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

Gomez, Christophe; Pinaud, Olivier

ESAIM: Mathematical Modelling and Numerical Analysis , Multiscale problems and techniques. Special Issue, Tome 48 (2014) no. 2, pp. 411-431.

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.

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