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Methods for the numerical solution of the nonlinear Schrödinger equation. (English) Zbl 0555.65061

The paper considers approximate solutions to the nonlinear Schrödinger equation \(iu_ t+\Delta u+\lambda | u|^{p-1}u=0\) together with two invariants for the solution u. Two methods are considered. The first is a finite difference scheme which the author calls a ”leap frog” scheme. For this method a bound for the difference between the approximate and exact solution is proved assuming that \(k<C_ 1\) and \(h<C_ 2\) where h and k are the grid lengths and \(C_ 1\) and \(C_ 2\) are positive constants.
The second method may be described as a Crank-Nicolson-Galerkin method and has the additional advantage that the invariants are conserved by the approximate solutions. In this method differences are used for the time integration and finite elements are used for the space discretion. A parameter \(h\in (0,1)\) is used as an index for the finite element functions and given that \(k=o(h)\) where k is the step size in time a bound is derived for the error difference whenever \(h<C_ 1\) where \(C_ 1\) is a positive constant.
Reviewer: B.Burrows

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin 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
35Q99 Partial differential equations of mathematical physics and other areas of application
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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