A positive and asymptotic preserving filtered PN method for the gray radiative transfer equations

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Highlights

Abstract

This paper presents a positive and asymptotic preserving scheme for the nonlinear gray radiative transfer equations. The scheme is constructed by combining the filtered spherical harmonics (FPN) method for the discretization of angular variable and with the framework of the unified gas kinetic scheme (UGKS) for the spatial- and time-discretization. The constructed scheme is almost free of ray effects and can also mitigate oscillations in the spherical harmonics (PN) approximation. Moreover, it can be shown that the current scheme is asymptotic preserving. Consequently, in the optically thick regimes the current scheme can exactly capture the solution of the diffusion limit equation without requiring the cell size being smaller than the photon's mean free path, while the solution in optically thin regimes can also be well resolved in a natural way. In addition, the FPN angular discretization induces a natural macro-micro decomposition, with this help we can obtain the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Then, a linear scaling limiter is given to enforce that sufficient conditions. With the process of such construction, we finally obtain a scheme, called the PPFPN-based UGKS scheme, that is positive and asymptotic preserving. Various numerical experiments are included to validate the robustness, positive- and asymptotic-preserving property as well as the property of almost ray effect free.

Introduction

In many branches of science and technology, such as high-energy astrophysics, supernova and inertial/magnetic confinement fusion researches, accurate and efficient numerical solution of the radiative transfer equations, which describe the radiation photon transport and energy exchange with the background material, are required. The challenges in the numerical simulation of such equations lie in the treatment of high dimensionality due to the time-, spatial- and particle-variables, and multi-scale features characterized by wide-ranging optical thicknesses of background materials. Indeed, the optical thicknesses of background materials have a great impact on the behavior of radiation transfer. For a material with low opacity, the radiation propagates in a transparent way, while for a material with high opacity, the radiation behaves like a diffusion process. In order to resolve the kinetic-scale-based radiative transfer equations in numerical simulations, the spatial mesh size in many numerical methods usually should be comparable to the photon's mean-free path, which is very small in the optically thick regions, leading to huge computational costs. Therefore, numerical methods should take into account this multi-scale nature and accurately capture the solution of different optical thickness regimes, with affordable computational costs. We should point out that in this paper, the notions of “optically thick” and “optically thin” are discriminated by the photon's mean-free path of the background material. For example, if the length of a physical medium is many photon's mean free paths, we call the medium an optically thick region. On the contrary, if the length of a physical medium is few photon's mean free paths, we call the medium an optically thin region.

In the last decades, a number of numerical methods have been proposed to solve the radiative transfer equations, see [1], [2], [7], [19] for example, where the finite volume method, finite difference method and Monte Carlo method, etc. are used. However, most of them are essentially single scale methods that usually require the cell size and time step be comparable to the photon's mean free path and mean collision time. As aforementioned, this may incurs huge computational costs in the optically thick regimes. To circumvent this difficulty, one strategy is to design so-called asymptotic preserving (AP) multi-scale methods to greatly reduce the computational costs, see [9], [10], [11], [23] and among others. The unified gas kinetic scheme (UGKS) [31], [32], developed recently for rarefied gases, happens to fall into this AP category. Based on the UGKS framework, an asymptotic preserving scheme has been developed for the linear radiation equation [23], and then extended to the multi-frequency radiative transfer equations on both structured and unstructured meshes [26], [27], [28], [29], [30]. The main idea in UGKS is to couple the photons' transport process with their collision process by using a multi-scale flux function obtained from the local (exact) integral solution of the original transfer equation, thus the constrains on cell size and time step can be released.

The UGKS schemes developed in [26], [27], [28], [29], [30] are based on the discrete ordinate (SN) method for the angular discretization. They unavoidably suffer from the ray effects, in particular, when they are used to solve problems involving isolated sources within optically thin media. To cope with drawback, we proposed an angular finite element based UGKS in [33], which can significantly mitigate the ray effects of the SN method. However, for problems with strongly angular dependence, the ray effects still exist. In this paper, we continue our efforts to develop numerical schemes that can be free of or efficiently mitigate the ray effects.

It is well-known that the spherical harmonic (PN) method [24] can preserve the rotational invariance for transport equations, thus it is free of ray effects. And also, for smooth solutions, the PN approximation can achieve spectral convergence. On the other hand, for not sufficiently smooth solutions, the PN approximation can produce spurious oscillations. This can make the radiative energy density negative. Moreover, in the numerical solution of the coupled system of radiative transfer and material temperature equations, the negative radiative energy density could cause a negative material temperature [20], which may make the system unstable.

In order to reduce the effect of the above oscillatory (Gibbs) phenomena in the PN approximation, McClarren and Hauck employed filtering techniques to propose the so-called filtered PN (FPN) in [21], [22], while in [25] this filtering method was generalized and a more general framework was given. Although it can significantly suppress spurious oscillations, the filtering can also result in a negative numerical radiative energy density (numerical solution). So, to maintain the positivity of the numerical solution, Hauck and McClarren introduced the positive PN closures to develop the so-called PPN method in [8]. But, this method is computationally much more expensive than the original PN method and the computed solutions could be quite oscillatory. Moreover, the positive preserving FPN method was also constructed in [16], [18] for the linear radiative transfer equation by imposing a positivity limiter or via a positive moment closure. We remark that an additional expensive optimization problem has to be solved in the positive preserving FPN method. Recently, through a linear scaling limiter, a much cheaper positive- and asymptotic-preserving scheme is constructed for the linear kinetic equation, see [17].

In this paper, inspired by the idea of the linear scaling limiter in [17] and based on the framework of UGKS in [26], [27], [28], [29], [30], [33] for the radiative transfer equations, we shall propose a both positive- and asymptotic-preserving FPN scheme for the nonlinear gray radiative transfer system which is a coupled system of the radiative transfer and material temperature equations. To our best knowledge, there seems no such a scheme that preserves the positivity of both radiation energy density and material temperature for the gray radiative transfer system, and is also asymptotic preserving simultaneously.

The reminder of this paper is organized as follows. In Section 2 we introduce the model system of gray radiative transfer, while in Section 3 we present the detailed construction of a FPN-based UGKS. Section 4 is devoted to analysis of the asymptotic preserving property of the proposed scheme. In Section 5 we give the sufficient conditions to preserve the positivity and impose these conditions by using a linear scaling limiter. As a result, a positive and asymptotic preserving filtered spherical harmonics method (PPFPN-based UGKS) is hence proposed in this paper. Section 6 presents a number of numerical tests to validate the current scheme. Finally, a conclusion is given in Section 7.

Section snippets

Gray radiative transfer system

The gray radiative transfer system is a simplified but physically meaningful model of the radiative transfer process, which can be written in the following scaled form:{ϵ2cIt+ϵΩrI=σ(14πacT4I),ϵ2CνTt=σ(S2IdΩacT4). Here I(r,Ω,t) is the radiation intensity, T(r,t) is the material temperature, σ(r,T) is the scaled opacity, a is the scaled radiation constant, and c is the scaled speed of light, ϵ>0 is the Knudsen number, and Cν(r,t) is the scaled heat capacity. r=(x,y,z)R3 and t are the

A FPN-based UGKS

In this section, we shall construct an asymptotic preserving unified gas kinetic scheme with the angular variable discretized by the filtered spherical harmonics (FPN) method and spatial variables by the finite volume method for the system (2.1). In the following subsections, we give the angular, spatial and time discretization in details, respectively.

The asymptotic-preserving property

In this section, we analyze the asymptotic preserving (AP) property of the numerical scheme (3.10)–(3.11) by adapting the idea from [23]. In fact, the behavior of the algorithm in the small ϵ limit is completely determined by the property of the coefficient functions, which are given by

Proposition 4.1

Let σ be positive and Δt=O(1). Then as ϵ0, we have

  • α˜(Δt,ϵ,σ,ν) tends to 0;

  • d˜(Δt,ϵ,σ,ν) tends to c4πσ;

  • ϵcb˜(Δt,ϵ,σ,ν) tends to 14π.

With Proposition 4.1 in hand, we can show the desired asymptotic preserving

The positive-preserving property

In this section we show that, under some reasonable conditions, the current scheme can preserve the positivity of the radiation energy density ρi,jn+1 and material temperature Ti,jn+1. Since the equations in (3.10) are nonlinear and solved by the iteration method, we prove the positivity of ρi,jn+1,s+1 and ϕi,jn+1,s+1 in every iteration of solving the linearized equations (3.24). In the rest of this section we first present the sufficient conditions in Subsection 5.1 (cf. Theorem 5.1) that

Numerical experiments

In this section, we present a number of numerical examples to validate the proposed AP and PP scheme (i.e., PPFPN-based UGKS).

As for the time step, it should be taken to satisfy the stability condition and the condition (5.1) to guarantee the positive-preserving property. Now, we determine the stability condition of the proposed scheme by following an argument to that used in [23]. For the optically thick case, since the proposed scheme is AP and the limiting diffusion scheme (4.2) is implicit,

Conclusion

In this paper, based on the filtered spherical harmonics method for the angular variable discretization and UGKS for the spatial and time variables discretization, we have proposed a positive and asymptotic preserving FPN-based UGKS for the nonlinear gray radiative transfer equations.

Due to the rotational invariance of the FPN method, the current scheme is almost free of ray effects. At the same time, it can reduce the Gibbs phenomena in the PN approximation. In addition, we have also shown

CRediT authorship contribution statement

Xiaojing Xu: Methodology, Software, Writing – original draft. Song Jiang: Conceptualization, Investigation, Supervision, Visualization. Wenjun Sun: Software, Validation, Writing – review & editing.

Declaration of Competing Interest

We declare that we have no known competing financial interests and personal relationships that would have appeared to influence the work reported in this paper.

Acknowledgements

The current research is supported by NSFC (Grant No. 12001451) for Xu, and by National Key R&D Program (2020YFA0712200), National Key Project (GJXM92579), and NSFC (Grant No. 11631008), the Sino-German Science Center (Grant No. GZ 1465) and the ISF-NSFC joint research program (Grant No. 11761141008) for Jiang, and by CAEP Foundation(No. CX20200026), NSFC (Grant Nos. 11671048, 91630310) and Science Challenge Project (No. TZ2016002) for Sun.

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