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

TL;DR

This paper introduces an efficient asymptotic-preserving Monte Carlo method for frequency-dependent radiative transfer equations, combining a reformulated model, hybrid solution techniques, and iterative decoupling to improve computational efficiency across frequencies.

Contribution

It develops a novel AP-MC method for frequency-dependent RTEs, incorporating a correction for free streaming and a hybrid solver with iterative decoupling, advancing computational efficiency.

Findings

Method achieves larger time steps independent of light speed and frequency.

Numerical experiments confirm high efficiency and AP property.

The approach effectively handles nonlinear coupling across multiple frequency groups.

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.