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A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. (English) Zbl 1206.65207
The authors show that the presented method improves the accuracy of the split-step finite difference method by introducing a compact scheme for discretization of the space variable while this improvement does not reduce the stability range and does not increase the computational cost. This method also preserves some conservation laws. Numerical tests are presented to confirm the theoretical results for the new numerical method by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and the Gross-Pitaevskii equation.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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