Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data

TL;DR

This paper proves strong convergence of a semi-discrete finite difference method for the KdV and mKdV equations with $L^2$ initial data, extending previous results to non-smooth data without size restrictions.

Contribution

It introduces a new stabilized discretization approach that ensures convergence for non-smooth $L^2$ data, including a conservative nonlinear term discretization.

Findings

Convergence is established using smoothing effects and energy estimates.

Numerical experiments confirm theoretical results and explore an open problem on uniqueness.

The method works without restrictions on the size of initial data.

Abstract

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.