Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas

TL;DR

This paper develops spectrally accurate, fully discrete numerical schemes for certain integrable PDEs, leveraging explicit formulas to achieve exact time integration and spectral spatial accuracy, with proven convergence and efficient long-term simulations.

Contribution

The paper introduces novel fully-discrete schemes based on explicit formulas for integrable PDEs, achieving spectral accuracy and linear-in-time error growth, enabling long-term numerical analysis.

Findings

Spectral convergence in L^2 norm for Benjamin-Ono and Calogero-Sutherland equations.

Error constant depends linearly on final time, not exponentially.

Numerical simulations demonstrate effectiveness at short and long time scales.

Abstract

We construct fully-discrete schemes for the Benjamin-Ono, Calogero-Sutherland DNLS, and cubic Szeg\H{o} equations on the torus, which are $\textit{exact in time}$ with $\textit{spectral accuracy}$ in space. We prove spectral convergence for the first two equations, of order $K^{-s+1}$ in $L^2$ norm for initial data in $H^s(\mathbb T)$, $s>1$, with an error constant depending $\textit{linearly}$ on the final time instead of exponentially. These schemes are based on $\textit{explicit formulas}$, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales, thanks to the remarkable fact that the computational cost of the method is independent of the final time. These schemes open doors for the understanding of the long-time dynamics of integrable equations.