Lattice Green's Functions for High Order Finite Difference Stencils

TL;DR

This paper develops fast, accurate methods for computing lattice Green's functions for high-order finite difference discretizations of the Laplace operator, enabling efficient solutions of elliptic PDEs on unbounded or partially unbounded domains.

Contribution

It introduces algorithms for evaluating LGFs for high-order discretizations and derives closed-form expressions for certain boundary conditions, extending the applicability of LGFs beyond second-order methods.

Findings

Achieves near machine-precision accuracy in LGF evaluations.

Enables consistent solutions to high-order Poisson problems on unbounded domains.

Provides computationally efficient methods for high-order discretizations.

Abstract

Lattice Green's Functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2D or 3D LGFs in linear time, avoiding the need for brute-force multi-dimensional quadrature of numerically unstable integrals. Here we focus on higher-order discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with high-order dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson's equation on fully or partially unbounded 3D domains.