Uniformly accurate time-splitting schemes for NLS in the semiclassical limit

Philippe Chartier (1, 2) , Loïc Le Treust (1, 2) , Florian Méhats (2, 1)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes

We construct new numerical methods for the nonlinear Schrödinger equation in the semiclassical limit. We introduce time-splitting schemes for a phase-amplitude reformulation of the equation where the dimensionless Planck constant is not a singular parameter anymore. Our methods have an accuracy which is spectral in space, of second or fourth-order in time, and independent of the Planck constant before the formation of caustics. The scheme of second-order preserves exactly the L^2 norm of the solution, as the flow of the nonlinear Schrödinger equation does. In passing, we introduce a new time-splitting method for the eikonal equation, whose precision is spectral in space and of second or fourth-order in time.

Mots clés: eikonal equation; splitting schemes; Madelung transform; semiclassical limit; numerical simulation; uniformly accurate; nonlinear Schr\"odinger equation

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Philippe Chartier, Loïc Le Treust, Florian Méhats. Uniformly accurate time-splitting schemes for NLS in the semiclassical limit. 2015. ⟨hal-01140880v1⟩