13629.html} }
TY - JOUR T1 - High-Order Finite Difference Schemes Based on Symmetric Conservative Metric Method: Decomposition, Geometric Meaning and Connection with Finite Volume Schemes AU - Deng , Xiaogang AU - Zhu , Huajun AU - Min , Yaobing AU - Liu , Huayong AU - Mao , Meiliang AU - Wang , Guangxue JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 436 EP - 479 PY - 2020 DA - 2020/01 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2017-0243 UR - https://global-sci.org/intro/article_detail/aamm/13629.html KW - High-order, finite difference schemes, symmetric conservative metric method, finite volume schemes, complex geometries. AB -

High-order finite difference schemes (FDSs) based on symmetric conservative metric method (SCMM) are investigated. Firstly, the decomposition and geometric meaning of the discrete metrics and Jacobian based on SCMM are proposed. Then, high-order central FDS based on SCMM is proved to be a weighted summation of second-order finite difference schemes (FDSs). Each second-order FDS has the same vectorized surfaces and cell volume as a second-order finite volume scheme (FVS), and the cell volume is uniquely determined by the vectorized surfaces. Moreover, the decomposition and connection with FVSs are also discussed for general high-order FDSs. SCMM can be applied for high-order weighted compact nonlinear scheme (WCNS). Numerical experiments show superiority of high-order WCNS based on SCMM in stability, accuracy and ability to compute flows around complex geometries. The results in this paper may to some extent explain why high-order FDSs based on SCMM can solve problems with complex geometries and may give some guidance in constructing high-order FDSs on curvilinear coordinates.

Xiaogang Deng, Huajun Zhu, Yaobing Min, Huayong Liu, Meiliang Mao & Guangxue Wang. (2020). High-Order Finite Difference Schemes Based on Symmetric Conservative Metric Method: Decomposition, Geometric Meaning and Connection with Finite Volume Schemes. Advances in Applied Mathematics and Mechanics. 12 (2). 436-479. doi:10.4208/aamm.OA-2017-0243
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