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.

翻译：格子格林函数（LGFs）是离散化线性算子的基本解，因而是求解在一个或多个方向无界区域上离散化椭圆型偏微分方程的有效工具。目前大多数利用LGFs的数值求解器均采用二阶离散格式，并在所有方向均具有自由空间边界条件的区域上运行。在这些条件下，可用的快速展开方法能够在线性时间内预计算二维或三维LGFs，从而避免了数值不稳定积分的暴力多维求积。本文聚焦于具有更一般边界条件区域上拉普拉斯算子的高阶离散格式，通过以下两方面开展工作：（1）提出一种算法，用于快速精确计算与无界区域上高阶维度分裂中心有限差分相关的LGFs；（2）推导出与维度分裂及Mehrstellen离散格式相关LGFs的闭式表达式，适用于含一个无界维度的区域。数值实验表明，这些技术能以接近机器精度的精度完成LGFs评估，且所得LGFs能够在完全或部分无界的三维区域上，为泊松方程的高阶离散格式提供数值一致解。