Splitting algorithms for paraxial and Itô-Schrödinger models of wave propagation in random media
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| Format: | Preprint |
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2025
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| _version_ | 1866916638704533504 |
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| author | Bal, Guillaume Nair, Anjali |
| author_facet | Bal, Guillaume Nair, Anjali |
| contents | 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 $Δz$ 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 $θ\llΔz$. More precisely, we obtain a convergence of order $Δz$ in mean-square sense while the errors on statistical moments are of order $(Δz)^2$ as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when $Δz$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splitting algorithms for paraxial and Itô-Schrödinger models of wave propagation in random media Bal, Guillaume Nair, Anjali Numerical Analysis Mathematical Physics Analysis of PDEs Probability 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 $Δz$ 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 $θ\llΔz$. More precisely, we obtain a convergence of order $Δz$ in mean-square sense while the errors on statistical moments are of order $(Δz)^2$ as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when $Δz$ 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. |
| title | Splitting algorithms for paraxial and Itô-Schrödinger models of wave propagation in random media |
| topic | Numerical Analysis Mathematical Physics Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2503.00633 |