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 $\mathcal{O}(\mathrm{\Delta }{t}^2+h_x^2)$ in $L^{\infty }$-, $H^1$- and $L^2$-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.

本文涉及通过一种线性能量和质量保持有限差分方法 (EMP-FDM) 的一维 (1D) 和二维 (2D) 非线性耦合薛定谔-布西涅斯方程 (CSBE) 的数值解，因为现有的用于 CSBE 的 EMP-FDM 是非线性和耗时的，相应的理论分析不容易推广高维问题。首先，创建线性化 EMP-FDM 以解决一维 CSBE。通过使用离散能量分析方法，表明该方案是唯一可解的，并且收敛于阶数为$\mathcal{哦}(\mathrm{\Delta }{吨}^2+H_X^2)$ 在 ${升}^{\infty }$-, $H^1$- 和 ${升}^2$-范数，相应的数值能量和质量是保守的。然后，通过推广这种类型的 EMP-FDM，开发了一个线性化的 EMP-FDM 来解决 2D CSBE。该数值算法的收敛性、离散守恒律和可解性等理论结论也是采用离散能量分析方法严格详细推导出来的。最后，数值结果证实了我们算法的效率和理论结果的准确性。