A numerical study of the nonlinear Schrödinger equation
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.
scientific article; zbMATH DE number 3885248
scientific article