Asymptotics of a Time-Splitting Scheme for the Random Schrödinger Equation with Long-Range Correlations

Abstract

This work is concerned with the asymptotic analysis of a time-splitting scheme for the Schrödinger equation with a random potential having weak amplitude, fast oscillations in time and space, and long-range correlations. Such a problem arises for instance in the simulation of waves propagating in random media in the paraxial approximation. The high-frequency limit of the Schrödinger equation leads to different regimes depending on the distance of propagation, the oscillation pattern of the initial condition, and the statistical properties of the random medium. We show that the splitting scheme captures these regimes in a statistical sense for a time stepsize independent of the frequency. Résumé. Nous nous intéressons au comportement asymptotique d'un schéma de time-splitting pour l'équation de Schrödinger avec potential aléatoire de faible amplitude, oscillant rapidement en temps et en espace, et présentant des corrélations longue portée. Cetteéquation décrit par exemple la propagation d'une onde dans un milieu aléatoire dans le cadre de l'approximation paraxiale. La limite haute-fréquence de l'équation de Schrödinger mèneà différents régimes selon la distance de propagation, la fréquence d'oscillation de la condition initiale, et la statistique du milieu aléatoire. Nous montrons que le schéma de splitting capture statistiquement ces régimes asymptotiques pour un pas de discrétisation en temps indépendant de la fréquence d'oscillation.

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

“…Hence, using Proposition 3.2, we obtain the convergence in law of p ε L to p 0 L in C 0 ([−T, T ], L 2 w (R 2 )). To conclude, we use the Skorohod's representation theorem [6, Theorem 6.7 pp.70]: there exist a probability space (Ω,T ,P) and random variables p ε L and p 0 L , with the same laws as p ε L and p 0 L , respectively, and such that lim Sincef 0 (ω, κ) has a compact support with respect to ω, so dop ε ω andΨ ω according to (52) and (22). The Jensen's inequality then yields sup t∈[−T,T ] p ε L (t, ·) − p 0 L (t, ·) L 2 (R 2 ) ≤ CI ε .…”

“…It is not always the case in practice, as is pointed out in [12,23,31] for geophysical problems, wave propagation in turbulent atmosphere, or medical imaging. This has then stimulated recent mathematical works on wave propagation in random media with long-range dependence [2,18,20,21,22,25,26]. It is shown there that the wave dynamics in such media can be in great contrast with that of waves in media with rapidly decaying correlations.…”

Analysis of a splitting scheme for a class of random nonlinear partial differential equations

“…There exists a lot of works dealing with Asymptotic-Preserving (AP) property in various problems (for instance [4,10,13,19,23,25]). In particular for splitting schemes for Schrödinger and/or random equations, we mention for instance [2,3,7,21,27].…”