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<p>This paper investigates the stochastic Galerkin method for solving the time-dependent radiative transfer equation (RTE) with random, highly oscillatory periodic scattering coefficients, addressing the coupled effects of the small-scale parameter and the randomness. We first prove the well-posedness and regularity of the solution in the random space, justifying the application of stochastic Galerkin methods. We then identify the homogenization limit of the numerical solution and establish rigorous error estimates to show that the stochastic Galerkin solution accurately captures its homogenization limit, and this limit approximates the theoretical homogenized solution with spectral convergence. Our analysis provides the first unified treatment of stochastic discretization and homogenization limits, bridging a critical gap in the numerical analysis of the RTE in random media. Numerical experiments validate our theoretical findings, demonstrating both the correct asymptotic behavior in the homogenization limit and the spectral convergence with respect to the expansion order.</p>

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Stochastic Galerkin Methods for Radiative Transfer Equations with Uncertain and Highly Oscillatory Scattering Coefficients

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • Chuqi Zheng,
  • Qin Li,
  • Xinghui Zhong

摘要

This paper investigates the stochastic Galerkin method for solving the time-dependent radiative transfer equation (RTE) with random, highly oscillatory periodic scattering coefficients, addressing the coupled effects of the small-scale parameter and the randomness. We first prove the well-posedness and regularity of the solution in the random space, justifying the application of stochastic Galerkin methods. We then identify the homogenization limit of the numerical solution and establish rigorous error estimates to show that the stochastic Galerkin solution accurately captures its homogenization limit, and this limit approximates the theoretical homogenized solution with spectral convergence. Our analysis provides the first unified treatment of stochastic discretization and homogenization limits, bridging a critical gap in the numerical analysis of the RTE in random media. Numerical experiments validate our theoretical findings, demonstrating both the correct asymptotic behavior in the homogenization limit and the spectral convergence with respect to the expansion order.