Volume 29, Issue 3
A Nonlinear Finite Volume Scheme Preserving Maximum Principle for Diffusion Equations

Jinjing Xu, Fei Zhao, Zhiqiang Sheng & Guangwei Yuan

Commun. Comput. Phys., 29 (2021), pp. 747-766.

Published online: 2021-01

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  • Abstract

In this paper we propose a new nonlinear cell-centered finite volume scheme on general polygonal meshes for two dimensional anisotropic diffusion problems, which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative.

  • Keywords

Maximum principle, finite volume scheme, diffusion equation.

  • AMS Subject Headings

65N08, 65M22

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-747, author = {Xu , Jinjing and Zhao , Fei and Sheng , Zhiqiang and Yuan , Guangwei}, title = {A Nonlinear Finite Volume Scheme Preserving Maximum Principle for Diffusion Equations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {3}, pages = {747--766}, abstract = {

In this paper we propose a new nonlinear cell-centered finite volume scheme on general polygonal meshes for two dimensional anisotropic diffusion problems, which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0047}, url = {http://global-sci.org/intro/article_detail/cicp/18565.html} }
TY - JOUR T1 - A Nonlinear Finite Volume Scheme Preserving Maximum Principle for Diffusion Equations AU - Xu , Jinjing AU - Zhao , Fei AU - Sheng , Zhiqiang AU - Yuan , Guangwei JO - Communications in Computational Physics VL - 3 SP - 747 EP - 766 PY - 2021 DA - 2021/01 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0047 UR - https://global-sci.org/intro/article_detail/cicp/18565.html KW - Maximum principle, finite volume scheme, diffusion equation. AB -

In this paper we propose a new nonlinear cell-centered finite volume scheme on general polygonal meshes for two dimensional anisotropic diffusion problems, which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative.

Jinjing Xu, Fei Zhao, Zhiqiang Sheng & Guangwei Yuan. (2021). A Nonlinear Finite Volume Scheme Preserving Maximum Principle for Diffusion Equations. Communications in Computational Physics. 29 (3). 747-766. doi:10.4208/cicp.OA-2020-0047
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