Linearized and decoupled structure‐preserving finite difference methods and their analyses for the coupled Schrödinger–Boussinesq equations

Abstract

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 𝒪false(τ2+hx2false) 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 𝒪false(τ2+hx2+hy2false) in L2‐ and H2‐norms. By the embedding relations H2false(normalΩfalse)⊂Cfalse(trueΩ‾false), we further derive that the numerical solutions are convergent with an order of 𝒪false(τ2+hx2+hy2false) in H1‐ and Lnormal∞‐norms. Finally, numerical results confirm the exactness of theoretical results, and the efficiency of the proposed algorithms.

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

“…Although FFT‐based implementations make these schemes appear efficient, the solutions for $$$ v $$$ and $$$ \phi $$$ remain coupled, making them time‐consuming and unsuitable for parallel simulations. In earlier studies [16, 17], semi‐implicit energy‐preserving finite difference schemes were proposed for one‐ and two‐dimensional CSBEs, achieving second‐order spatial and temporal convergence rates using the discrete energy method.…”