An efficient asymptotic preserving Monte Carlo method for frequency-dependent radiative transfer equations

TL;DR

The paper tackles frequency-dependent radiative transfer by developing an asymptotic-preserving Monte Carlo method that couples a low-dimensional macro-convection–diffusion system with a high-dimensional microscopic transport equation under a multi-group discretization. Using a semi-Lagrangian flux closure, a theta_g-based correction, and a Picard predictor-corrector, the authors decouple space and frequency in the nonlinear coupling and solve the macro part with a hybrid FV-MC scheme while the micro part remains MC-based with a known source. They establish formal AP analysis in the diffusive limit and demonstrate substantial efficiency gains over IMC, especially in optically thick regimes, across a suite of 1D and 2D tests, including Marshak waves, Larsen’s problem, and a hohlraum example. The framework naturally handles both FRTE and gray limits via reformulations that relate group quantities to the total radiation through $4\pi B_g=b_g\phi$ and $4\pi\nabla B_g=(b_g+\frac{T}{4}\partial_T b_g)\nabla\phi$, and provides a path toward variance reduction to manage MC noise in the diffusion-dominated regime.

Abstract

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.

Figures (11)

• Figure 5.1: Comparison of the material and radiation temperatures using the EMC and IMC methods at $t = 1.0\,\text{ns}$, with $\Delta t =0.0025\,\text{ns}$ ($\text{CFL} \approx 15$) for the infinite medium problem.
• Figure 5.2: Figure of Merit for the material and radiation temperatures using the EMC and IMC methods.
• Figure 5.3: Comparisons of the material and radiation temperatures using the EMC and IMC methods at $t = 1.0\,\text{ns}$, with $\Delta t =0.0025\,\text{ns}$ ($\text{CFL} \approx 15$) for homogeneous Marshark wave problem when $\sigma_0 = 10 \, \mathrm{keV}^{7 / 2} / \mathrm{cm}$.
• Figure 5.4: Comparisons of the material and radiation temperatures using the EMC and IMC methods at $t = 1.0\,\text{ns}$, with $\Delta t =0.0025\,\text{ns}$ ($\text{CFL} \approx 15$) for homogeneous Marshark wave problem when $\sigma_0 = 1000 \, \mathrm{keV}^{7 / 2} / \mathrm{cm}$.
• Figure 5.5: Comparisons of the material and radiation temperatures using the EMC and IMC methods at $t = 1.0\,\text{ns}$, with $\Delta t =0.00125\,\text{ns}$ ($\text{CFL} \approx 8$) for heterogeneous Marshark wave problem A.

Theorems & Definitions (14)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 3.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Proposition 4.1
• Lemma 4.2
• Proposition 4.3