An artificially-damped Fourier method for dispersive evolution equations

TL;DR

This paper introduces an artificially-damped Fourier method to suppress unwanted oscillations in numerical solutions of dispersive PDEs, improving efficiency while maintaining Fourier transform advantages.

Contribution

It develops two damping techniques outside a finite computational domain to enhance Fourier-based solutions of dispersive equations, addressing oscillation issues.

Findings

Heat equation damping effectively reduces high-frequency oscillations.

Exponential decay damping suppresses traveling waves and high-amplitude oscillations.

Damping methods significantly improve computational runtime.

Abstract

Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one side reappear on the other and for dispersive equations these are typically high-velocity, high-frequency waves. However, the fast Fourier transform is a very efficient numerical tool and it is important to find a way to damp these oscillations so that this transform can still be used. In this paper, we accurately model solutions to four nonlinear partial differential equations on an infinite domain by considering a finite interval and implementing two damping methods outside of that interval: one that solves the heat equation and one that simulates rapid exponential decay. Heat equation-based damping is best suited for small-amplitude, high-frequency oscillations while exponential decay is used to damp traveling waves and high-amplitude oscillations. We demonstrate significant improvements in the runtime of well-studied numerical methods when adding in the damping method.