Volume 27, Issue 5
A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes

Qiannan Dong, Shuai Su & Jiming Wu

Commun. Comput. Phys., 27 (2020), pp. 1378-1412.

Published online: 2020-03

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  • Abstract

We propose a decoupled and positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusionproblems on polyhedral meshes with star-shaped cells and planar faces. Under the generalized DDFV framework, two sets of finite volume (FV) equations are respectively constructed on the dual and primary meshes, where the ones on the dual mesh are derived from the ingenious combination of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it is conditionally maintained on the dual mesh while strictly preserved on the primary mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations aredecoupled. As aresult, nononlinear iteration is required for linear problems and a general nonlinear solver could be used for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by numerical experiments. The efficiency of the Newton method is also demonstrated by comparison with those of the fixed-point iteration method and its Anderson acceleration.

  • Keywords

DDFV, decoupled algorithm, diffusion problems, positivity-preserving, polyhedral meshes.

  • AMS Subject Headings

65M08, 65M22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

dongqiannan17@gscaep.ac.cn (Qiannan Dong)

sushuai@math.pku.edu.cn (Shuai Su)

wu jiming@iapcm.ac.cn (Jiming Wu)

  • BibTex
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@Article{CiCP-27-1378, author = {Dong , Qiannan and Su , Shuai and Wu , Jiming }, title = {A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {5}, pages = {1378--1412}, abstract = {

We propose a decoupled and positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusionproblems on polyhedral meshes with star-shaped cells and planar faces. Under the generalized DDFV framework, two sets of finite volume (FV) equations are respectively constructed on the dual and primary meshes, where the ones on the dual mesh are derived from the ingenious combination of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it is conditionally maintained on the dual mesh while strictly preserved on the primary mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations aredecoupled. As aresult, nononlinear iteration is required for linear problems and a general nonlinear solver could be used for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by numerical experiments. The efficiency of the Newton method is also demonstrated by comparison with those of the fixed-point iteration method and its Anderson acceleration.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0292}, url = {http://global-sci.org/intro/article_detail/cicp/15774.html} }
TY - JOUR T1 - A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes AU - Dong , Qiannan AU - Su , Shuai AU - Wu , Jiming JO - Communications in Computational Physics VL - 5 SP - 1378 EP - 1412 PY - 2020 DA - 2020/03 SN - 27 DO - http://dor.org/10.4208/cicp.OA-2018-0292 UR - https://global-sci.org/intro/cicp/15774.html KW - DDFV, decoupled algorithm, diffusion problems, positivity-preserving, polyhedral meshes. AB -

We propose a decoupled and positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusionproblems on polyhedral meshes with star-shaped cells and planar faces. Under the generalized DDFV framework, two sets of finite volume (FV) equations are respectively constructed on the dual and primary meshes, where the ones on the dual mesh are derived from the ingenious combination of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it is conditionally maintained on the dual mesh while strictly preserved on the primary mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations aredecoupled. As aresult, nononlinear iteration is required for linear problems and a general nonlinear solver could be used for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by numerical experiments. The efficiency of the Newton method is also demonstrated by comparison with those of the fixed-point iteration method and its Anderson acceleration.

Qiannan Dong, Shuai Su & Jiming Wu. (2020). A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes. Communications in Computational Physics. 27 (5). 1378-1412. doi:10.4208/cicp.OA-2018-0292
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