Numerical Solution Partial Differential Equations using the Discrete Fourier Transform

TL;DR

This paper demonstrates how to utilize the Fast Fourier Transform to efficiently solve various partial differential equations across different types, highlighting the method's implementation and limitations due to sampling constraints.

Contribution

It provides a detailed explanation of applying FFT to solve PDEs in multiple dimensions, addressing key limitations like aliasing and Nyquist frequency.

Findings

FFT can efficiently solve 2D PDEs like Poisson, diffusion, and wave equations.

Sampling theorem limits the method due to aliasing effects.

Implementation details for multidimensional FFT are provided.

Abstract

In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of the method arising from the Sampling Theorem, which defines the critical Nyquist frequency and the aliasing effect. We then define the Fourier Transform (FT) and the FFT in a way that can be implemented in one and more dimensions. Finally, we show how to apply the FFT in the solution of PDEs related to problems involving two spatial dimensions, specifically the Poisson equation, the diffusion equation and the wave equation for elliptic, parabolic and hyperbolic cases respectively.