The Lie–Trotter splitting method for nonlinear evolutionary problems involving critical parameters. An exact local error representation and application to nonlinear Schrödinger equations in the semi-classical regime

In the present work, we investigate the error behaviour of exponential operator splitting methods for nonlinear evolutionary problems of the form u ′(t) = A ( u(t) ) +B ( u(t) ) , 0≤ t ≤ T , u(0) given . In particular, our concern is to deduce an exact local error representation that is suitable in the presence of critical parameters. Essential tools in the theoretical analysis including time-dependent nonlinear Schrödinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients are an abstract formulation of differential equations on function spaces and the formal calculus of Lie-derivatives. We expose the general mechanism on the basis of the least technical example method, the first-order Lie–Trotter splitting. Our conjecture that exponential operator splitting methods are favourable for the time integration of a nonlinear Schrödinger equation in the semi-classical regime, provided that the time stepsizes are suitably chosen in dependence of the magnitude of the critical parameter 0 < ε << 1, is confirmed by a numerical example for the time-dependent Gross–Pitaevskii equation iε ∂ tψ(x, t) =− 2 ε 2∆ ψ(x, t)+U(x)ψ(x, t)+θ ∣∣ψ(x, t)∣∣2 ψ(x, t) , ψ(x,0) given , x ∈ Rd , 0≤ t ≤ T , and substantiated by theoretical considerations for the Lie–Trotter splitting method. Moreover, we illustrate the ability of an embedded 4(3) splitting pair to serve as a reliable basis for a local error control. Stéphane Descombes, Laboratoire J. A. Dieudonné, Université de Nice – Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France. E-mail: sdescomb@unice.fr Mechthild Thalhammer, Institut für Mathematik, Leopold–Franzens Universität Innsbruck, Technikerstraße 13/7, 6020 Innsbruck, Austria. E-mail: mechthild.thalhammer@uibk.ac.at 2 Stéphane Descombes, Mechthild Thalhammer

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