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Zbl 0762.65070
Akrivis, Georgios D.
Finite difference discretization of the cubic Schrödinger equation. (English)
[J] IMA J. Numer. Anal. 13, No.1, 115-124 (1993). ISSN 0272-4979; ISSN 1464-3642/e

The initial-boundary value problem for the cubic Schrödinger equation in one space dimension is investigated in the framework in the finite difference approach. The author proposes a Crank-Nicolson type discretization for which he proves the existence theorem and establishes error estimates as well as some conservation and convergence properties.\par The essential tools used are a Brouwer-type fixed point theorem and the discrete version of Gronwall's inequality. The corresponding nonlinear discrete equations are then linearized at each time level by Newton's method and solved by an algorithm developed for this purpose, for which an error estimate is established.
[O.Titow (Berlin)]

MSC 2000:: *65Z05 Applications to physics
65M06 Finite difference methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
65M15 Error bounds (IVP of PDE)
35Q55 NLS-like (nonlinear Schroedinger) equations

Keywords: cubic Schrödinger equation; finite difference; Crank-Nicolson type discretization; existence; conservation; convergence; Brouwer-type fixed point theorem; Gronwall's inequality; Newton's method

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