High-order, Compact, and Symmetric Finite Difference Methods for a $d$-Dimensional Hypercube

Abstract: This paper presents compact, symmetric, and high-order finite difference methods (FDMs) for the variable Poisson equation on a $d$-dimensional hypercube. Our scheme produces a symmetric linear system: an important property that does not immediately hold for a high-order FDM. Since the model problem is coercive, the linear system is in fact symmetric positive definite, and consequently many fast solvers are applicable. Furthermore, the symmetry combined with the minimum support of the stencil keeps the storage requirement minimal. Theoretically speaking, we prove that a compact, 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, 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 finite compact, 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.