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In this paper, two conservative finite difference schemes for fractional Schrödinger–Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi‐implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis. [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]

In this article, we formulate two orthogonal spline collocation schemes, which consist of a nonlinear and a linear scheme for solving the coupled Schrödinger–Boussinesq equations numerically. Firstly, the conservation laws of our schemes are derived. Secondly, the existence solutions of our schemes are investigated. Thirdly, the convergence and stability of the nonlinear scheme are analyzed by means of discrete energy methods, while the convergence of the linear scheme is proved by cut-off function technique. Finally, numerical results are reported to verify our theoretical analysis for the numerical methods. [ABSTRACT FROM AUTHOR]

In this article, two conserved compact finite difference schemes for solving the nonlinear coupled Schrödinger-Boussinesq equation are proposed. The conservative property, existence, convergence, and stability of the difference solutions are theoretically analyzed. The numerical results are reported to demonstrate the accuracy and efficiency of the methods and to confirm our theoretical analysis. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1667-1688, 2016 [ABSTRACT FROM AUTHOR]