Volume 14, Issue 2
An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes

Cheng DongTong Kang

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 405-437.

Published online: 2021-01

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  • Abstract

In this paper, we suggest a new vertex interpolation algorithm to improve an existing cell-centered finite volume scheme for nonlinear diffusion problems on general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability of the corresponding local system is proved under some assumptions. Additionally, we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The final scheme allows arbitrary continuous or discontinuous diffusion coefficients and can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux are observed in numerical experiments. In the comparative experiments with some existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes.

  • Keywords

Cell-centered scheme, nonlinear diffusion equation, vertex interpolation algorithm, linearity-preserving criterion, numerical heat-barrier issue, finite volume scheme.

  • AMS Subject Headings

65M08, 65M22

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-405, author = {Dong , Cheng and Kang , Tong}, title = {An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {2}, pages = {405--437}, abstract = {

In this paper, we suggest a new vertex interpolation algorithm to improve an existing cell-centered finite volume scheme for nonlinear diffusion problems on general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability of the corresponding local system is proved under some assumptions. Additionally, we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The final scheme allows arbitrary continuous or discontinuous diffusion coefficients and can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux are observed in numerical experiments. In the comparative experiments with some existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0006}, url = {http://global-sci.org/intro/article_detail/nmtma/18605.html} }
TY - JOUR T1 - An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes AU - Dong , Cheng AU - Kang , Tong JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 405 EP - 437 PY - 2021 DA - 2021/01 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0006 UR - https://global-sci.org/intro/article_detail/nmtma/18605.html KW - Cell-centered scheme, nonlinear diffusion equation, vertex interpolation algorithm, linearity-preserving criterion, numerical heat-barrier issue, finite volume scheme. AB -

In this paper, we suggest a new vertex interpolation algorithm to improve an existing cell-centered finite volume scheme for nonlinear diffusion problems on general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability of the corresponding local system is proved under some assumptions. Additionally, we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The final scheme allows arbitrary continuous or discontinuous diffusion coefficients and can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux are observed in numerical experiments. In the comparative experiments with some existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes.

Cheng Dong & Tong Kang. (2021). An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes. Numerical Mathematics: Theory, Methods and Applications. 14 (2). 405-437. doi:10.4208/nmtma.OA-2020-0006
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