Monotone finite difference schemes for anisotropic diffusion problems via nonnegative directional splittings

TL;DR

This paper develops conditions for nonnegative directional splittings of anisotropic diffusion operators and constructs monotone finite difference schemes that adapt to the diffusion matrix's properties, ensuring stability and accuracy.

Contribution

It introduces new conditions for nonnegative directional splittings and provides a method to construct monotone schemes based on the diffusion matrix's characteristics.

Findings

Monotone schemes are possible if the diffusion matrix is continuous, symmetric, and positive definite.

The stencil size depends on the anisotropy level of the diffusion matrix.

A three-by-three stencil suffices for strictly diagonally dominant matrices.

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.