# American Institute of Mathematical Sciences

October  2016, 9(5): 1327-1349. doi: 10.3934/dcdss.2016053

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

Received  October 2014 Revised  July 2015 Published  October 2016

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).
Citation: Philippe Chartier, Norbert J. Mauser, Florian Méhats, Yong Zhang. Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1327-1349. doi: 10.3934/dcdss.2016053
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