Plane Waves Numerical Stability of Some Explicit Exponential Methods for Cubic Schrödinger Equation

Abstract

Numerical stability when integrating plane waves of cubic Schrödinger equation is thoroughly analysed for some explicit exponential methods. We center on the following secondorder methods: Strang splitting and Lawson method based on a one-parameter family of 2-stage 2nd-order explicit Runge-Kutta methods. Regions of stability are plotted and numerical results are shown which corroborate the theoretical results. Besides, a technique is suggested to avoid the possible numerical instabilities which do not correspond to continuous ones. Mathematics subject classification: 65M12,65M15, 65M99

Plane Wave Stability of the Split-Step Fourier Method for the Nonlinear Schrödinger Equation

“…This enables us to use the technique of modulated Fourier expansions for its long-time analysis, see Sect. 4. It is likely that normal form techniques in the spirit of [9,10] would lead to similar conclusions, but we have not worked out the details.…”

“…This is done by examining the eigenvalues of the linearization around a plane wave of a numerical method applied to (1). Such a linear stability analysis has recently been extended to different numerical methods [4,6,21,22].…”

“…In the present paper, we take up this line of research. In contrast to previous work [4,6,21,22,25], however, we are interested in the long-time behaviour of a numerical method near plane waves, and hence a linear stability analysis is of limited use. We pursue the question as to whether the remarkably stable behaviour on long time intervals of the exact solution near plane waves [7] is shared by one of the most popular numerical methods for the nonlinear Schrödinger equation, the split-step Fourier method [19].…”

Metastable energy strata in numerical discretizations of weakly nonlinear wave equations

“…In particular and in contrast to the nonlinear wave equation considered in the present paper, there is no formation of energy strata in the other modes. The stability in numerical discretizations of these plane wave solutions under small perturbations of the initial value is an old question [22] and has been studied on short time intervals [4,7,19,20,22] and on long time intervals [10].…”

Adaptive splitting methods for nonlinear Schrödinger equations in the semiclassical regime

“…The seminal work [26] provides a rigorous convergence analysis for the Strang splitting method applied to the Schrödinger-Poisson and cubic Schrödinger equations; extensions to Gross-Pitaevskii equations and high-order splitting methods as well as a study of the effect of spatial discretization by spectral methods are given for instance in [20,25,29]. The question of long-time integration, with view on near-conservation of invariants under time discretization by splitting methods, is considered in [18,19,21], see also references given therein and [11,14] for the analysis of related classes of methods.…”