Split-step methods for the solution of the nonlinear Schrödinger equation. (English) Zbl 0597.76012

Weideman, J. A. C.; Herbst, B. M.

SIAM J. Numer. Anal. 23, 485-507 (1986).

Summary: A split-step method is used to discretize the time variable for the numerical solution of the nonlinear Schrödinger equation (NLS). The space variable is discretized by means of a finite difference and a Fourier method. These methods are analyzed with respect to various physical properties represented in the NLS. In particular it is shown how a conservation law, dispersion and instability are reflected in the numerical schemes. Analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived. These results are demonstrated by numerical examples.

MSC:

76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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
76M99	Basic methods in fluid mechanics

Keywords:

weakly nonlinear periodic wave trains on deep water; split-step method; numerical solution; nonlinear Schrödinger equation; finite difference; Fourier method; conservation law; dispersion; instability; numerical instabilities; wave train solutions