Convergence error estimates at low regularity for time discretizations of KdV

TL;DR

This paper establishes convergence error estimates for various filtered time discretizations of the periodic KdV equation at low regularity, using discrete Bourgain spaces to prove convergence in $L^2$ for rough initial data.

Contribution

It introduces new filtered discretization schemes for KdV and provides explicit convergence rates at low regularity levels.

Findings

Convergence in $L^2$ for initial data in $H^s$, $s>0$

Explicit error estimates for filtered schemes

Applicability to rough initial data

Abstract

We consider various filtered time discretizations of the periodic Korteweg--de Vries equation: a filtered exponential integrator, a filtered Lie splitting scheme as well as a filtered resonance based discretisation and establish convergence error estimates at low regularity. Our analysis is based on discrete Bourgain spaces and allows to prove convergence in $L^2$ for rough data $u_{0} \in H^s,$ $s>0$ with an explicit convergence rate.