In this paper, a three-level finite difference method (FDM), which preserves energy and mass conservative laws, is first derived for one-dimensional (1D) nonlinear coupled Schrödinger–Boussinesq equations (NCSBEs). Using the discrete energy analysis method, error estimations have been proven to be in L²-, H¹-, and L^∞-norms, respectively. Secondly, this energy- and mass-preserving FDM (EM-FDM) is generalized to solve NCSBEs in two dimensions. Also, by the discrete energy method, it is shown that numerical solutions provided by the generalized EM-FDM method for two-dimensional (2D) NCSBEs converge to exact solutions with a convergent rate of in L²- and H²-norms. By the embedding relations , we further derive that the numerical solutions are convergent with an order of in - and -norms. Finally, numerical results confirm the exactness of theoretical results, and the efficiency of the proposed algorithms.

中文翻译：

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耦合薛定谔-Boussinesq方程的线性化解耦保结构有限差分法及其分析

在本文中，首先针对一维 (1D) 非线性耦合薛定谔-布西涅斯克方程 (NCSBE)推导出了一种保持能量和质量保守定律的三级有限差分法 (FDM )。使用离散能量分析方法，误差估计已被证明分别为L ² -、H ¹ - 和L ^{∞ -}范数。其次，将这种能量和质量保持不变的 FDM (EM-FDM) 推广到解决二维 NCSBE。另外，通过离散能量的方法，它示出了由广义EM-FDM方法用于二维（2D）NCSBEs提供数值解收敛于精确解用的收敛速度在大号² - 和H ^{2 -}范数。通过嵌入关系，我们进一步推导出数值解以in -and- norms的顺序收敛。最后，数值结果证实了理论结果的准确性，以及所提出算法的效率。