The Lie-Trotter splitting for nonlinear evolutionary problems with critical parameters: a compact local error representation and application to nonlinear Schrodinger equations in the semiclassical regime

Abstract

In the present work, the error behaviour of the first-order Lie–Trotter splitting method for nonlinear evolutionary problems is analysed. In particular, a local error representation that is suitable in the presence of unbounded nonlinear operators and critical parameters is deduced. Such local error expansions together with stability bounds are the basic ingredients in the derivation of convergence estimates. Essential tools in the theoretical analysis are an abstract formulation of differential equations on function spaces and the formal calculus of Lie derivatives. In order to illustrate the general approach, the application of the Lie–Trotter splitting method to Schrödinger equations in the semiclassical regime is studied. From numerical computations presented in the literature, it is expected that exponential operator splitting methods are favourable for the time integration of nonlinear Schrödinger equations, provided that the time-step size is suitably chosen depending on the magnitude of the critical parameter. For the least technical example method, the first-order Lie–Trotter splitting method, this is substantiated by theoretical considerations for the time-dependent Gross–Pitaevskii equation and confirmed by numerical examples. Numerical illustrations for higher-order exponential operator splitting methods complement the considerations.