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

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. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.