An asymptotic-preserving method for the three-temperature radiative transfer model

TL;DR

This work addresses the computational challenge of the three-temperature radiative transfer model by introducing an asymptotic-preserving (AP) time-splitting scheme. The method decomposes the model into a microscopic GRTE-like part and a macroscopic part, solved via a fully implicit alternating iteration that preserves energy. It provably recovers the diffusion limit as $\\epsilon\\to0$ and the two-temperature limit as $\\kappa\\to\\infty$, and is validated through comprehensive benchmarks including homogeneous problems, Marshak waves, and a 2D Riemann problem, showing robust accuracy and efficiency across multiscale regimes. The approach offers a practical, scalable framework for complex radiative-transfer simulations in inertial confinement fusion and suggests pathways for extensions to radiation-fluid coupling and plasma contexts.

Abstract

We present an asymptotic-preserving (AP) numerical method for solving the three-temperature radiative transfer model, which holds significant importance in inertial confinement fusion. A carefully designedsplitting method is developed that can provide a general framework of extending AP schemes for the gray radiative transport equation to the more complex three-temperature radiative transfer model. The proposed scheme captures two important limiting models: the three-temperature radiation diffusion equation (3TRDE) when opacity approaches infinity and the two-temperature limit when the ion-electron coupling coefficient goes to infinity. We have rigorously demonstrated the AP property and energy conservation characteristics of the proposed scheme and its efficiency has been validated through a series of benchmark tests in the numerical part.

Figures (17)

• Figure 1: (Tests of the AP property in Sec. \ref{['sec:AP']}) The $l_2$ error between the numerical solution with mesh size $N=50,100,200,400$ and $800$ and the reference solution. Here, the reference solution is obtained by the same numerical method with $N=1600$. The parameter $\epsilon$ is set as $\epsilon = 1, 0.1$ and $0.001$ and $\kappa = 1$.
• Figure 2: (Tests of the AP property in Sec. \ref{['sec:AP']}) The $l_2$ error between the numerical solution with grid size $N=50,100,200,400$ and $800$ and the reference solution. Here, the reference solution is obtained by the same numerical method with $N = 1600$. The parameter $\kappa$ is set as $\kappa = 1, 10$ and $100$ and $\epsilon = 1$.
• Figure 3: (Homogeneous model problem in Sec. \ref{['sec:ex1']}) The time evolution of the numerical solution, the reference solution by the same method with a time step length of $\Delta t=0.01\times 2^{-10}$ns, and the reference solution from EVANS20071695. The dot-dashed lines are the numerical solution by the AP $P_N$ method. The solid lines are the reference solution by the same method with a time step length of $\Delta t=0.01\times 2^{-10}$ns, and the symbols are the reference solutions from EVANS20071695. (a) Solutions of Problem I, (b) Solutions of Problem II.
• Figure 4: (Homogeneous model problem in Sec. \ref{['sec:ex1']}) The $l_2$ error of the numerical solution by the AP $P_N$ method and the reference solution by the same method with a time step length of $\Delta t=0.01\times 2^{-10}$ns. (a) The error for the first-order $P_N$ method, (b) The error for the second-order $P_N$ method.
• Figure 5: (1D Marshak wave problem without conduction terms in Sec. \ref{['sec:marshak_ex2']}) The numerical solution of $T_r, T_e$ and $T_i$ for $\kappa = 1$ at different times. The symbols are the numerical solution by the AP $P_N$ method, and the solid lines are the reference solution obtained from the method in enaux2020numerical.

Theorems & Definitions (2)

• Remark 1
• Remark 2