Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings (Communications in Computational Physics, 2018)
Nonnegative directional splittings of anisotropic diffusion operators in thedivergence form are investigated. Conditions are established for nonnegative directionalsplittings to hold in a neighborhood of an arbitrary interior point. The resultis used to construct monotone finite difference schemes for the boundary value problemof anisotropic diffusion operators. It is shown that such a monotone scheme can beconstructed if the underlying diffusion matrix is continuous on the closure of the physicaldomain and symmetric and uniformly positive definite on the domain, the meshspacing 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 thediffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, thelarger stencil is required. An exception is the situation with a strictly diagonally dominantdiffusion matrix where a three-by-three stencil is sufficient for the construction ofa monotone finite difference scheme. Numerical examples are presented to illustratethe theoretical findings.