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On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. (English) Zbl 0739.65096
For approximating the solution of a weakly nonlinear Schrödinger equation $$L^ 2$$ conservative schemes are studied. The schemes are based on a space-discretization by the Galerkin method (in $$H^ 1_ 0$$). For the time-discretization two Crank-Nicolson type methods are used. Existence, uniqueness and convergence of the approximate solution to the exact one are proved. Existence, uniqueness and convergence of the approximate solution to the exact one are proved. The Newton method of “inner” iterations for solving the system of complex nonlinear equations is discussed. Numerical results are given.

##### MSC:
 65Z05 Applications to the sciences 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations)
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