Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior

Philippe  Chartier; Norbert J.  Mauser; Florian  Méhats; Yong  Zhang

doi:10.3934/dcdss.2016053

Article Contents

2016, Volume 9, Issue 5: 1327-1349. Doi: 10.3934/dcdss.2016053

This issue Previous Article A multiscale finite element method for Neumann problems in porous microstructures Next Article The IDSA and the homogeneous sphere: Issues and possible improvements

Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior

1.
INRIA-Rennes, IRMAR and ENS Bruz, Campus de Beaulieu, 35042 Rennes Cedex
2.
Wolfgang Pauli Institute c/o Fak. Mathematik, University Wien, Nordbergstrasse 15, 1090 Vienna
3.
IRMAR, Université de Rennes 1 and INRIA-Rennes, Campus de Beaulieu, 35042 Rennes Cedex

Received: October 2014

Revised: July 2015

Published online: October 2016

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we present the Stroboscopic Averaging Method (SAM), recently introduced in [7,8,10,13], which aims at numerically solving highly-oscillatory differential equations. More specifically, we first apply SAM to the Schrödinger equation on the $1$-dimensional torus and on the real line with harmonic potential, with the aim of assessing its efficiency: as compared to the well-established standard splitting schemes, the stiffer the problem is, the larger the speed-up grows (up to a factor $100$ in our tests). The geometric properties of SAM are also explored: on very long time intervals, symmetric implementations of the method show a very good preservation of the mass invariant and of the energy. In a second series of experiments on $2$-dimensional equations, we demonstrate the ability of SAM to capture qualitatively the long-time evolution of the solution (without spurring high oscillations).

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Mathematics Subject Classification: 34K33, 37L05, 35Q55.

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