A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering

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Abstract

We describe a semi-analytical numerical method for coherent isotropic scattering time-dependent radiative transfer problems in slab geometry. This numerical method is based on a combination of two classes of numerical methods: the spectral methods and the Laplace transform (LTSN) methods applied to the radiative transfer equation in the discrete ordinates (SN) formulation. The basic idea is to use the essence of the spectral methods and expand the intensity of radiation in a truncated series of Laguerre polynomials in the time variable and then solve recursively the resulting set of “time-independent” SN problems by using the LTSN method. We show some numerical experiments for a typical model problem.

Introduction

It is well known that obtaining accurate numerical solutions to time-dependent neutral particle transport problems is not an easy task. One of the reasons for the difficulty is the nonlinearities due to the temperature dependencies of the Planck function, the material heat capacity, and the cross sections. Therefore, a number of alternative numerical methods have been applied to approximate mathematical models, e.g. diffusion methods [1], [2], [3], [4].

In this paper, we present a semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent (no energy exchange) and isotropic scattering. In this method, we combine a spectral method [5] with the Laplace transform (LTSN) method [6] as follows. We expand the intensity of radiation in a truncated series of Laguerre polynomials in the time variable. By substituting this expansion into the radiative transfer equation, we calculate the moments in order to obtain a set of “time-independent” discrete ordinates (SN) problems in slab geometry that we solve by the LTSN method [6], i.e., we solve the one-dimensional SN equations by using the LTSN in the space variable with analytical inversion. Therefore, if we neglect heat conduction and fluid motion, we solve recursively the radiative transfer equation coupled only to a simple material energy balance equation. Furthermore, if we assume a gray description of the problem, we define the blackbody energy density as being proportional to the fourth power of the material temperature. Bearing these assumptions in mind, we solve the radiative transfer SN equations for the intensity of radiation and we use the result to calculate the temperature by integration of the energy balance equation. We proceed further by evaluating the blackbody energy density and we go back to solving the radiative transfer equation until a prescribed convergence criterion for the material temperature is satisfied.

An outline of the remainder of this paper follows. In Section 2, we present the mathematical formulation of the physical problem we are interested in modeling. In Section 3, we describe the offered semi-analytical numerical method, and in Section 4, we show some numerical experiments to a typical model problem and present a brief discussion with suggestions for future work.

Section snippets

Mathematical formulation

Let us consider the slab-geometry time-dependent radiative transfer equation with coherent isotropic scattering:1c∂tI(x,μ,t)+μ∂xI(x,μ,t)+σI(x,μ,t)=c2aB(x,t)+σsE(x,t))0⩽x⩽L.Here I(x,μ,t) is the specific intensity of radiation, t is the time variable, c is the speed of light, μ is the cosine of the angle between the photon direction of motion and the x (spatial variable) axis. Moreover, we follow the standard definitions: σa(x,t) is the macroscopic absorption cross section, σs(x,t) is the

Semi-analytical numerical method

At this point, we describe a semi-analytical numerical method to solve the auxiliary problem given in , , , . Therefore, we expand function Φ(x,μ,t,τ) in a truncated series of Laguerre polynomials in the time variable [9]. That is, we consider the approximationΦ(x,μ,t,τ)=k=0MΦk(x,μ,τ)Lk(t)where Lk(t), k=1,…,M, is the kth degree Laguerre polynomial and the coefficients Φk(x,μ,τ) are the moments of the expansion.

Now, we substitute Eq. (10) into Eq. (7) and multiply the result by etLm(t). By

Numerical results and discussion

Let us consider a homogeneous slab, 20cm thick, where the radiation is driven by a time-independent, isotropic intensity incident upon the left side of the slab for all times t>0, and no radiation is incident upon the right side of the slab, i.e., we consider the boundary conditionsI(0,μ,t)=Γ0(μ,t)=2,μ>0,I(20,μ,t)=Γ1(μ,t)=0,μ<0.To discuss the behavior of this problem for finite times, we have chosen the initial conditionsΦ0(x,μ)=0.5×10−10T0(x)=10−5/2.Moreover, the parameters c, a, σ and σs,

Acknowledgements

This work was partially supported by Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico (CNPq-Brazil), Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (FAPERGS-Brazil) and by Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior (CAPES-Brazil).

References (10)

  • G.C. Pomraning

    The equations of radiation hydrodynamics.

    (1973)
  • E.W. Larsen et al.

    Asymptotic solution of neutron transport problem for small mean free paths

    J Math Phys

    (1974)
  • C.D. Levermore et al.

    A flux-limited diffusion theory

    Astrophys J

    (1981)
  • R.H. Szilard et al.

    Numerical transport and diffusion methods in radiative transfer

    Nucl Sci Eng

    (1992)
  • D. Gottlieb et al.

    Numerical analysis of spectral methods: theory and applications.

    (1977)
There are more references available in the full text version of this article.

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