泊松方程的广义有限差分方法

Generalized Finite Difference Method for Poisson Equation

有限差分方法是求解泊松方程的一种常见的数值方法,其思想是通过对计算区域进行网格剖分,将微分算子进行离散化,建立以网格节点上的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.但是有限差分方法对计算区域及边界条件具有局限性,为此,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,文中基于重心对偶体,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,从而激发起他们运用数学工具解决实际问题的兴趣.

The finite difference method is a common numerical method for solving Poisson equation.Its idea is to discretize the differential operator by meshing the calculation area,and establish an algebraic equation system with the value on the grid node as the degree of freedom,so as to transform the problem of determining the solution of the differential equation into the problem of solving the algebraic equation system.However,the finite difference method has limitations on the calculation region and boundary conditions.Therefore,in the textbook of numerical solution of partial differential equations,the finite volume discrete scheme is usually constructed by using the outer central dual body of primitive mesh.However,the centroid duality is strongly dependent on the original mesh.In order to overcome this problem,based on the barycentric duality,a unified generalized finite difference scheme for two-dimensional Poisson equation is given in this paper.This scheme is helpful for students to deepen their understanding of the difference method,so as to stimulate their interest in using mathematical tools to solve practical problems.