A numerical study of the nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: A numerical study of the nonlinear Schrödinger equation
zbMATH Open document ID: 0555.65060
DOI: 10.1016/0045-7825(84)90156-7
author: David F. Griffiths
author: A. R. Mitchell
author: John Ll. Morris
published in: Computer Methods in Applied Mechanics and Engineering
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3885248
zbMATH Keywords: nonlinear cubic Schrödinger equation
zbMATH Keywords: finite element Galerkin method
zbMATH Keywords: mass matrix
zbMATH Keywords: stiffness matrix
zbMATH Keywords: Numerical results
zbMATH Keywords: Crank-Nicolson scheme
reviewed by: Brian L. Burrows
MaRDI profile type: Publication
full work available at URL: 0045-7825(84)90156-7
OpenAlex ID: W2063721225
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cites work: Solitons and the Korteweg-de Vries equation: Integrable systems in 1834--1995
cites work: Product Approximation for Non-linear Problems in the Finite Element Method
cites work: The propagation of a weak nonlinear wave
cites work: Finite-difference solutions of a non-linear Schrödinger equation
cites work: Numerical study of the regularized long-wave equation. I: Numerical methods
cites work: Method for Solving the Korteweg-deVries Equation
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cites work: Nonlinear deep water waves: Theory and experiment
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description: scientific article; zbMATH DE number 3885248