An artificially-damped Fourier method for dispersive evolution equations

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.TT is partially supported by NSF DMS-1945652. 1 All computations in this paper, with the exception of computing reference solutions for the KdV, NLS and Kawahara equations, were done on a Microsoft Surface Pro 6 Laptop with 8GB RAM, 256GB SSD, a 1.60GHz CPU, the Intel® Core™ i5-8250U Processor, and the Windows 10 operating system.