Lattice Green’s Functions for High-Order Finite Difference Stencils
Name
2309.13503v1.pdf
Description
Accepted version
Size
522.42 KB
Format
Adobe PDF
Checksum (MD5)
fccb1348d87956561b6520823c42f3e0
Author(s)
Gabbard, James
van Rees, Wim M.
Date Issued
January 4, 2024
Journal
SIAM Journal on Numerical Analysis
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Gabbard, James and van Rees, Wim M. 2024. "Lattice Green’s Functions for High-Order Finite Difference Stencils." SIAM Journal on Numerical Analysis, 62 (1).
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Terms of Use
Creative Commons Attribution-Noncommercial-ShareAlike
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DOI of Published Version
https://doi.org/10.1137/23m1573872
