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

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.

A study on moving mesh finite element solution of the porous medium equation

“…A standard finite element discretization like the one we used here typically leads to solutions with oscillations for this type of problems (also see Zhang and Wu [58] for the case with the one-dimensional PME). The oscillations may be suppressed using, for instance, monotone schemes (e.g., see [8,44,46]) or structure-preserving schemes (e.g., see [41,42,48,57,58,59]). These schemes and their combination with adaptive mesh movement for PME are worth future investigations.…”

“…4. How to suppress these oscillations using a monotone or structure-preserving scheme (e.g., see [8,41,42,44,46,48,57,58,59]) and to combine them with adaptive mesh movement for PME are worth future investigations.…”