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Volume 30, Issue 4
Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes

Menghuan Liu, Shi Shu, Guangwei Yuan & Xiaoqiang Yue

Commun. Comput. Phys., 30 (2021), pp. 1185-1215.

Published online: 2021-08

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  • Abstract

In this article we present two types of nonlinear positivity-preserving finite volume (PPFV) schemes for a class of three-dimensional heat conduction equations on general polyhedral meshes. First, we present a new parameter selection strategy on the one-sided flux and establish a nonlinear PPFV scheme based on a two-point flux with higher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai, H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes our interpolation formulae suitable to be constructed by existing approaches with high accuracy and well robustness (e.g., the finite element method), thus enhancing the adaptability to distorted meshes with large deformations. Then we derive a linear multi-point flux involving combination coefficients and, via the Patankar trick, obtain another nonlinear PPFV scheme that is concise and easy to implement. The selection strategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preserving properties of these two nonlinear PPFV solutions are proved. Numerical experiments with the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that our schemes both can achieve ideal-order accuracy even on severely distorted meshes.

  • AMS Subject Headings

65M08, 65N55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-30-1185, author = {Liu , MenghuanShu , ShiYuan , Guangwei and Yue , Xiaoqiang}, title = {Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {4}, pages = {1185--1215}, abstract = {

In this article we present two types of nonlinear positivity-preserving finite volume (PPFV) schemes for a class of three-dimensional heat conduction equations on general polyhedral meshes. First, we present a new parameter selection strategy on the one-sided flux and establish a nonlinear PPFV scheme based on a two-point flux with higher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai, H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes our interpolation formulae suitable to be constructed by existing approaches with high accuracy and well robustness (e.g., the finite element method), thus enhancing the adaptability to distorted meshes with large deformations. Then we derive a linear multi-point flux involving combination coefficients and, via the Patankar trick, obtain another nonlinear PPFV scheme that is concise and easy to implement. The selection strategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preserving properties of these two nonlinear PPFV solutions are proved. Numerical experiments with the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that our schemes both can achieve ideal-order accuracy even on severely distorted meshes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0011}, url = {http://global-sci.org/intro/article_detail/cicp/19398.html} }
TY - JOUR T1 - Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes AU - Liu , Menghuan AU - Shu , Shi AU - Yuan , Guangwei AU - Yue , Xiaoqiang JO - Communications in Computational Physics VL - 4 SP - 1185 EP - 1215 PY - 2021 DA - 2021/08 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2021-0011 UR - https://global-sci.org/intro/article_detail/cicp/19398.html KW - General polyhedral mesh, heat conduction equation, nonlinearity, positivity-preserving, existence. AB -

In this article we present two types of nonlinear positivity-preserving finite volume (PPFV) schemes for a class of three-dimensional heat conduction equations on general polyhedral meshes. First, we present a new parameter selection strategy on the one-sided flux and establish a nonlinear PPFV scheme based on a two-point flux with higher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai, H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes our interpolation formulae suitable to be constructed by existing approaches with high accuracy and well robustness (e.g., the finite element method), thus enhancing the adaptability to distorted meshes with large deformations. Then we derive a linear multi-point flux involving combination coefficients and, via the Patankar trick, obtain another nonlinear PPFV scheme that is concise and easy to implement. The selection strategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preserving properties of these two nonlinear PPFV solutions are proved. Numerical experiments with the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that our schemes both can achieve ideal-order accuracy even on severely distorted meshes.

Menghuan Liu, Shi Shu, Guangwei Yuan & Xiaoqiang Yue. (2021). Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes. Communications in Computational Physics. 30 (4). 1185-1215. doi:10.4208/cicp.OA-2021-0011
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