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 It\^o-Schr\"odinger 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 Δ 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 θ of the propagating signal satisfies θΔ z. More precisely, we obtain a convergence of order Δ z in mean-square sense while the errors on statistical moments are of order (Δ z)² as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when Δ z is not the smallest scale of the problem. The analysis is based on equations satisfied by statistical moments in the It\^o-Schr\"odinger case and on integral (Duhamel) expansions for the paraxial model. Several numerical simulations illustrate and confirm the theoretical findings.