Solving highly-oscillatory NLS with SAM: numerical efficiency and geometric properties

TL;DR

This paper introduces the Stroboscopic Averaging Method (SAM) for efficiently solving highly-oscillatory differential equations, demonstrating significant speed-ups and good geometric property preservation in long-time simulations.

Contribution

The paper applies SAM to Schrödinger equations, showing its efficiency and geometric property preservation, and extends its application to 2D equations for long-time evolution analysis.

Findings

SAM achieves up to 100x speed-up over standard methods.

Symmetric SAM preserves mass and energy over long times.

SAM accurately captures long-term solution behavior in 2D cases.

Abstract

In this paper, we present the Stroboscopic Averaging Method (SAM), recently introduced in [7,8,10,12], which aims at numerically solving highly-oscillatory differential equations. More specifically, we first apply SAM to the Schr\"odinger 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).