Volume 13, Issue 1
A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Shuai Su & Jiming Wu

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 220-252.

Published online: 2019-12

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  • Abstract

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations  on general polygonal meshes. The vertex unknowns are treated as primary ones  for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is  not convex in general, leading to  a fixed stencil of  the flux  approximation  and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.

  • Keywords

2-D 3-T, radiation diffusion equations, vertex-centered scheme, positivity-preserving, finite volume.

  • AMS Subject Headings

65M08, 65M22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

sushuai16@gscaep.ac.cn (Shuai Su)

wu_jiming@iapcm.ac.cn (Jiming Wu)

  • BibTex
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@Article{NMTMA-13-220, author = {Su , Shuai and Wu , Jiming }, title = {A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {13}, number = {1}, pages = {220--252}, abstract = {

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations  on general polygonal meshes. The vertex unknowns are treated as primary ones  for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is  not convex in general, leading to  a fixed stencil of  the flux  approximation  and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0121}, url = {http://global-sci.org/intro/article_detail/nmtma/13438.html} }
TY - JOUR T1 - A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes AU - Su , Shuai AU - Wu , Jiming JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 220 EP - 252 PY - 2019 DA - 2019/12 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2018-0121 UR - https://global-sci.org/intro/article_detail/nmtma/13438.html KW - 2-D 3-T, radiation diffusion equations, vertex-centered scheme, positivity-preserving, finite volume. AB -

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations  on general polygonal meshes. The vertex unknowns are treated as primary ones  for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is  not convex in general, leading to  a fixed stencil of  the flux  approximation  and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.

Shuai Su & Jiming Wu. (2019). A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes. Numerical Mathematics: Theory, Methods and Applications. 13 (1). 220-252. doi:10.4208/nmtma.OA-2018-0121
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