A numerical study of variational discretizations of the Camassa-Holm equation

TL;DR

This paper introduces and compares two variational semidiscretizations of the Camassa-Holm equation in periodic domains, emphasizing energy conservation and computational efficiency through novel algorithms and Lagrangian-based methods.

Contribution

It presents a new variational discretization for the two-component Camassa-Holm system and adapts existing methods for periodic domains, with a focus on energy conservation and computational algorithms.

Findings

The new discretization effectively conserves energy in numerical simulations.

The periodic multipeakon method is computationally efficient and accurate.

Comparative analysis shows advantages over existing methods in specific scenarios.

Abstract

We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa-Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.