Sparse Non-Negative Stencils for Anisotropic Diffusion

TL;DR

This paper presents AD-LBR, a new discretization scheme for anisotropic diffusion that is non-negative, computationally efficient, and maintains good spectral properties, suitable for grids in 2D and 3D.

Contribution

The paper introduces AD-LBR, a novel discretization method for anisotropic diffusion that uses lattice basis reduction to achieve bounded stencil size and spectral advantages.

Findings

AD-LBR is non-negative and has bounded stencil size in 2D and 3D.

It allows larger time steps and avoids artifacts like chessboard patterns.

Numerical experiments confirm the scheme's effectiveness.

Abstract

We introduce a new discretization scheme for Anisotropic Diffusion, AD-LBR, on two and three dimensional cartesian grids. The main features of this scheme is that it is non-negative, and has a stencil cardinality bounded by 6 in 2D, by 14 in 3D, despite allowing diffusion tensors of arbitrary anisotropy. Our scheme also has good spectral properties, which permits larger time steps and avoids e.g. chessboard artifacts. AD-LBR relies on Lattice Basis Reduction, a tool from discrete mathematics which has recently shown its relevance for the discretization on grids of strongly anisotropic Partial Differential Equations. We prove that AD-LBR is in 2D asymptotically equivalent to a finite element discretization on an anisotropic Delaunay triangulation, a procedure more involved and computationally expensive. Our scheme thus benefits from the theoretical guarantees of this procedure, for a fraction of its cost. Numerical experiments in 2D and 3D illustrate our results.