An explicit finite difference scheme for the Camassa-Holm equation

TL;DR

This paper introduces and analyzes an explicit finite difference scheme for the Camassa-Holm equation, capable of handling complex initial data and peakon interactions, with proven convergence under certain conditions.

Contribution

It presents a new explicit finite difference scheme for the Camassa-Holm equation with convergence proof for general initial data.

Findings

Scheme converges strongly in H^1 to a dissipative weak solution

Handles general H^1 initial data and peakon-antipeakon interactions

Requires a specific time step condition for convergence

Abstract

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.