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Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings

Part of: Partial differential equations, boundary value problems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  01 February 2016

Cuong Ngo  and
Weizhang Huang
Cuong Ngo
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA.
Weizhang Huang*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA.
*
*Corresponding author. Email addresses:[email protected] (C. Ngo), [email protected] (W. Huang)
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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.

Keywords

Anisotropic diffusionanisotropic coefficientdiscrete maximum principlemonotone schemefinite differencenonnegative directional splitting

MSC classification

Secondary: 65N06: Finite difference methods 65M06: Finite difference methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 2 , February 2016 , pp. 473 - 495
Copyright
Copyright © Global-Science Press 2016 

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