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Volume 42, Issue 5
Monotonicity Corrections for Nine-Point Scheme of Diffusion Equations

Wang Kong, Zhenying Hong, Guangwei Yuan & Zhiqiang Sheng

DOI: 10.4208/jcm.2303-m2022-0139

J. Comp. Math., 42 (2024), pp. 1305-1327.

Published online: 2024-07

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  • Abstract

In this paper, we present a nonlinear correction technique to modify the nine-point scheme proposed in [SIAM J. Sci. Comput., 30:3 (2008), 1341-1361] such that the resulted scheme preserves the positivity. We first express the flux by the cell-centered unknowns and edge unknowns based on the stencil of the nine-point scheme. Then, we use a nonlinear combination technique to get a monotone scheme. In order to obtain a cell-centered finite volume scheme, we need to use the cell-centered unknowns to locally approximate the auxiliary unknowns. We present a new method to approximate the auxiliary unknowns by using the idea of an improved multi-points flux approximation. The numerical results show that the new proposed scheme is robust, can handle some distorted grids that some existing finite volume schemes could not handle, and has higher numerical accuracy than some existing positivity-preserving finite volume schemes.

  • Keywords

Monotonicity corrections, Diffusion equation, Improved MPFA, Distorted meshes.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1305, author = {Kong , WangHong , ZhenyingYuan , Guangwei and Sheng , Zhiqiang}, title = {Monotonicity Corrections for Nine-Point Scheme of Diffusion Equations}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {5}, pages = {1305--1327}, abstract = {

In this paper, we present a nonlinear correction technique to modify the nine-point scheme proposed in [SIAM J. Sci. Comput., 30:3 (2008), 1341-1361] such that the resulted scheme preserves the positivity. We first express the flux by the cell-centered unknowns and edge unknowns based on the stencil of the nine-point scheme. Then, we use a nonlinear combination technique to get a monotone scheme. In order to obtain a cell-centered finite volume scheme, we need to use the cell-centered unknowns to locally approximate the auxiliary unknowns. We present a new method to approximate the auxiliary unknowns by using the idea of an improved multi-points flux approximation. The numerical results show that the new proposed scheme is robust, can handle some distorted grids that some existing finite volume schemes could not handle, and has higher numerical accuracy than some existing positivity-preserving finite volume schemes.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2303-m2022-0139}, url = {http://global-sci.org/intro/article_detail/jcm/23279.html} }
TY - JOUR T1 - Monotonicity Corrections for Nine-Point Scheme of Diffusion Equations AU - Kong , Wang AU - Hong , Zhenying AU - Yuan , Guangwei AU - Sheng , Zhiqiang JO - Journal of Computational Mathematics VL - 5 SP - 1305 EP - 1327 PY - 2024 DA - 2024/07 SN - 42 DO - http://doi.org/10.4208/jcm.2303-m2022-0139 UR - https://global-sci.org/intro/article_detail/jcm/23279.html KW - Monotonicity corrections, Diffusion equation, Improved MPFA, Distorted meshes. AB -

In this paper, we present a nonlinear correction technique to modify the nine-point scheme proposed in [SIAM J. Sci. Comput., 30:3 (2008), 1341-1361] such that the resulted scheme preserves the positivity. We first express the flux by the cell-centered unknowns and edge unknowns based on the stencil of the nine-point scheme. Then, we use a nonlinear combination technique to get a monotone scheme. In order to obtain a cell-centered finite volume scheme, we need to use the cell-centered unknowns to locally approximate the auxiliary unknowns. We present a new method to approximate the auxiliary unknowns by using the idea of an improved multi-points flux approximation. The numerical results show that the new proposed scheme is robust, can handle some distorted grids that some existing finite volume schemes could not handle, and has higher numerical accuracy than some existing positivity-preserving finite volume schemes.

Kong , WangHong , ZhenyingYuan , Guangwei and Sheng , Zhiqiang. (2024). Monotonicity Corrections for Nine-Point Scheme of Diffusion Equations. Journal of Computational Mathematics. 42 (5). 1305-1327. doi:10.4208/jcm.2303-m2022-0139
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