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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 +Δu+λ|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(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:
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35Q99PDE of mathematical physics and other areas
81Q05Closed and approximate solutions to quantum-mechanical equations