Lattice Green's Functions for High Order Finite Difference Stencils

TL;DR

This work extends lattice Green's function methods to high-order finite difference discretizations on unbounded domains and on domains with a single unbounded direction. It delivers fast, accurate evaluation algorithms for dimension-split stencils on fully unbounded domains and closed-form, numerically stable LGFs for both Mehrstellen and dimension-split discretizations with one unbounded dimension, enabling near machine-precision solutions of high-order Poisson problems. The authors provide rigorous asymptotic expansions, contour-based stability techniques, and a practical open-source implementation, demonstrating accurate residuals and expected convergence across 2D/3D test cases. Collectively, these advances enable exact, high-accuracy Poisson solves with high-order discretizations on a broad class of unbounded or semi-unbounded domains, with significant implications for FFT/FMM-based solvers and immersed-interface methods.

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.

Figures (1)

• Figure 1: Convergence data for the 3D Poisson problem with stencils LGF2 (), LGF4 (), MEH4 (), LGF6(), MEH6 (), LGF8 (). Dashed lines indicate the expected order of convergence for each difference scheme.