General conservative sixth- and eighth-order compact finite difference schemes for the N-coupled Schrödinger-Boussinesq equations

Abstract

In this paper, general conservative sixth- and eighth-order compact finite difference schemes are presented to solve the N-coupled nonlinear Schrödinger-Boussinesq equations numerically. The existence of the difference solution is proved by fixed-point theorem. By utilizing the discrete energy methods, the proposed difference schemes are proved to be unconditionally convergent at the order $O({\tau }^2+h^8)$ with mesh-size $h$ and time step $\tau$ in the discrete $L_{\infty }$-norm. By using the Yoshida’s composition method, we improve the scheme (3.1)-(3.3) with a group of given time-step increments repeatedly and then obtain a temporal fourth-order difference scheme. Numerical experiments confirm the theoretical results and verify the accuracy and efficiency of our method.