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This paper is concerned with numerical solution of the two-dimensional Klein-Gordon-Dirac equations by a fourth-order compact finite difference method in space and an energy-preserving Crank-Nicolson-type discretization in time. For convenience of illustrating the conservative properties and investigating the convergence results, we convert the component-wise form of the proposed scheme into an equivalent matrix-vector form. By using the mathematical induction argument and standard energy method, we establish the optimal error estimates under condition τ ≤ 1 | ln (h) | with time step τ and mesh size h. Compared with the condition τ = O (h 1 2) required by existing results in literature for two-dimensional case, this greatly relaxes the dependence of the time step on the grid size. The convergence order of the scalar ϕ and the 2-spinor ψ is of O (τ 2 + h 4) in the maximum norm and the discrete H 1 -norm, respectively. In addition, by using the orthogonal diagonalization technique, a fast solver is designed to solve the proposed scheme. Several numerical results are reported to verify the error estimates and the discrete conservation laws. [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, 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, 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]