基于任意多边形网格剖分的泊松方程广义有限差分方法

Generalized Finite Difference Method for Poisson Equation Based on Arbitrary Polygon Mesh

求解泊松方程的一种常见数值方法是有限差分方法,其思想是对区域进行网格剖分,利用差商代替导数将微分算子离散,建立以网格节点值为自由度的代数方程组,从而把微分方程的定解问题转化为求解代数方程组的问题.有限差分方法对计算区域、边界条件都具有局限性,在偏微分方程数值解教材中,通常利用原始网格剖分的外心对偶体,构造其有限体积离散格式,但外心对偶体对原始网格依赖性较强.为了克服此问题,基于任意多边形网格的重心对偶体,给出二维泊松方程的广义有限差分统一格式,该格式有利于学生加深对差分方法的理解,从而提高学生运用数学工具解决实际问题的能力.

A common numerical method for solving Poisson’s equations is the finite difference method.Its idea is to mesh the region,use difference quotient instead of derivative to discrete the differential operator,and establish algebraic equation with grid node values as degrees of freedom,thus transforming the problem of determining solutions of differential equations into the problem of solving algebraic equation.The finite difference method has limitations in terms of computational regions and boundary conditions.In textbooks for numerical solutions of partial differential equations,it is common to construct a finite volume discretization scheme using the centroid dual body generated from the original mesh.However,the centroid dual body has a strong dependence on the original mesh.In this paper,in order to overcome this problem,based on the barycentric dual body of any polygon mesh,the generalized finite difference unified format of the two-dimensional Poisson’s equation is given.This format is conducive to students’understanding of the difference method,thus improving their ability to solve practical problems with mathematical tools.