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

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.