In this paper, we develop an efficient asymptotic-preserving (AP) Monte Carlo (MC) method for frequency-dependent radiative transfer equations (RTEs), which is based on the AP-MC method proposed for the gray RTEs in \cite{shi2023efficient}. We follow the characteristics-based approach by Zhang et al. \cite{zhang2023asymptotic} to get a reformulated model, which couples a low dimension convection-diffusion-type equation for macroscopic quantities with a high dimension transport equation for the radiative intensity. To recover the correct free streaming limit due to frequency-dependency, we propose a correction to the reformulated macroscopic equation. The macroscopic system is solved using a hybrid method: convective fluxes are handled by a particle-based MC method, while diffusive fluxes are treated implicitly with central difference. To address the nonlinear coupling between radiative intensity and the Planck function across multiple frequency groups, we adopt a Picard iteration with a predictor-corrector procedure, which decouples a global nonlinear system into a linear system restricted to spatial dimension (independent of frequency) with scalar algebraic nonlinear equations. Once the macroscopic update is done, the transport equation, with a known emission source provided by the macroscopic variables, is efficiently solved using an implicit MC method. This approach enables larger time steps independent of the speed of light and also the frequency across a wide range, significantly enhancing computational efficiency, especially for frequency-dependent RTEs. Formal AP analysis in the diffusive scaling is established. Numerical experiments are performed to demonstrate the high efficiency and AP property of the proposed method.

翻译：本文针对频率依赖辐射传输方程，发展了一种高效的渐近保持蒙特卡洛方法。该方法基于文献\\cite{shi2023efficient}中针对灰体辐射传输方程提出的AP-MC方法。我们遵循Zhang等人\\cite{zhang2023asymptotic}的特征线方法，得到一个重构模型，该模型将宏观量的低维对流-扩散型方程与辐射强度的高维传输方程相耦合。为恢复因频率依赖性导致的正确自由流极限，我们对重构的宏观方程提出了修正。宏观系统采用混合方法求解：对流通量通过基于粒子的蒙特卡洛方法处理，而扩散通量则采用中心差分隐式处理。为解决辐射强度与普朗克函数在多个频率组间的非线性耦合，我们采用带有预测-校正过程的皮卡迭代，将全局非线性系统解耦为仅限于空间维度（与频率无关）的线性系统与标量代数非线性方程。一旦完成宏观更新，传输方程（其发射源由宏观变量已知提供）即可使用隐式蒙特卡洛方法高效求解。该方法允许采用与光速无关且跨越宽频率范围的更大时间步长，显著提升了计算效率，尤其适用于频率依赖辐射传输方程。我们建立了扩散尺度下的形式化渐近保持分析。数值实验验证了所提方法的高效性和渐近保持特性。