High order uniform in time schemes for weakly nonlinear Schr\"odinger equation and wave turbulence

TL;DR

This paper develops high-order multiscale numerical schemes for weakly nonlinear Schrödinger equations that maintain uniform accuracy over long times and are effective in studying wave turbulence dynamics.

Contribution

The paper introduces novel multiscale, high-order time integration schemes based on Picard iterates, with proven long-time uniform accuracy for weakly nonlinear Schrödinger equations.

Findings

Schemes achieve high precision with small nonlinearity parameter

Uniform accuracy maintained over long time horizons

Numerical simulations confirm theoretical properties

Abstract

We introduce two multiscale numerical schemes for the time integration of weakly nonlinear Schr\"odinger equations, built upon the discretization of Picard iterates of the solution. These high-order schemes are designed to achieve high precision with respect to the small nonlinearity parameter under particular CFL condition. By exploiting the scattering properties of these schemes thanks to a low-frequency projected linear flow, we also establish its uniform accuracy over long time horizons. Numerical simulations are provided to illustrate the theoretical results, and these schemes are further applied to investigate dynamics in the framework of wave turbulence.