High-order, Compact, and Symmetric Finite Difference Methods for $d$-Dimensional Elliptic Equations

Abstract

This paper presents compact, symmetric, and high-order finite difference methods (FDMs) for the variable Poisson equation on a $d$-dimensional hypercube. Our schemes produce symmetric linear systems: an important property that does not immediately hold for a high-order FDM. This symmetry, combined with the stencil's minimal support, keeps the storage requirements to a minimum. For the model problem considered here, the resulting linear systems are, in fact, symmetric positive definite, allowing a wide range of efficient solvers to be applied. Designing compact, symmetric, and high-order FDMs is challenging, because all overlapping stencils have to satisfy highly specific relations and central differences alone are not enough. We prove that a compact 3-point, symmetric 1D FDM on a uniform grid can achieve arbitrary consistency order. On the other hand, in the $d$-dimensional setting, where $d\ge 2$, the maximum consistency order that a compact $3^d$-point, symmetric FDM on a uniform grid can achieve is 4. If $d=2$ and the diffusion coefficient satisfies a certain derivative condition, the maximum consistency order is 6. Moreover, the compact $3^d$-point, symmetric, 4th-order FDMs for $d\ge 3$, can be conveniently expressed as a linear combination of two types of FDMs: one that depends on partial derivatives along one axis, and the other along two axes. All finite difference stencils are explicitly provided for ease of reproducibility.