拉普拉斯算子及其方格离散化:平面 2-Torus 及其他相关二维流形的格林函数与格网格林函数

IF 2 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL
Malik Mamode
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引用次数: 0

摘要

本文研究了定义在 2-Torus 上的拉普拉斯算子的格林函数和晶格格林函数 (LGF) 之间的截断误差及其在规则正方形晶格上的离散化。此外,还提出了自由(或诺伊曼)边界条件下圆柱体和矩形域的扩展。在上述每种情况下,格林函数及其离散类似物都以精确的分析闭合形式给出,当晶格间距趋于零时,可以推断出精确的估计值。不出所料,在每种情况下,LGF 的连续极限都与格林函数非常吻合。特别是,通过两个有关矩形域的应用实例,解决了格点离散化对格林函数的对数奇异正则化问题,这两个实例专门用于计算导电正方形的角到角电阻和标准布朗运动的正方形对角顶点之间的平均首次通过时间,两者都是考虑连续极限而得出的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds
The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.
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来源期刊
CiteScore
4.10
自引率
14.30%
发文量
542
审稿时长
1.9 months
期刊介绍: Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.
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