Volume 13, Issue 5
A Combination of High-OrdCompact Finite Difference Schemes and a Splitting Method that Preserves Accuracy for the Multi-Dimensional Burgers' Equation

Shengfeng Wang, Xiaohua Zhang, Julian Koellermeier & Daobin Ji

Adv. Appl. Math. Mech., 13 (2021), pp. 1261-1292.

Published online: 2021-06

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  • Abstract

A class of high-order compact finite difference schemes combined with a splitting method that preserves accuracy are presented for numerical solutions of the multi-dimensional Burgers' equation. Firstly, the implicit high-order compact difference scheme is used to discretize Burgers' equation by non-linear weights that are required to be calculated at each time stage. Secondly, the sixth-order compact difference scheme in space and the fourth-order Runge-Kutta in time are applied to solve the 1D Burgers' equation. Meanwhile a linear stability analysis indicates the scheme is conditionally stable. Thirdly, the 2D and 3D Burgers' equations are divided into 1D subsystems by the splitting method, then these sub-equations' spatial terms are discretized by the fourth-order compact difference scheme, whereas the time discretizations are unchanged. The analysises of stability and accuracy of the splitting method are given to prove the accuracy of splitting without a significant loss. Finally, the accuracy and reliability of the proposed method are tested by comparing our experimental results with others selected from the available literature. It is shown that the new method has high-resolution properties and can effectively calculate Burgers' equation at large Reynolds number.

  • Keywords

Multi-dimensional Burgers' equation, high-order compact difference scheme, splitting method.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-13-1261, author = {Shengfeng Wang , and Xiaohua Zhang , and Koellermeier , Julian and Daobin Ji , }, title = {A Combination of High-OrdCompact Finite Difference Schemes and a Splitting Method that Preserves Accuracy for the Multi-Dimensional Burgers' Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {5}, pages = {1261--1292}, abstract = {

A class of high-order compact finite difference schemes combined with a splitting method that preserves accuracy are presented for numerical solutions of the multi-dimensional Burgers' equation. Firstly, the implicit high-order compact difference scheme is used to discretize Burgers' equation by non-linear weights that are required to be calculated at each time stage. Secondly, the sixth-order compact difference scheme in space and the fourth-order Runge-Kutta in time are applied to solve the 1D Burgers' equation. Meanwhile a linear stability analysis indicates the scheme is conditionally stable. Thirdly, the 2D and 3D Burgers' equations are divided into 1D subsystems by the splitting method, then these sub-equations' spatial terms are discretized by the fourth-order compact difference scheme, whereas the time discretizations are unchanged. The analysises of stability and accuracy of the splitting method are given to prove the accuracy of splitting without a significant loss. Finally, the accuracy and reliability of the proposed method are tested by comparing our experimental results with others selected from the available literature. It is shown that the new method has high-resolution properties and can effectively calculate Burgers' equation at large Reynolds number.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0277}, url = {http://global-sci.org/intro/article_detail/aamm/19261.html} }
TY - JOUR T1 - A Combination of High-OrdCompact Finite Difference Schemes and a Splitting Method that Preserves Accuracy for the Multi-Dimensional Burgers' Equation AU - Shengfeng Wang , AU - Xiaohua Zhang , AU - Koellermeier , Julian AU - Daobin Ji , JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1261 EP - 1292 PY - 2021 DA - 2021/06 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0277 UR - https://global-sci.org/intro/article_detail/aamm/19261.html KW - Multi-dimensional Burgers' equation, high-order compact difference scheme, splitting method. AB -

A class of high-order compact finite difference schemes combined with a splitting method that preserves accuracy are presented for numerical solutions of the multi-dimensional Burgers' equation. Firstly, the implicit high-order compact difference scheme is used to discretize Burgers' equation by non-linear weights that are required to be calculated at each time stage. Secondly, the sixth-order compact difference scheme in space and the fourth-order Runge-Kutta in time are applied to solve the 1D Burgers' equation. Meanwhile a linear stability analysis indicates the scheme is conditionally stable. Thirdly, the 2D and 3D Burgers' equations are divided into 1D subsystems by the splitting method, then these sub-equations' spatial terms are discretized by the fourth-order compact difference scheme, whereas the time discretizations are unchanged. The analysises of stability and accuracy of the splitting method are given to prove the accuracy of splitting without a significant loss. Finally, the accuracy and reliability of the proposed method are tested by comparing our experimental results with others selected from the available literature. It is shown that the new method has high-resolution properties and can effectively calculate Burgers' equation at large Reynolds number.

Shengfeng Wang, Xiaohua Zhang, Julian Koellermeier & Daobin Ji. (1970). A Combination of High-OrdCompact Finite Difference Schemes and a Splitting Method that Preserves Accuracy for the Multi-Dimensional Burgers' Equation. Advances in Applied Mathematics and Mechanics. 13 (5). 1261-1292. doi:10.4208/aamm.OA-2020-0277
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