On error estimates for Galerkin finite element methods for the Camassa-Holm equation

TL;DR

This paper analyzes the error estimates of Galerkin finite element methods applied to the Camassa-Holm equation, providing optimal-order $L^{2}$-error bounds and demonstrating the effectiveness of a fully discrete scheme through numerical experiments.

Contribution

It establishes optimal-order $L^{2}$-error estimates for Galerkin finite element discretizations of the Camassa-Holm equation and develops a highly accurate, stable fully discrete scheme tested on wave solutions.

Findings

Optimal-order $L^{2}$-error estimates proved for semidiscrete Galerkin methods.

A stable, highly accurate fully discrete scheme constructed and validated.

Numerical experiments confirm effectiveness on smooth and nonsmooth wave solutions.

Abstract

We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We also consider an initial-boundary-value problem on a finite interval for the system form of CH and analyze the convergence of its standard Galerkin semidiscretization. Using the fourth-order accurate, explicit, "classical" Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the `peakon' type.