Improved Finite Difference Method with a Compact Correction Term for Solving Poisson's Equations

TL;DR

This paper introduces an enhanced finite difference method incorporating a compact correction term to improve the accuracy of solving Poisson's equations, applicable to multiple dimensions with verified numerical efficiency.

Contribution

A novel finite difference approach combining high-order compact and classical formulations to significantly improve solution accuracy for Poisson's equations.

Findings

Improved accuracy over classical finite difference methods.

Method is easily extendable to multi-dimensional problems.

Numerical experiments confirm efficiency and effectiveness.

Abstract

An improved finite difference method with compact correction term is proposed to solve the Poisson equations. The compact correction term is developed by a coupled high-order compact and low-order classical finite difference formulations. The numerical solutions obtained by the classical finite difference method are considered as fundamental solutions with lower accuracy, whereas compact correction term is added into source term of classical discrete formulation to improve the accuracy of numerical solutions. The proposed method can be extended from two- to multi-dimensional cases straightforwardly. Numerical experiments are carried out to verify the accuracy and efficiency of this method.