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In this article, an exponential wave integrator sine pseudospectral method is formulated and analysed for discretizing the coupled Schrödinger–Boussinesq system. Our method is based on the utilizations of sine pseudospectral discretization for spatial derivatives followed by exponential wave integrator for temporal derivatives in phase space. The scheme is fully explicit and very efficient due to discrete sine transform (DST), and it is of spectral accuracy in space and second-order accuracy in time. Numerical analysis of the proposed method is carried out and rigorous error estimates are established via discrete energy and induction argument method. The numerical results are reported to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this study, we propose an efficient split-step compact finite difference (SSCFD) method for computing the coupled Gross–Pitaevskii (CGP) equations. The coupled equations are divided into two parts, nonlinear subproblems and linear ones. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, the midpoint and trapezoidal rules are applied approximately. At the same time, the split order is not reduced. For the linear ones, compact finite difference cannot be designed directly. To circumvent this problem, a linear transformation is introduced to decouple the system, which can make the split-step method be used again. Additionally, the proposed SSCFD method also holds for the coupled nonlinear Schrödinger (CNLS) system with time-dependent potential. Finally, numerical experiments for CGP equations and CNLS equations are well simulated, conservative properties and convergence rates are demonstrated as well. It is shown from the numerical tests that the present method is efficient and reliable. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the convergence of explicit finite difference schemes which contain a Richardson scheme and a leap-frog scheme for computing the coupled Gross--Pitaevskii equations in high space dimensions. We establish the optimal error estimates of our schemes at the order of O(τ² + h²) in the l∞-norm with the time step τ and the mesh size h. Besides the standard techniques of the energy analysis method, the key techniques in the analysis is to use the method of induction argument and order reduction. The numerical results are reported to confirm our theoretical analysis for the numerical methods. [ABSTRACT FROM AUTHOR]

In this article, a decoupled and linearized compact difference scheme is investigated to solve the coupled Schrödinger-Boussinesq equations numerically. We establish the convergence rates for the error at the order of in the -norm with the time step τ and mesh size h. The linear scheme is proved to conserve the total energy which is defined as a recursion relationship. Due to the difficulty in obtaining the priori estimate from the discrete energy, we utilize cut-off function technique to prove the convergence. The numerical results are reported to verify the theoretical analysis, and the numerical comparison between our scheme with previous methods are conducted to show the efficiency of our scheme. [ABSTRACT FROM AUTHOR]