Energy preserving methods for nonlinear Schrödinger equations

Christophe Besse (1) , Stephane Descombes (2, 3) , Guillaume Dujardin (4, 5) , Ingrid Lacroix-Violet (6, 5)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 NACHOS - Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 Paradyse - Systèmes de particules et systèmes dynamiques
5 LPP - Laboratoire Paul Painlevé - UMR 8524
6 RAPSODI - Reliable numerical approximations of dissipative systems

This paper is concerned with the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analog of it. The Crank-Nicolson method is a well known method of order 2 but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in [10] for the cubic nonlinear Schrödinger equation. This method is also an energy preserving method and numerical simulations have shown that its order is 2. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.

Mots clés: Nonlinear Schrödinger equation; Relaxation methods; Numerical methods; Gross-Pitaevskii equation

Domaines: Analyse numérique [math.NA]; Optique [physics.optics]; Physique Quantique [quant-ph]

Christophe Besse, Stephane Descombes, Guillaume Dujardin, Ingrid Lacroix-Violet. Energy preserving methods for nonlinear Schrödinger equations. IMA Journal of Numerical Analysis, 2021, 41 (1), pp.618-653. ⟨10.1093/imanum/drz067⟩. ⟨hal-01951527⟩