Splitting algorithms for paraxial and It\^o-Schr\"odinger models of wave propagation in random media

This paper introduces a full discretization procedure to solve wave beam propagation in random media modeled by a paraxial wave equation or an Itô-Schrödinger stochastic partial differential equation.

This method bears similarities with the phase screen method used routinely to solve such problems.

The main axis of propagation is discretized by a centered splitting scheme with step $\Delta z$ while the transverse variables are treated by a spectral method after appropriate spatial truncation.

The originality of our approach is its theoretical validity even when the typical wavelength $\theta$ of the propagating signal satisfies $\theta\ll\Delta z$.

More precisely, we obtain a convergence of order $\Delta z$ in mean-square sense while the errors on statistical moments are of order $(\Delta z)^2$ as expected for standard centered splitting schemes.

This is a surprising result as splitting schemes typically do not converge when $\Delta z$ is not the smallest scale of the problem.

The analysis is based on equations satisfied by statistical moments in the Itô-Schrödinger case and on integral (Duhamel) expansions for the paraxial model.

Several numerical simulations illustrate and confirm the theoretical findings.