Volume 24, Issue 5
A Positivity-Preserving Finite Volume Scheme for Heat Conduction Equation on Generalized Polyhedral Meshes

Hui Xie, Xuejun Xu, Chuanlei Zhai & Heng Yong

Commun. Comput. Phys., 24 (2018), pp. 1375-1408.

Published online: 2018-06

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  • Abstract

In this paper we present a nonlinear finite volume scheme preserving positivity for heat conduction equations. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are eliminated by our newly developed second-order explicit interpolation formula on generalized polyhedral meshes. With the help of the additional parameters, it is not necessary to choose the stencil adaptively to obtain the convex decomposition of the co-normal vector and also is not required to replace the interpolation formula with positivity-preserving but usually low-order accurate ones whenever negative interpolated auxiliary unknowns appear. Moreover, the new flux approximation has a fixed stencil. These features make our scheme more efficient compared with other existing methods based on Le Potier's nonlinear two-point approximation, especially in 3D. Numerical experiments show that the scheme maintains the positivity of the continuous solution and has nearly second-order accuracy for the solution on the distorted meshes where the diffusion tensor may be anisotropic and discontinuous.

  • Keywords

Heat conduction, anisotropic diffusion tensor, distorted mesh, positivity-preserving, cell-centered scheme, vertex elimination.

  • AMS Subject Headings

65M08, 35R05, 76S05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-1375, author = {}, title = {A Positivity-Preserving Finite Volume Scheme for Heat Conduction Equation on Generalized Polyhedral Meshes}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {5}, pages = {1375--1408}, abstract = {

In this paper we present a nonlinear finite volume scheme preserving positivity for heat conduction equations. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are eliminated by our newly developed second-order explicit interpolation formula on generalized polyhedral meshes. With the help of the additional parameters, it is not necessary to choose the stencil adaptively to obtain the convex decomposition of the co-normal vector and also is not required to replace the interpolation formula with positivity-preserving but usually low-order accurate ones whenever negative interpolated auxiliary unknowns appear. Moreover, the new flux approximation has a fixed stencil. These features make our scheme more efficient compared with other existing methods based on Le Potier's nonlinear two-point approximation, especially in 3D. Numerical experiments show that the scheme maintains the positivity of the continuous solution and has nearly second-order accuracy for the solution on the distorted meshes where the diffusion tensor may be anisotropic and discontinuous.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0183}, url = {http://global-sci.org/intro/article_detail/cicp/12482.html} }
TY - JOUR T1 - A Positivity-Preserving Finite Volume Scheme for Heat Conduction Equation on Generalized Polyhedral Meshes JO - Communications in Computational Physics VL - 5 SP - 1375 EP - 1408 PY - 2018 DA - 2018/06 SN - 24 DO - http://dor.org/10.4208/cicp.OA-2017-0183 UR - https://global-sci.org/intro/article_detail/cicp/12482.html KW - Heat conduction, anisotropic diffusion tensor, distorted mesh, positivity-preserving, cell-centered scheme, vertex elimination. AB -

In this paper we present a nonlinear finite volume scheme preserving positivity for heat conduction equations. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are eliminated by our newly developed second-order explicit interpolation formula on generalized polyhedral meshes. With the help of the additional parameters, it is not necessary to choose the stencil adaptively to obtain the convex decomposition of the co-normal vector and also is not required to replace the interpolation formula with positivity-preserving but usually low-order accurate ones whenever negative interpolated auxiliary unknowns appear. Moreover, the new flux approximation has a fixed stencil. These features make our scheme more efficient compared with other existing methods based on Le Potier's nonlinear two-point approximation, especially in 3D. Numerical experiments show that the scheme maintains the positivity of the continuous solution and has nearly second-order accuracy for the solution on the distorted meshes where the diffusion tensor may be anisotropic and discontinuous.

Hui Xie, Xuejun Xu, Chuanlei Zhai & Heng Yong. (2020). A Positivity-Preserving Finite Volume Scheme for Heat Conduction Equation on Generalized Polyhedral Meshes. Communications in Computational Physics. 24 (5). 1375-1408. doi:10.4208/cicp.OA-2017-0183
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