Linear and Hamiltonian-conserving Fourier pseudo-spectral schemes for the Camassa–Holm equation

Abstract

In this paper, we develop two linear conservative Fourier pseudo-spectral schemes for the Camassa–Holm equation. We first apply the Fourier pseudo-spectral method in space for the Camassa–Holm equation to arrive at a spatial semi-discretized system in which a corresponding discrete momentum conservation law is preserved. Then we employe the linear-implicit Crank–Nicolson scheme and the leap-frog scheme for the semi-discrete system, respectively. The two new fully discrete methods are proved to conserve the discrete momentum conservation law of the original system, which implies the numerical solutions are bounded in the discrete L∞ norm. Furthermore, the proposed methods are unconditionally stable, second order in time and high order in space, and uniquely solvable. Numerical experiments are presented to show the convergence property as well as the efficiency and accuracy of the new schemes. The proposed methods in this paper could be readily utilized to design linear momentum-preserving numerical approximations for many other Hamiltonian PDEs.