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.

Stéphane Descombes (1) , Mechthild Thalhammer (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 Department of Mathematics, Faculty of Mathematics, Computer Science and Physics

In the present work, we investigate the error behaviour of exponential operator splitting methods for nonlinear evolutionary problems. 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

Mots clés: Convergence; Nonlinear evolutionary problems; Time-dependent nonlinear Schrödinger equations; Semi-classical regime; Exponential operator splitting methods; Local error representation; Convergence.

Domaines: Analyse numérique [math.NA]

Stéphane Descombes, Mechthild Thalhammer. 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.. IMA Journal of Numerical Analysis, 2013, 33 (2), pp.722-745. ⟨hal-00557593⟩