Volume 13, Issue 1
A Conservative Gradient Discretization Method for Parabolic Equations

Huifang Zhou, Zhiqiang Sheng & Guangwei Yuan

Adv. Appl. Math. Mech., 13 (2021), pp. 232-260.

Published online: 2020-10

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  • Abstract

In this paper, we propose a new conservative gradient discretization method (GDM) for one-dimensional parabolic partial differential equations (PDEs). We use the implicit Euler method for the temporal discretization and conservative gradient discretization method  for spatial discretization. The method is based on a new cell-centered meshes, and it is locally conservative. It has smaller truncation error than the classical finite volume method on uniform meshes. We use the framework of the gradient discretization method to analyze the stability and convergence. The numerical experiments show that the new method has second-order convergence. Moreover, it is more accurate than the classical finite volume method in flux error, $L^2$ error and $L^{\infty}$ error.

  • Keywords

Gradient discretization method, mass conservation, parabolic equations.

  • AMS Subject Headings

65M08, 35K10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-13-232, author = {Huifang Zhou , and Zhiqiang Sheng , and Guangwei Yuan , }, title = {A Conservative Gradient Discretization Method for Parabolic Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {13}, number = {1}, pages = {232--260}, abstract = {

In this paper, we propose a new conservative gradient discretization method (GDM) for one-dimensional parabolic partial differential equations (PDEs). We use the implicit Euler method for the temporal discretization and conservative gradient discretization method  for spatial discretization. The method is based on a new cell-centered meshes, and it is locally conservative. It has smaller truncation error than the classical finite volume method on uniform meshes. We use the framework of the gradient discretization method to analyze the stability and convergence. The numerical experiments show that the new method has second-order convergence. Moreover, it is more accurate than the classical finite volume method in flux error, $L^2$ error and $L^{\infty}$ error.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0047}, url = {http://global-sci.org/intro/article_detail/aamm/18349.html} }
TY - JOUR T1 - A Conservative Gradient Discretization Method for Parabolic Equations AU - Huifang Zhou , AU - Zhiqiang Sheng , AU - Guangwei Yuan , JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 232 EP - 260 PY - 2020 DA - 2020/10 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0047 UR - https://global-sci.org/intro/article_detail/aamm/18349.html KW - Gradient discretization method, mass conservation, parabolic equations. AB -

In this paper, we propose a new conservative gradient discretization method (GDM) for one-dimensional parabolic partial differential equations (PDEs). We use the implicit Euler method for the temporal discretization and conservative gradient discretization method  for spatial discretization. The method is based on a new cell-centered meshes, and it is locally conservative. It has smaller truncation error than the classical finite volume method on uniform meshes. We use the framework of the gradient discretization method to analyze the stability and convergence. The numerical experiments show that the new method has second-order convergence. Moreover, it is more accurate than the classical finite volume method in flux error, $L^2$ error and $L^{\infty}$ error.

Huifang Zhou, Zhiqiang Sheng & Guangwei Yuan. (2020). A Conservative Gradient Discretization Method for Parabolic Equations. Advances in Applied Mathematics and Mechanics. 13 (1). 232-260. doi:10.4208/aamm.OA-2020-0047
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