Direct/split invariant-preserving Fourier pseudo-spectral methods for the rotation-two-component Camassa–Holm system with peakon solitons

Abstract

The Fourier pseudo-spectral method is well suited to solve PDEs under the periodic boundary condition due to its high-order accuracy and easy-to-implement feature. In this paper, we explore as well as comparatively study four classes of Fourier pseudo-spectral schemes for solving the rotation-two-component Camassa–Holm system which possibly owns peakon solitons. Via exploiting inherent structural properties of the system, we reformulate it into two kinds of different equivalent forms and then apply the Fourier pseudo-spectral method to derive two spatial semi-discrete systems, both of which are proved to preserve the corresponding invariants including mass, momentum and energy. Subsequently, we construct two linearly implicit schemes based on Strang splitting technique and two nonlinear schemes, respectively, for both semi-discrete systems. Owing to the different equivalent forms in the structure, one of the nonlinear schemes preserves discrete mass and momentum, while the other one is shown to preserve all three invariants. Numerical results under the situation of smooth/nonsmooth initial values are provided for distinct types of solutions to test the accuracy in long time simulation and to verify the capacity of predicting water wave propagation, as well as advantages in preserving these invariants. For instance, the present schemes are shown to be at least 14 significant digits, improving upon 10 from ones in previous references.