# Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation

## TL;DR

This paper introduces a new class of high-order energy-preserving numerical schemes for the Camassa-Holm equation, combining invariant energy quadratization, Fourier pseudo-spectral methods, and symplectic Runge-Kutta integrators to ensure energy conservation.

## Contribution

The paper presents a novel framework that achieves arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation using invariant energy quadratization and spectral methods.

## Key findings

- Schemes exactly preserve semi-discrete energy conservation law.
- Numerical results confirm high accuracy and efficiency.
- The approach is applicable to high-order discretizations.

## Abstract

In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system, which inherits a quadratic energy. {The new system is then discretized} by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, { a symplectic Runge-Kutta method such as the Gauss collocation method is applied} for the resulting semi-discrete system to arrive at an arbitrarily high-order fully discrete scheme. {We prove that the obtained schemes can conserve the discrete energy conservation law}. Numerical results are addressed to confirm accuracy and efficiency of the proposed schemes.