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A numerical study of the nonlinear Schrödinger equation. (English) Zbl 0555.65060
The paper studies the numerical solution of the nonlinear cubic Schrödinger equation \(i(\partial u/\partial t)+(\partial^ 2u/\partial x^ 2)+u| u|^ 2=0\). By writing \(u=v+iw\) this can be transformed to the vector equation \(\partial u/\partial t+A(\partial^ 2u/\partial x^ 2)+f(\underline u)=0\) where A is a \(2\times 2\) matrix. The method of solution proposed is a finite element Galerkin method where an approximation U is written in the form \(U(x,t)=\sum^{N}_{j=1}\alpha_ j(t)\psi_ j(x)\) for some specified functions \(\psi_ j\) and the \(\alpha_ j\) are found from the iterative scheme \((M+rS)\alpha^{m+1}=[M-rS]\alpha^ m-\tau MF((\alpha^*+\alpha^ m))\). The step lengths for x and t are h and \(\tau\) and \(r=\tau /h^ 2\). M is a mass matrix, S a stiffness matrix and F is a nonlinear vector function. This scheme uses an intermediate time \(t^*=(m+\beta)\tau\), \(\beta >0\) to aid the treatment of F. It is found that the best choice of \(\beta\) is \(1+0(\tau)\) and that the scheme is stable to small local perturbations if \(\beta \leq 1+0(\tau)\). Numerical results are given for three example problems and results are also given for the finite difference scheme when M is replaced by I. This corresponds to the usual Crank-Nicolson scheme. The analysis uses the discrete least squares norm but it is pointed out that some nodal values may grow as t increases.
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
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q99 Partial differential equations of mathematical physics and other areas of application
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