Efficient eighth‐order accurate energy‐preserving compact difference schemes for the coupled Schrödinger–Boussinesq equations

Abstract

In this study, efficient eighth‐order accurate energy‐preserving compact finite difference schemes are constructed for solving the two‐dimensional coupled Schrödinger–Boussinesq equations (CSBEs) with periodic boundary conditions. The temporal discretization of the first scheme is carried out by a second‐order fully implicit scheme, which requires an iterative method. Thanks to the circulant matrix of spatial discretization, we significantly reduce the computational costs of matrix‐array multiplications and memory requirements via the discrete Fourier transform. The considered scheme is shown to preserve the total mass and energy in a discrete sense, and the rate of convergence is proved, without any restriction on the grid ratio, to be of the order of in the discrete ‐norm with time step and mesh size . To overcome the challenge of nonlinearity, we also constructed two other schemes based on improved scalar auxiliary variable approaches by transforming the CSBEs into an equivalent new system that involves solving linear systems with constant coefficients at each time step. Furthermore, their algorithms are supplied, and the time‐consuming challenge resulting from the coupled problem is addressed. Finally, numerical examples are given to illustrate the excellent long‐time conservation behaviors of the presented schemes and to verify their effectiveness and correctness.