Convergence rates for dispersive approximation schemes to nonlinear Schr\"odinger equations

TL;DR

This paper analyzes the convergence rates of numerical schemes for linear and nonlinear Schrödinger equations, showing dispersive schemes with Strichartz estimates outperform non-dispersive ones especially for less regular data.

Contribution

The authors establish explicit convergence rates for dispersive numerical schemes and demonstrate their superior performance over non-dispersive schemes for certain initial data regularities.

Findings

Dispersive schemes achieve polynomial convergence rates.

Non-dispersive schemes only attain logarithmic decay rates.

Dispersive schemes better handle less regular initial data.

Abstract

This article is devoted to the analysis of the convergence rates of several nu- merical approximation schemes for linear and nonlinear Schr\"odinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schr\"odinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L2(R), a fact that can not be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for Hs(R) data if 0 < s < 1/2. Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic decay rates.