Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings
Year: 2016
Communications in Computational Physics, Vol. 19 (2016), Iss. 2 : pp. 473–495
Abstract
Nonnegative directional splittings of anisotropic diffusion operators in the divergence form are investigated. Conditions are established for nonnegative directional splittings to hold in a neighborhood of an arbitrary interior point. The result is used to construct monotone finite difference schemes for the boundary value problem of anisotropic diffusion operators. It is shown that such a monotone scheme can be constructed if the underlying diffusion matrix is continuous on the closure of the physical domain and symmetric and uniformly positive definite on the domain, the mesh spacing is sufficiently small, and the size of finite difference stencil is sufficiently large. An upper bound for the stencil size is obtained, which is determined completely by the diffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the larger stencil is required. An exception is the situation with a strictly diagonally dominant diffusion matrix where a three-by-three stencil is sufficient for the construction of a monotone finite difference scheme. Numerical examples are presented to illustrate the theoretical findings.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.280315.140815a
Communications in Computational Physics, Vol. 19 (2016), Iss. 2 : pp. 473–495
Published online: 2016-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
-
A monotone finite volume element scheme for diffusion equations on arbitrary polygonal grids
Nie, Cunyun | Fang, Jianglin | Shu, ShiComputers & Mathematics with Applications, Vol. 153 (2024), Iss. P.225
https://doi.org/10.1016/j.camwa.2023.11.030 [Citations: 0] -
A monotone finite volume element scheme for diffusion equations on triangular grids
Nie, Cunyun | Yu, HaiyuanComputers & Mathematics with Applications, Vol. 105 (2022), Iss. P.1
https://doi.org/10.1016/j.camwa.2021.11.011 [Citations: 3] -
A study on moving mesh finite element solution of the porous medium equation
Ngo, Cuong | Huang, WeizhangJournal of Computational Physics, Vol. 331 (2017), Iss. P.357
https://doi.org/10.1016/j.jcp.2016.11.045 [Citations: 30]