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arXiv · 2503.00633

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

Guillaume Bal·Anjali Nair
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Abstract

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 $\Delta z$ 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 It\^o-Schr\"odinger case and on integral (Duhamel) expansions for the paraxial model. Several numerical simulations illustrate and confirm the theoretical findings.

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Guillaume Bal, Anjali Nair. 2025-03-01. Splitting algorithms for paraxial and It\^o-Schr\"odinger models of wave propagation in random media. https://arxiv.org/abs/2503.00633

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