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Multigrid and adaptive algorithm for solving the nonlinear Schrödinger equation. (English) Zbl 0708.65111
An application of a multigrid and adaptive algorithm for solving the initial-boundary value problem \(iu_ t-(\partial /\partial x)A(x)(\partial u/\partial x)+2| u|^ 2u+F(x,t)u=G(x,t),\quad u\in {\mathbb{R}}^ m,\) \(x_ L<x<x_ R,\quad u|_{x_ L}=u|_{x_ R}=0,\quad u|_{t=0}=u_ 0(x),\) where A(x) is a real diagonal matrix, is considered. There are estimates for the solution of the difference scheme. Some numerical examples are presented and compared with corresponding results obtained by a traditional method. There is a discussion of the method and the numerical results.
Reviewer: L.P.Lebedev

65Z05 Applications to the sciences
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI
[1] Chang, Q., Sci. sinica (ser. A), 16, 687, (1983)
[2] Chang, Q.; Xu, L., J. comput. math., 4, 191, (1986)
[3] Brandt, A., Math. comput., 31, 333, (1977)
[4] Hackbusch, W., Multigrid methods and applications, (1985), Springer-Verlag New York · Zbl 0577.65118
[5] Tana, T.R.; Ablowitz, M.J., J. comput. phys., 55, 203, (1984)
[6] Brandt, A., Multigrid techniques: guide with applications to fluid dynamics, GMD-stidier no. 85, (1984), Bonn · Zbl 0581.76033
[7] Sjoberg, A., J. math. anal. appl., 29, 569, (1970)
[8] Manikoff, A., Commun. pure appl. math., 25, 407, (1972)
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