High Order Accurate Solution of Poisson's Equation in Infinite Domains for Smooth Functions

TL;DR

This paper introduces a high-order accurate, FFT-based numerical method for solving Poisson's equation in infinite domains for smooth functions, achieving 4th and 6th order accuracy efficiently.

Contribution

The paper presents a novel FFT-based method for high-order accurate solutions of Poisson's equation in infinite domains, applicable to smooth functions with efficient computational complexity.

Findings

Achieves 4th and 6th order accuracy for smooth functions.

Computational cost is O(N log N) using FFTs.

Effective for solving Poisson's equation in infinite domains.

Abstract

In this paper a method is presented for evaluating the convolution of the Green's function for the Laplace operator with a specified function $\rho(\vec x)$ at all grid points in a rectangular domain $\Omega \subset {\mathrm R}^{d}$ ($d = 1,2,3$), i.e. a solution of Poisson's equation in an infinite domain. 4th and 6th order versions of the method achieve high accuracy when $\rho ( \vec x )$ possesses sufficiently many continuous derivatives. The method utilizes FFT's for computational efficiency and has a computational cost that is $\rm O (N \log N)$ where $\rm N$ is the total number of grid points in the rectangular domain.