Analysis of the linearly energy- and mass-preserving finite difference methods for the coupled Schrödinger-Boussinesq equations

Abstract

This paper is concerned with numerical solutions of one-dimensional (1D) and two-dimensional (2D) nonlinear coupled Schrödinger-Boussinesq equations (CSBEs) by a type of linearly energy- and mass- preserving finite difference methods (EMP-FDMs) because the existing EMP-FDMs for CSBEs are nonlinear and time-consuming, and corresponding theoretical analyses are not easy to generalize high-dimensional problems. Firstly, a linearized EMP-FDM is created for solving 1D CSBEs. By using the discrete energy analysis method, it is shown that this scheme is uniquely solvable and convergent with an order of O(Δt2+hx2) in L∞-, H1- and L2-norms, and corresponding numerical energy and mass are conservative. Then, by generalizing this type of EMP-FDM, a linearized EMP-FDM is developed for solving 2D CSBEs. Theoretical findings including the convergence, the discrete conservative laws, and the solvability of this numerical algorithm are strictly derived in detail by using the discrete energy analysis method as well. Finally, numerical results confirm the efficiency of our algorithms and the exactness of the theoretical results.