Uniformly accurate time-splitting methods for the semiclassical Schrödinger equation Part 1 : Construction of the schemes and simulations

Philippe Chartier 1, 2 Loïc Le Treust 1, 2 Florian Méhats 2, 1
1 IPSO - Invariant Preserving SOlvers
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : This article is devoted to the construction of new numerical methods for the semiclassical Schrödinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter. This allows to build splitting schemes whose accuracy is spectral in space, of up to fourth order in time, and independent of epsilon before the caustics. The second-order method additionally preserves the L^2-norm of the solution just as the exact flow does. In this first part of the paper, we introduce the basic splitting scheme in the nonlinear case, reveal our strategy for constructing higher-order methods, and illustrate their properties with simulations. In the second part, we shall prove a uniform convergence result for the first-order splitting scheme applied to the linear Schrödinger equation with a potential.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-01140880
Contributor : Loïc Le Treust <>
Submitted on : Monday, January 18, 2016 - 11:05:31 AM
Last modification on : Tuesday, February 5, 2019 - 2:40:03 PM

Files

UAmethodPart1.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01140880, version 2
  • ARXIV : 1605.03446

Citation

Philippe Chartier, Loïc Le Treust, Florian Méhats. Uniformly accurate time-splitting methods for the semiclassical Schrödinger equation Part 1 : Construction of the schemes and simulations. 2015. ⟨hal-01140880v2⟩

Share

Metrics

Record views

512

Files downloads

329