Volume 19, Issue 2
Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings

Cuong Ngo & Weizhang Huang

Commun. Comput. Phys., 19 (2016), pp. 473-495.

Published online: 2018-04

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  • 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.

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@Article{CiCP-19-473, author = {Cuong Ngo and Weizhang Huang}, title = {Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings}, journal = {Communications in Computational Physics}, year = {2018}, volume = {19}, number = {2}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.280315.140815a}, url = {http://global-sci.org/intro/article_detail/cicp/11097.html} }
TY - JOUR T1 - Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings AU - Cuong Ngo & Weizhang Huang JO - Communications in Computational Physics VL - 2 SP - 473 EP - 495 PY - 2018 DA - 2018/04 SN - 19 DO - http://dor.org/10.4208/cicp.280315.140815a UR - https://global-sci.org/intro/cicp/11097.html KW - AB -

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.

Cuong Ngo & Weizhang Huang. (1970). Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings. Communications in Computational Physics. 19 (2). 473-495. doi:10.4208/cicp.280315.140815a
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