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Zbl 0739.65096
Akrivis, Georgios D.; Dougalis, Vassilios A.; Karakashian, Ohannes A.
On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. (English)
[J] Numer. Math. 59, No.1, 31-53 (1991). ISSN 0029-599X; ISSN 0945-3245/e

For approximating the solution of a weakly nonlinear Schrödinger equation $L\sp 2$ conservative schemes are studied. The schemes are based on a space-discretization by the Galerkin method (in $H\sp 1\sb 0$). 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.
[R.Farzan (Budapest)]

MSC 2000:: *65Z05 Applications to physics
65M06 Finite difference methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
65M60 Finite numerical methods (IVP of PDE)
35Q55 NLS-like (nonlinear Schroedinger) equations

Keywords: weakly nonlinear Schrödinger equation; conservative schemes; Galerkin method; Crank-Nicolson type methods; convergence; Newton method of ``inner'' iterations; numerical results

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