A self-adaptive mesh method for the Camassa-Holm equation

TL;DR

This paper introduces a self-adaptive moving mesh numerical scheme for the Camassa-Holm equation that automatically adjusts the mesh based on the solution, maintaining integrability and efficiently capturing soliton solutions.

Contribution

The paper presents a novel self-adaptive mesh method for the Camassa-Holm equation that preserves integrability and improves computational efficiency.

Findings

Accurately captures soliton solutions with few grid points

Maintains integrability with exact N-soliton solutions

Provides satisfactory results in test problems

Abstract

A self-adaptive moving mesh method is proposed for the numerical simulations of the Camassa-Holm equation. It is an integrable scheme in the sense that it possesses the exact N-soliton solution. It is named a self-adaptive moving mesh method, because the non-uniform mesh is driven and adapted automatically by the solution. Once the non-uniform mesh is evolved, the solution is determined by solving a tridiagonal linear system. Due to these two superior features of the method, several test problems give very satisfactory results even if by using a small number of grid points.