An asymptotic-preserving IMEX PN method for the gray model of the radiative transfer equation

Abstract

An asymptotic-preserving (AP) implicit-explicit PN numerical scheme is proposed for the gray model of the radiative transfer equation, where the first- and second-order numerical schemes are discussed for both the linear and nonlinear models. The AP property of this numerical scheme is proved theoretically and numerically, while the numerical stability of the linear model is verified by Fourier analysis. Several classical benchmark examples are studied to validate the efficiency of this numerical scheme.

Figures (31)

• Figure 1: (Numerical stability by Fourier analysis in Sec. \ref{['sec:Fourier']}) Numerical stability of the numerical scheme for the linear system \ref{['eq:1D_linear_RTE']} by Fourier analysis for the expansion number $M = 7, 15$ and $31$. Here the blue area is the stable region and the yellow area is the unstable region.
• Figure 2: (The verification of the AP property in Sec. \ref{['sec:ex1']}) The $l_2$ error of the numerical solution obtained by the first-order scheme \ref{['eq:first_RTE']} with mesh sizes $N = 32, 64, 128$ and $256$ and the reference solution. The reference solution is obtained by the same numerical method with mesh size $N = 1024$. The parameters $\epsilon$ tested are $\epsilon = 1, 10^{-1}, 10^{-3}$ and $10^{-6}$. (a) The $l_2$ error of the radiative temperature $T_r$. (b) The $l_2$ error of the material temperature $T$.
• Figure 3: (The verification of the AP property in Sec. \ref{['sec:ex1']}) The $l_2$ error of the numerical solution obtained by the second-order scheme \ref{['eq:second_RTE']} with mesh sizes $N = 32, 64, 128$ and $256$ and the reference solution. The reference solution is obtained by the same numerical method with mesh size $N = 1024$. The parameters $\epsilon$ tested are $\epsilon = 1, 10^{-1}, 10^{-3}$ and $10^{-6}$. (a) The $l_2$ error of the radiative temperature $T_r$. (b) The $l_2$ error of the material temperature $T$.
• Figure 4: (1D plane source problem in Sec. \ref{['sec:ex2']}) $I_{0}$ of the plane source problem for different times. Here, the blue circle lines are obtained from IMEX-IM, the orange diamond lines are the reference solution obtained by the $S_N$ method, and the yellow square lines are the analytical solution in Ganapol2001. (a) Initial condition $t = 0$. (b) $t = 1$. (c) $t = 5$.
• Figure 5: (1D plane source problem in Sec. \ref{['sec:ex2']}) $I_{0}$ of the plane source problem for different $\epsilon$. Here, the blue circle lines are obtained from IMEX-IM, the orange diamond lines are the reference solution obtained by the $S_N$ method, and the yellow square lines are the numerical solution of the diffusion limit equation \ref{['eq:1D_linear_limit']}. (a) $\epsilon = 1$. (b) $\epsilon = 0.8$. (c) $\epsilon = 0.6$. (d) $\epsilon = 0.4$. (e) $\epsilon = 10^{-3}$. (f) $\epsilon = 10^{-6}$.

Theorems & Definitions (6)

• Lemma 1
• Remark 1
• Theorem 2: AP property of the IMEX scheme for linear RTE \ref{['eq:1D_linear_RTE']}
• Theorem 3: AP property of the IMEX scheme for RTE \ref{['eq:1D_RTE']}
• proof : Proof of Thm. \ref{['thm:AP_RTE']}
• Remark 2