Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js
Research article Special Issues

Bidirectional monte carlo method for thermal radiation transfer in participating medium

  • A bidirectional Monte Carlo (BDMC) method based on reversibility of bundle trajectory and reciprocity of thermal radiative energy exchange was developed to solve radiative heat transfer in absorbing and scattering medium. Two types of sampling models were introduced into the Monte Carlo (MC) simulation, namely the equivalent sampling and the weight sampling, respectively. Mathematical formula for the sampling models and the statistical calculation of sampling bundles were derived. Furthermore, the reciprocity error correlation of radiative exchange factors between the BDMC method and the traditional Monte Carlo (TMC) method were demonstrated and analyzed. Radiative heat transfer in a two-dimensional rectangular domain with absorbing and scattering media was solved by using both the BDMC method and the TMC method. Radiative exchange factors and radiative equilibrium temperature profiles predicted by the BDMC method were compared with those predicted by the TMC method. The performance parameter P, defined to evaluate the performance of MC methods, was computed and compared between the BDMC and the TMC methods. The results showed the superiority of BDMC method compared with the TMC method for radiative heat transfer, in addition, the weight sampling was proved to be more flexible than the equivalent sampling in the BDMC method.

    Citation: Xiaofeng Zhang, Qing Ai, Kuilong Song, Heping Tan. Bidirectional monte carlo method for thermal radiation transfer in participating medium[J]. AIMS Energy, 2021, 9(3): 603-622. doi: 10.3934/energy.2021029

    Related Papers:

    [1] T. A. Shaposhnikova, M. N. Zubova . Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3(3): 675-689. doi: 10.3934/nhm.2008.3.675
    [2] Ken-Ichi Nakamura, Toshiko Ogiwara . Periodically growing solutions in a class of strongly monotone semiflows. Networks and Heterogeneous Media, 2012, 7(4): 881-891. doi: 10.3934/nhm.2012.7.881
    [3] Junlong Chen, Yanbin Tang . Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure. Networks and Heterogeneous Media, 2023, 18(3): 1118-1177. doi: 10.3934/nhm.2023049
    [4] Benjamin Contri . Fisher-KPP equations and applications to a model in medical sciences. Networks and Heterogeneous Media, 2018, 13(1): 119-153. doi: 10.3934/nhm.2018006
    [5] Thomas Geert de Jong, Georg Prokert, Alef Edou Sterk . Reaction–diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions. Networks and Heterogeneous Media, 2025, 20(1): 1-14. doi: 10.3934/nhm.2025001
    [6] Iryna Pankratova, Andrey Piatnitski . Homogenization of convection-diffusion equation in infinite cylinder. Networks and Heterogeneous Media, 2011, 6(1): 111-126. doi: 10.3934/nhm.2011.6.111
    [7] Feiyang Peng, Yanbin Tang . Inverse problem of determining diffusion matrix between different structures for time fractional diffusion equation. Networks and Heterogeneous Media, 2024, 19(1): 291-304. doi: 10.3934/nhm.2024013
    [8] Xavier Blanc, Claude Le Bris . Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks and Heterogeneous Media, 2010, 5(1): 1-29. doi: 10.3934/nhm.2010.5.1
    [9] Avner Friedman . PDE problems arising in mathematical biology. Networks and Heterogeneous Media, 2012, 7(4): 691-703. doi: 10.3934/nhm.2012.7.691
    [10] Bendong Lou . Self-similar solutions in a sector for a quasilinear parabolic equation. Networks and Heterogeneous Media, 2012, 7(4): 857-879. doi: 10.3934/nhm.2012.7.857
  • A bidirectional Monte Carlo (BDMC) method based on reversibility of bundle trajectory and reciprocity of thermal radiative energy exchange was developed to solve radiative heat transfer in absorbing and scattering medium. Two types of sampling models were introduced into the Monte Carlo (MC) simulation, namely the equivalent sampling and the weight sampling, respectively. Mathematical formula for the sampling models and the statistical calculation of sampling bundles were derived. Furthermore, the reciprocity error correlation of radiative exchange factors between the BDMC method and the traditional Monte Carlo (TMC) method were demonstrated and analyzed. Radiative heat transfer in a two-dimensional rectangular domain with absorbing and scattering media was solved by using both the BDMC method and the TMC method. Radiative exchange factors and radiative equilibrium temperature profiles predicted by the BDMC method were compared with those predicted by the TMC method. The performance parameter P, defined to evaluate the performance of MC methods, was computed and compared between the BDMC and the TMC methods. The results showed the superiority of BDMC method compared with the TMC method for radiative heat transfer, in addition, the weight sampling was proved to be more flexible than the equivalent sampling in the BDMC method.



    In this paper, we study the positive solutions of the periodic-parabolic problem

    {ut=μk(x,t)Δu+m(x,t)uc(x,t)up, in Ω×R,uν=0, on Ω×R,u(x,0)=u(x,T), in Ω, (1.1)

    where Ω is a bounded domain of RN(N1) with smooth boundary Ω, ν is the outward normal vector of Ω, μ>0 and p>1 is constant, m(x,t)Cα,α2(ˉΩ×R)(0<α<1) is T-periodic in t, k(x,t), c(x,t)Cα,1(ˉΩ×R) are positive and T-periodic in t. It is known that the periodic reaction-diffusion equation (1.1) can be accurately used to describe different diffusion phenomena in infectious diseases, microbial growth, and population ecology, see [1,2,3,4]. From a biological point of view, Ω represents the habitat of species u and μk(x,t) stands for the diffusion rate, which is time and space dependent. The function m(x,t) represents the growth rate of species. In this situation, in the subset {(x,t)Ω×R:m(x,t)>0}, the species will increase, while in {(x,t)Ω×R:m(x,t)<0}, species will decrease. The coefficient c(x,t) means that environment Ω can accommodate species u. There are many interesting conclusions about the study of the reaction-diffusion equation, see [5,6,7,8] for the elliptic problems and [9,10,11,12,13,14] for the periodic problems.

    In particular, if k(x,t)k(t) for xˉΩ, problem (1.1) has been well investigated by Hess [2], Cantrell and Cosner [1]. Let λ(μ) be the unique principal eigenvalue of the eigenvalue problem

    {utμk(t)Δum(x,t)u=λ(μ)u, in Ω×R,uν=0, on Ω×R,u(x,0)=u(x,T), in Ω.

    It follows from [2,15] that Eq (1.1) has a positive periodic solution θμ(x,t) if and only if λ(μ)<0. In addition, Dancer and Hess [16] and Daners and López-Gómez [17] studied the effect of μ on the positive periodic solution of Eq (1.1) with various boundary conditions. The most interesting conclusion of [11,16,17] is that

    limμ0+θμ(x,t)=θ(x,t) locally uniformly  in Ω×[0,T],

    here θ(x,t) is the maximum nonnegative periodic solution of

    {ut=m(x,t)uc(x,t)up,tR,u(x,0)=u(x,T).

    However, there is little result on the associated large diffusion and the effect of large diffusion on positive solutions.

    Our goal is to study the existence and uniqueness of positive periodic solutions of Eq (1.1) and the asymptotic behavior of positive periodic solutions when the diffusion rate μ is large. To this end, let λ(μ;m) be the principal eigenvalue of

    {utμk(x,t)Δum(x,t)u=λ(μ;m)u, in Ω×R,uν=0, on Ω×R,u(x,0)=u(x,T), in Ω. (1.2)

    {It is well known that λ(μ;m) plays a major role in the study of the positive periodic solution of Eq (1.1). The properties of λ(μ;m) will be established in Section 2. In addition, let W2,pν(Ω)={uW2,p(Ω):uν=0}(N<p<). If u0W2,pν(Ω), it follows from [2] that the semilinear initial value problem}

    {ut=μk(x,t)Δu+m(x,t)uc(x,t)up, in Ω×R,uν=0, on Ω×R,u(x,0)=u0(x), in Ω,

    has a unique solution U(x,t)=U(x,t;u0) satisfying

    U(x,t)C1+α,1+α2(ˉΩ×[0,T])C2+α,1+α2(ˉΩ×(0,T]).

    Our first result is the existence and uniqueness of positive periodic solutions of Eq (1.1). For simplicity, in the rest of this paper, we use the following notations:

    k(t)=Ω1k(x,t)dx,m(t)=Ωm(x,t)k(x,t)dx,c(t)=Ωc(x,t)k(x,t)dx.

    Theorem 1.1. Suppose that T0m(t)dt>0. Then Eq (1.1) admits a unique positive periodic solution θμ(x,t) for all μ>0.

    Remark 1.1. By the result of Section 3, we know that there exists a unique positive solution to Eq (1.1) if and only if λ(μ;m)<0. In the case T0m(t)dt>0, we obtain that λ(μ;m)<0.

    Next, we study the asymptotic behavior of positive periodic solutions when the diffusion rate is large.

    Theorem 1.2. Suppose that

    m(t)>0 for t[0,T]. (1.3)

    Let θμ(x,t) be the unique positive periodic solution of Eq (1.1) for μ>0. Then we have

    limμθμ(x,t)=ω(t) in C1,12(ˉΩ×[0,T]), (1.4)

    where ω(t) is the unique positive periodic solution of

    {ut=m(t)k(t)uc(t)k(t)up,tR,u(0)=u(T). (1.5)

    Remark 1.2. With the approach of local upper-lower solutions developed by Daners and López-Gómez [17] in the study of classical periodic-parabolic logistic equations, we can prove that

    limμ0+θμ(x,t)=θ(x,t) locally uniformly  in Ω×[0,T],

    provided maxΩT0m(x,t)dt>0. It also shows that when m(t)<0<T0maxΩm(x,t)dt, populations with small dispersal rates survive, while populations with large dispersal rates perish. This means that a small diffusion rate is a better strategy than a large diffusion rate under appropriate circumstances.

    The rest of this paper is arranged as follows: In Section 2, we study the properties of principal eigenvalues for the periodic eigenvalue problems. In Section 3, we mainly study the existence, uniqueness and stability of the positive solution to Eq (1.1). Moreover, we investigate the asymptotic profiles of the positive periodic solution to Eq (1.1) as μ in Section 4.

    In this section, we consider the principal eigenvalue of Eq (1.2). To this end, we first study the linear initial value problem

    {utμk(x,t)Δua(x,t)u=0, in Ω×(τ,T],uν=0, on Ω×(τ,T],u(x,τ)=u0(x), in Ω, (2.1)

    where 0τ<T, u0W2,pν(Ω)(N<p<) and a(x,t)Cα,α2(ˉΩ×[τ,)). It is well known that there is a one-to-one correspondence between Eq (2.1) and the evolution operator Uμ(t,τ). Then we can define that u(x,t)=Uμ(t,τ)u0 is the solution of Eq (2.1). For simplicity, let X=Lp(Ω)(N<p<), X1=W2,pν(Ω) and

    F={uCα,α2(ˉΩ×R):u(,t+T)=u(,t) in R}.

    Inspired by the classical works of Hess [2], we first give some important results of Eq (2.1), which will be used in the rest of this paper.

    Lemma 2.1. If u0X is positive, then Uμ(t,τ)u0>0 in C1ν(ˉΩ) for 0τ<tT.

    Proof. Note that X1 is compactly embedded in X. The operator Uμ(t,τ)/X1:X1X1 can be continuously extended to the positive operator Uμ(t,τ)L(X,X1). Thus Uμ(t,τ)u00 in X1. Since sUμ(s,τ)u0 is continuous from [τ,T] to X1 and Uμ(τ,τ)u0=u00, we can get that Uμ(s,τ)u0>0 in X1 as s>τ goes to τ. In addition, we have

    Uμ(t,τ)u0=Uμ(t,s)Uμ(s,τ)u0,

    for τ<s<t. Thus it can be obtained that Uμ(t,τ)u0>0 for 0τ<tT.

    We now study the periodic-parabolic eigenvalue problem

    {utμk(x,t)Δua(x,t)u=λ(μ;a)u, in Ω×(0,T],uν=0, on Ω×(0,T],u(x,0)=u(x,T), in Ω. (2.2)

    If there is a nontrivial solution u(x,t) of Eq (2.2), then λ(μ;a) is called the eigenvalue. In particular, if u(x,t) is positive, then λ(μ;a) is the principal eigenvalue.

    Theorem 2.1. Let Kμ:=Uμ(T,0) and r be the spectral radius of Kμ. Then r is the principal eigenvalue of Kμ with positive eigenfunction u0 if and only if λ(μ;a)=1Tlnr is the principal eigenvalue of Eq (2.2) with positive eigenfunction u(x,t)=eλ(μ;a)tUμ(t,0)u0.

    Proof. It can be proved by the similar arguments as in [2, Proposition 14.4]. For the completeness, we provide a proof in the following. Suppose that r is the principal eigenvalue of Kμ with positive eigenfunction u0X1. Let u(x,t)=eλ(μ;a)tUμ(t,0)u0. Then u(x,t) satisfies

    {utμk(x,t)Δua(x,t)u=λ(μ;a)u, in Ω×(0,T],uν=0, on Ω×(0,T],u(x,0)=u0=1rKμu0=eλ(μ;a)TKμu0=u(x,T), in Ω.

    According to the regularity results, we have u(x,t)C2+α,1+α2(ˉΩ×R). This means that μ=1Tlnr is the principal eigenvalue of Eq (2.2), while u(x,t)=eλ(μ;a)tUμ(t,0)u0 is the corresponding positive eigenfunction.

    On the contrary, suppose that λ(μ;a)=1Tlnr is the eigenvalue of Eq (2.2) with positive eigenfunction u(x,t). Set v(x,t)=eλ(μ;a)tu(x,t). Then v(x,t) is the solution of

    {vtμk(x,t)Δva(x,t)v=0, in Ω×(0,T],vν=0, on Ω×(0,T],v(x,0)=u(x)=:u0, in Ω.

    Thus, we obtain v(x,t)=Uμ(t,0)u0 for 0tT and u0X1 is positive. Hence,

    v(T)=eλ(μ;a)Tu0=Kμu0.

    It follows from Krein-Rutman theorem that eλ(μ;a)T=r.

    Remark 2.1. For τ<t, it follows that Uμ(t,τ) is a compact and strongly positive operator on X1. Moreover, by Krein-Rutman theorem, we obtain r>0, and r is the unique principal eigenvalue of Kμ. This implies that Eq (2.2) has the unique principal eigenvalue λ(μ;a) for any μ>0.

    Lemma 2.2. Let a1(x,t), a(x,t)F satisfy

    a1(x,t)<a2(x,t) in ˉΩ×[0,T].

    Then λ(μ;a2)<λ(μ;a1) for any μ>0.

    Proof. Assume that there exists μ1>0 such that λ(μ1;a2)λ(μ1;a1). Let u1(x,t) and u2(x,t) be corresponding positive eigenfunctions, chosen in such a way that

    0<u1(x,t)<u2(x,t) in ˉΩ×[0,T].

    Then ω(x,t)=u2(x,t)u1(x,t) satisfies

    {ωtμ1k(x,t)Δωa1(x,t)ω>λ(μ1;a1)ω, in Ω×(0,T],ων=0, on Ω×(0,T],ω(x,0)=ω(x,T), in Ω.

    Set ϕ(x,t)=eλ(μ1;a1)tω(x,t), then we have

    {ϕtμ1k(x,t)Δϕa1(x,t)ϕ>0, in Ω×(0,T],ϕν=0, on Ω×(0,T],ϕ(x,0)=ω(x,0)=ω(x,T), in Ω.

    Thus, for any xΩ, we can obtain

    ϕ(x,T)>Kμ1ω(x,0) and ϕ(x,T)=eλ(μ1;a1)Tω(x,0).

    Hence, we obtain

    (eλ(μ1;a1)TKμ1)ω(x,0)>0 in X1.

    Note that ω(x,0)>0, it follows from [2, Theorem 7.3] that

    eλ(μ1;a1)T=rμ1<eλ(μ1;a1)T,

    where rμ1 is the principal eigenvalue of Kμ1. This is a contradiction.

    Lemma 2.3. Suppose that for any nN, an(x,t)F satisfies

    limnan(x,t)=a(x,t) in C1(ˉΩ×[0,T]).

    Then for fixed μ>0, we have

    limnλ(μ;an)=λ(μ;a).

    Proof. For any given ε>0, there exists nεN such that for any n>nε, there holds

    a(x,t)ε<an(x,t)<a(x,t)+ε in ˉΩ×[0,T].

    Notice that λ(μ;a±ε)=λ(μ;a)ε. From Lemma 2.2, we have

    λ(μ;a)ε<λ(μ;an)<λ(μ;a)+ε,

    for any n>nε.

    Lemma 2.4. Let λ(μ;a) be the principal eigenvalue of Eq (2.2) for μ>0. Then we have

    λ(μ;a)T0a(t)dtT0k(t)dt, (2.3)

    here a(t)=Ωa(x,t)k(x,t)dx.

    Proof. First, we consider the case

    T0Ωkt(x,t)k2(x,t)dxdt0.

    Let φ(x,t) be the positive eigenfunction corresponding to the principal eigenvalue λ(μ;a). Taking α>0 satisfies

    lnα=T0Ωkt(x,t)lnφ(x,t)k2(x,t)dxdtT0Ωkt(x,t)k2(x,t)dxdt.

    Then φα:=αφ(x,t) is also the principal eigenfunction of Eq (2.2). It is easy to obtain

    λ(μ;a)T0Ω1k(x,t)dxdt=T0Ωa(x,t)k(x,t)dxdtμT0ΩΔφαφαdxdt=T0Ωa(x,t)k(x,t)dxdtμT0Ω|Dφα|2φ2αdxdt. (2.4)

    This implies that Eq (3.2) holds.

    Next, we consider the case of

    T0Ωkt(x,t)k2(x,t)dxdt=0.

    We can find smooth T-periodic functions {kn(x,t)} such that

    limnkn(x,t)=k(x,t) in C(ˉΩ×[0,T]),

    and

    T0Ω(kn(x,t))tk2n(x,t)dxdt0.

    It follows from Lemma 2.3 that

    limnλn(μ;a)=λ(μ;a),

    where λn(μ;a) is the principal eigenvalue of Eq (2.2) with k(x,t) replaced by kn(x,t). It is clear from Eq (2.4) that

    λn(μ;a)=T0Ωa(x,t)kn(x,t)dxdtμT0Ω|Dφα|2φ2αdxdt.

    Letting n, we have Eq (3.2).

    Remark 2.2. In Eq (3.2), we obtain upper estimates for the principal eigenvalue of the Neumann problem Eq (2.2). Indeed, let λD be the principal eigenvalue of the eigenvalue problem

    {utμk(x,t)Δua(x,t)u=λDu, in Ω×(0,T],u=0, on Ω×(0,T],u(x,0)=u(x,T), in Ω.

    By a similar way as in [2], we can show

    λD1TT0[μk(x,s)+a(s)]ds,

    for any μ>0.

    In this section, we study the existence and uniqueness of positive solutions of Eq (1.1). First, we show that if Eq (1.2) has negative principal eigenvalues, then Eq (1.1) has a unique positive solution. To this end, we recall the upper-lower solutions of Eq (1.1). For the sake of convenience, let

    QT=Ω×(0,T],Q1=Ω×(0,T].

    Definition 3.1. The continuous function ˉu(x,t) is called the upper-solution of Eq (1.1), if

    {utμk(x,t)Δu+m(x,t)uc(x,t)up,inQT,uν0,onQ1,u(x,0)u(x,T),inΩ,

    is satisfied.

    The definition of the lower-solution is similar. We then can prove the following result, see [2,4,5,15].

    Theorem 3.1. Suppose that ˉu(x,t), u_(x,t) are a pair of ordered bounded upper-lower solutions of Eq (1.1). Then Eq (1.1) has a unique positive periodic solution θμ(x,t)C1+α,(1+α)/2(ˉQT) that satisfies

    u_(x,t)θμ(x,t)ˉu(x,t) in ˉQT.

    Proof. Let

    f(x,t,u)=m(x,t)uc(x,t)up.

    Then there exists a constant K>0 such that

    |f(x,t,u2)f(x,t,u1)|K|u2u1|,

    for any (x,t,ui)ˉQT×[u_(x,t),ˉu(x,t)], i=1,2. It follows from Lp theory that for any uCα,α2(ˉQT) satisfying [u_(x,t),ˉu(x,t)], the linear initial value problem

    {vtμk(x,t)Δv+Kv=Ku+f(x,t,u), in QT,vν=0, on Q1,v(x,0)=u(x,T), in Ω,

    admits a unique solution v. Thus, a nonlinear operator v=Fu is defined. We will prove that there is a fixed point for F in four steps.

    Step1. In this step, we prove that if u_u1u2ˉu, then u_v1=Fu1v2=Fu2ˉu.

    Take ω1=v2v1, then ω1 satisfies

    {[ω1]tμk(x,t)Δω1+Kω1=K(u2u1)+f(x,t,u2)f(x,t,u1)0, in QT,ω1ν=0, on Q1,ω1(x,0)=u2(x,T)u1(x,T)0, in Ω.

    By the comparison principle, we obtain ω10. This implies Fu2Fu1. Similarly, let ω2=v1u_, then ω2 satisfies

    {[ω2]tμk(x,t)Δω2+Kω2=K(u1u_)+f(x,t,u1)f(x,t,u_)0, in QT,ω2ν=0, on Q1,ω1(x,0)=u1(x,T)u_(x,T)0, in Ω.

    Thus, u_v1. Similarly, v2ˉu.

    Step2. In this step, we construct a convergent monotone sequence.

    The iterative sequences {un} and {vn} are constructed as follows:

    u1=Fˉu,u2=Fu1,,un=Fun1,
    v1=Fu_,v2=Fv1,,vn=Fvn1.

    Since u_ˉu and F is monotonically non-decreasing, then

    u_v1u1ˉu.

    Similarly, we obtain

    u_vnunˉu.

    And since u_u1ˉu,

    u_u2u1ˉu.

    By induction, we have un+1un. In the same way, we obtain vnvn+1. Thus, we have

    u_v1v2u2u1ˉu.

    This also implies that {un} and {vn} are monotonically bounded sequences, so there are u0(x,t) and v0(x,t) such that

    limnun(x,t)=u0(x,t),limnvn(x,t)=v0(x,t).

    Thus

    u_(x,t)v0(x,t)u0(x,t)ˉu(x,t) in ˉQT.

    Step3. In this step, we prove that u0(x,t) and v0(x,t) are solutions of Eq (1.1).

    Take E=W2,1p(QT)(p>n+2). First, we prove that F:DC(ˉQT) is a compact operator, where

    D={u(x,t)E:u_(x,t)u(x,t)ˉu(x,t) in ˉQT}.

    For u1, u2E, let v1=Fu1 and v2=Fu2, then ω3=v2v1 satisfies

    {[ω3]tμk(x,t)Δω3+Kω3=K(u2u1)+f(x,t,u2)f(x,t,u1), in QT,ω3ν=0, on Q1,ω1(x,0)=u2(x,T)u1(x,T), in Ω.

    By the Lp estimate and embedding theorem, it follows that

    here C and C_1 are positive constants. Thus \mathcal{F}:D\to C(\bar Q_T) is continuous. It is known from the embedding theorem that if u is bounded in W^{2, 1}_p(Q_T) , then \mathcal{F}u is bounded in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) . This means that \mathcal{F}:D\to C(\bar Q_T) is a compact operator.

    Since u_n is bounded, u_n = \mathcal{F}u_{n-1} has a convergent subsequence in C(\bar Q_T) . By the monotonicity of u_n ,

    \begin{equation*} \lim\limits_{n\to\infty}u_n(x,t) = u_0(x,t)\,\,\text{ in }\,\,C(\bar Q_T). \end{equation*}

    Thus u_0(x, t) is the solution of Eq (1.1) in W^{2, 1}_p(Q_T) . The embedding theorem is used again to get u_0(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) . In the same way, we get that v_0(x, t) is the classical solution of Eq (1.1).

    \bf{Step\; 4.} In this step, we prove the uniqueness and periodicity of the solution of Eq (1.1).

    Since k(x, t) , m(x, t) and c(x, t) are periodic about t , then \tau(x, t) = u_0(x, t+T)-u_0(x, t) satisfies

    \begin{equation} \begin{cases} \tau_t(x,t)-\mu k(x,t)\Delta\tau(x,t) = m(x,t)\tau(x,t)-pc(x,t)\tilde{u}^{p-1}(x,t)\tau(x,t),&\text{ in } Q_T,\\ \frac{\partial\tau}{\partial\nu} = 0,&\text{ on } Q_1,\\ \tau(x,0) = 0,&\text{ in } \Omega, \end{cases} \end{equation} (3.1)

    here \tilde{u}(x, t) is between u_0(x, t+T) and u_0(x, t) . It is well known that the solution of Eq (3.1) is unique, thus u_0(x, t+T)\equiv u_0(x, t) in \bar Q_T .

    The uniqueness of the solution is based on the results of [2,4] and can also be found in recent research results [11,15]. Assume that v_1 and v_2 are two positive periodic solutions of Eq (1.1). We first prove that there exists a large constant M > 1 such that

    \begin{equation*} M^{-1}v_1 < v_2 < Mv_1\,\,\text{ in }\,\,Q_T. \end{equation*}

    Indeed, it is clear that there exists M_1 > 1 such that

    \begin{equation*} v_2(x,0) = v_2(x,T) < M_1v_1(x,T) = M_1v_1(x,0)\,\,\text{ in }\,\,\Omega. \end{equation*}

    This implies v_2(x, 0)\not\equiv M_1v_1(x, 0) on \bar\Omega . Let \eta(x, t): = M_1v_1(x, t)-v_2(x, t) , then \eta(x, t) satisfies

    \begin{align*} \eta_t-\mu k(x,t)\Delta\eta = &m(x,t)\eta-c(x,t)[M_1v_1^p-v_2^p]\\ > &m(x,t)\eta-c(x,t)[(M_1v_1)^p-v_2^p]\\ = &m(x,t)\eta-pc(x,t)\varsigma^{p-1}(x,t)\eta, \end{align*}

    where \varsigma(x, t) is lying between M_1v_1(x, t) and v_2(x, t) . Notice that \frac{\partial\eta}{\partial\nu} = 0 on Q_1 . By the maximum principle, we have \eta > 0 in \bar Q_T . Similarly, we can obtain that there exists M_2 > 0 such that v_1 < M_2v_2 in \bar Q_T . Take M = \max\{M_1, M_2\} , then we have

    \begin{equation*} M^{-1}v_1 < v_2 < Mv_1\,\,\text{ in }\,\,Q_T. \end{equation*}

    We know that Mv_1(x, t) and M^{-1}v_1(x, t) are a pair of upper-lower solutions of Eq (1.1). According to the second step, Eq (1.1) has a minimum solution u_* and a maximum solution u^* , which satisfies u_*\leq v\leq u^* in \bar Q_T for all solution v satisfying M^{-1}v_1\leq v\leq Mv_1 . Thus, we obtain u_*\leq v_i\leq u^* for i = 1, 2 . Hence, u_*\equiv u^* implies the uniqueness of the solution to Eq (1.1). Set

    \begin{equation*} \alpha_{*} = \inf\left\{\alpha > 0\,|\,u^*(x,t)\leq \alpha u_*(x,t)\,\text{ in } \bar Q_T \right\}. \end{equation*}

    It is clear that \alpha_{*}\geq1 . Note that if \alpha_{*} = 1 , then u_*(x, t)\equiv u^*(x, t) in \bar Q_T . Assume that \alpha_{*} > 1 . Let \pi(x, t) = \alpha_{*} u_*-u^* . It is known from the maximum principle that \pi(x, t) > 0 in \bar Q_T . On the other hand, we know that

    \begin{equation*} \pi(x,0) = \pi(x,T)\geq\alpha_1 u_*(x,T) = \alpha_1 u_*(x,0)\,\,\text{ on }\,\,\bar\Omega, \end{equation*}

    for some small \alpha_1 > 0 . We can use the previous method to prove the existence of M to show that

    \begin{equation*} \pi(x,t)\geq\alpha_1 u_*(x,t)\,\,\text{ on }\,\,\bar Q_T. \end{equation*}

    Then we have u^*(x, t)\leq (\alpha_*-\alpha) u_*(x, t) . This is in contradiction with the definition of \alpha_* . Thus, we obtain \alpha_{*} = 1 . The uniqueness is proved.

    Lemma 3.1. If \lambda(\mu; m) < 0 , then Eq (1.1) admits a unique positive periodic solution \theta_\mu(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) . Moreover, \theta_\mu(x, t) is globally asymptotically stable.

    Proof. Let \theta(x, t) be a principal eigenfunction of Eq (1.2) normalized by \|\theta(x, t)\|_{C(\bar Q_T)} = 1 . Then \underline{u} = \varepsilon\theta(x, t) is a lower-solution of Eq (1.1) for any

    \begin{equation*} 0 < \varepsilon\leq\left[\frac{-\lambda(\mu;m)}{\max_{\bar Q_T}c(x,t)}\right]^{1-p}. \end{equation*}

    Take

    \begin{equation*} M > \max\left\{1,\left[\frac{-\lambda(\mu;m)}{\min_{\bar Q_T}c(x,t)}\right]^{1-p}\right\}. \end{equation*}

    Then we have \bar u = M\theta(x, t) is an upper-solution of Eq (1.1). From Theorem 3.1, we get that Eq (1.1) has a unique positive solution \theta_\mu(x, t) .

    Since \theta_\mu(x, t) is the solution of Eq (1.1), then \lambda(\mu; m-c\theta_\mu^{p-1}) = 0 . Let \lambda_1 be the eigenvalue of the linear problem

    \begin{equation*} \begin{cases} u_t-\mu k(x,t)\Delta u-\left[m(x,t)-pc(x,t)\theta_\mu^{p-1}\right]u = \lambda_1u,&\text{ in } \Omega\times(0,T],\\ \frac{\partial u}{\partial\nu} = 0,&\text{ on } \partial\Omega\times(0,T],\\ u(x,0) = u(x,T),&\text{ in } \Omega. \end{cases} \end{equation*}

    Due to

    \begin{equation*} m(x,t)-c(x,t)\hat{u}^{p-1} > m(x,t)-pc(x,t)\hat{u}^{p-1}, \end{equation*}

    for u > 0 . Then \lambda_1 > 0 . It follows from Theorem 2.1 that r < 1 . Thus, \theta_\mu(x, t) is locally asymptotically stable. In addition, we can choose \varepsilon small enough and M large enough such that \varepsilon\theta(x, t) and M\theta(x, t) are a pair of ordered bounded upper-lower solutions of Eq (1.1). Then we know that \theta_\mu(x, t) is globally asymptotically stable by the standard iteration argument as in [2].

    Lemma 3.2. If (1.1) has a positive periodic solution, then \lambda(\mu; m) < 0 .

    Proof. Let \theta_\mu(x, t) be a positive periodic solution of Eq (1.1). Thanks to [2], we can get that Eq (1.1) is equivalent to

    \begin{equation*} (I-K_\mu)\theta_\mu(x,0) = -\int_0^TU_\mu(T,\tau)c(x,\tau)\theta_\mu^p(x,\tau)\,d\tau\,\,\text{ in }\,\,X_1. \end{equation*}

    Notice that \theta_\mu(x, t) > 0 . We now apply [2, Theorem 7.3] to obtain

    \begin{equation*} e^{-\lambda(\mu;m)T} > 1. \end{equation*}

    Thus, \lambda(\mu; m) < 0 .

    Proposition 3.1. If \int_0^Tm_*(t)\, dt > 0 , then Eq (1.1) admits a unique positive periodic solution \theta_\mu(x, t) for all \mu > 0 .

    Proof. According to Lemma 2.4, we know that

    \begin{equation} \lambda(\mu;m)\leq-\frac{\int_0^T m_*(t)\,dt}{\int_0^T k_*(t)\,dt}. \end{equation} (3.2)

    Due to \int_0^Tm_*(t)\, dt > 0 , \lambda(\mu; m) < 0 . This together with Lemma 3.1 implies that Eq (1.1) admits a unique positive periodic solution for all \mu > 0 .

    In this section, we study the asymptotic behavior of the positive periodic solution of Eq (1.1) when the diffusion rate is large. Here we use regularity estimates together with the perturbation technique to prove our main result. To do this, we first consider the perturbation equation

    \begin{equation} \begin{cases} u_t = \mu k(x,t)\Delta u+m(x,t)(u+\varepsilon)-c(x,t)u^{p},&\text{ in } Q_T,\\ \frac{\partial u}{\partial\nu} = 0,&\text{ on } Q_1,\\ u(x,0) = u(x,T),&\text{ in } \Omega, \end{cases} \end{equation} (4.1)

    where the parameter \varepsilon > 0 .

    Lemma 4.1. Assume that Eq (1.3) holds. Then Eq (4.1) has a positive periodic solution \theta^{\varepsilon}_{\mu}(x, t) for \mu > 0 , provided \varepsilon > 0 is small. Moreover, we can find \mu_1 > 0 such that

    \begin{equation} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \theta_\mu(x,t)\,\,\mathit{\text{ in }}\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} (4.2)

    for \mu\geq\mu_1 .

    Proof. Through a similar argument as in Theorem 3.1, we know the existence of the positive solution \theta^{\varepsilon}_{\mu}(x, t) to Eq (4.1). We only need to prove Eq (4.2). Let \sigma(x, t) = \theta^{\varepsilon}_{\mu}(x, \frac{t}{\mu}) . Then \sigma(x, t) satisfies

    \begin{equation*} \begin{cases} \sigma_t = k(x,\frac{t}{\mu})\Delta \sigma+\frac{1}{\mu}[m(x,\frac{t}{\mu})(\sigma+\varepsilon) -c(x,\frac{t}{\mu})\sigma^{p}],&\text{ in } Q_T,\\ \frac{\partial \sigma}{\partial\nu} = 0,&\text{ on } Q_1,\\ \sigma(x,0) = \theta^{\varepsilon}_{\mu}(x,0),&\text{ in } \Omega. \end{cases} \end{equation*}

    It is known from the L^p estimate that there exists \mu_1 > 0 such that \sigma(x, t) is bounded in W^{2, 1}_p(\Omega\times(0, \mu T]) for any \mu > \mu_1 . It then follows that \theta^{\varepsilon}_{\mu}(x, t) is bounded in W^{2, 1}_p(Q_T) for any \mu > \mu_1 . Then, taking p large enough, we know from the embedding theorem that \theta^{\varepsilon}_{\mu}(x, t) is compact in C^{1, \frac{1}{2}}(\bar Q_T) . Thus there is a subsequence, still denoted by \theta_\mu^{\varepsilon}(x, t) , such that

    \begin{equation} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \nu(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} (4.3)

    for some nonnegative periodic function \nu(x, t)\in C(\bar Q_T) . It follows from the argument of Lemma 3.1 that \varepsilon\theta(x, t) is a lower-solution of Eq (4.1). Thus we have \nu(x, t) > 0 for (x, t)\in\bar\Omega\times[0, T] . Since \theta^{\varepsilon}_{\mu}(x, t) is bounded and Eq (4.3), \nu satisfies

    \begin{equation*} \nu(x,t) = \nu(x,0)+\mu\int_0^t[k(x,s)\Delta\nu(x,s)+m(x,s)\nu -c(x,s)\nu^p]\,ds. \end{equation*}

    It is easy to obtain

    \begin{equation*} \begin{cases} \nu_t = \mu k(x,t)\Delta\nu+m(x,t)\nu-c(x,t)\nu^{p},&\text{ in } Q_T,\\ \frac{\partial\nu}{\partial\nu} = 0,&\text{ on } Q_1,\\ \nu(x,0) = \nu(x,T),&\text{ in } \Omega. \end{cases} \end{equation*}

    By standard parabolic regularity, we know that \nu(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) . The uniqueness of the solution means that Eq (4.2) holds.

    At the end of this section, we prove Theorem 1.2.

    Proof of Theorem 1.2. We divide the proof into the following three steps.

    \bf{Step\; 1.} In this step, we prove that \theta_\mu(x, t) has a convergent subsequence as \mu\to\infty .

    It follows from a similar argument to Lemma 4.1 that there exists \mu_1 > 0 such that \theta_\mu(x, t) is compact in C^{1, \frac{1}{2}}(\bar Q_T) for any \mu > \mu_1 . Thus, by passing to a subsequence, there is a nonnegative periodic function \theta\in C(\bar Q_T) such that

    \begin{equation*} \lim\limits_{\mu\to\infty}\theta_\mu(x,t) = \theta(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T). \end{equation*}

    \bf{Step\; 2.} In this step, we show that \theta(x, t) is independent of t .

    Let f(t) be a smooth T -periodic function, then we have

    \begin{equation*} \begin{aligned} -\int_0^T\theta_\mu(x,t) f_t(t)\,dt = &\mu \int_0^T k(x,t)f(t)\Delta\theta_\mu(x,t)\,dt\\ &+\int_0^Tm(x,t)\theta_\mu(x,t)f(t)\,dt-\int_0^T c(x,t)\theta^{p}_\mu(x,t)f(t)\,dt. \end{aligned} \end{equation*}

    By dividing \mu and making \mu\to\infty , we have

    \begin{equation*} \int_0^T k(x,t)f(t)\Delta\theta(x,t)\,dt = 0. \end{equation*}

    Since f(t) is arbitrary, we obtain

    \begin{equation*} \Delta\theta(x,t) = 0. \end{equation*}

    Then we derive

    \begin{equation*} \int_{\Omega}|\nabla\theta(x,t)|^2\,dx = 0. \end{equation*}

    Thus we have \theta(x, t)\equiv\theta(t) for x\in\bar\Omega .

    \bf{Step\; 3.} In this step, we show that \theta(t) = \omega(t) in [0, T] .

    First, we assert that \theta(t)\in C^{1}((0, \infty)) . Indeed, it is easy to obtain from Eq (1.1) that

    \begin{equation*} \int_t^{t+\varepsilon}\int_{\Omega}\frac{u_s(x,s)}{k(x,s)}\,dxds = \int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}u(x,s)\,dxds -\int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}u^p(x,s)\,dxds. \end{equation*}

    Then we have

    \begin{equation*} \begin{aligned} &\int_{\Omega}\frac{u(x,t+\varepsilon)}{k(x,t+\varepsilon)}\,dx -\int_{\Omega}\frac{u(x,t)}{k(x,t)}\,dx+\int_t^{t+\varepsilon} \int_{\Omega}\frac{k_{t}(x,s)}{k^2(x,s)}u(x,s)\,dxds\\ = &\int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}u(x,s)\,dxds- \int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}u^p(x,s)\,dxds. \end{aligned} \end{equation*}

    Taking \mu\to\infty , we obtain

    \begin{equation*} \begin{aligned} &\int_{\Omega}\frac{\theta(t+\varepsilon)}{k(x,t+\varepsilon)}\,dx -\int_{\Omega}\frac{\theta(t)}{k(x,t)}\,dx+\int_t^{t+\varepsilon} \int_{\Omega}\frac{k_{t}(x,s)}{k^2(x,s)}\theta(s)\,dxds\\ = &\int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}\theta(s)\,dxds- \int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}\theta^p(s)\,dxds. \end{aligned} \end{equation*}

    Thus, we derive

    \begin{equation*} \left[\theta(t)\int_{\Omega}\frac{1}{k(x,t)}\,dx\right]_{t} = \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx\theta(t)-\int_{\Omega} \frac{c(x,t)}{k(x,t)}\,dx\theta^p(t)-\int_{\Omega}\frac{k_{t}(x,t)} {k^2(x,t)}\,dx\theta(t). \end{equation*}

    Hence,

    \begin{equation*} \begin{aligned} \theta_{t}(t) = &\frac{1}{k_*(t)}\int_{\Omega} \frac{m(x,t)}{k(x,t)}\,dx\theta(t) -\frac{1}{k_*(t)}\int_{\Omega}\frac{c(x,t)}{k(x,t)}\,dx\theta^p(t)\\ &-\frac{1}{k_*(t)}\int_{\Omega}\frac{k_{t}(x,t)} {k^2(x,t)}\,dx\theta(t)-\frac{[k_*(t)]_t}{k_*(t)}\theta(t), \end{aligned} \end{equation*}

    for t > 0 . Thus \theta(t)\in C^{1}((0, \infty)) holds.

    We then prove that \theta(t) satisfies Eq (1.5). It is obvious from Eq (1.1) that

    \begin{equation} \int_{\Omega}\frac{u_t(x,t)}{k(x,t)}\,dx = \int_{\Omega}\frac{m(x,t)}{k(x,t)}u(x,t)\,dx -\int_{\Omega}\frac{c(x,t)}{k(x,t)}u^p(x,t)\,dx. \end{equation} (4.4)

    Similarly, suppose that f(t) is a smooth T -periodic function. Multiplying f(t) on both sides of Eq (4.4) and integrating over [0, T] gives

    \begin{equation*} \begin{aligned} &-\int_0^T\int_{\Omega}u(x,t)\left[\frac{f(t)}{k(x,t)}\right]_{t}\,dxdt\\ = &\int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}u(x,t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}u^p(x,t)f(t)\,dxdt. \end{aligned} \end{equation*}

    Letting \mu\to\infty , we know

    \begin{equation*} -\int_0^T\int_{\Omega}\theta(t)\left[\frac{f(t)}{k(x,t)}\right]_{t}\,dxdt = \int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}\theta(t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}\theta^p(t)f(t)\,dxdt. \end{equation*}

    This implies

    \begin{equation*} \int_0^T\int_{\Omega}\frac{f(t)}{k(x,t)}\theta_{t}(t)\,dxdt = \int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}\theta(t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}\theta^p(t)f(t)\,dxdt. \end{equation*}

    By the arbitrary of f(t) , it follows that

    \begin{equation*} \int_{\Omega}\frac{1}{k(x,t)}\,dx\theta_{t}(t) = \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx\theta(t) -\int_{\Omega}\frac{c(x,t)}{k(x,t)}\,dx\theta^p(t). \end{equation*}

    Thus, we have

    \begin{equation*} \begin{cases} \theta_t = \frac{\tilde{M}(t)}{\tilde{k}(t)}\theta -\frac{\tilde{C}(t)}{\tilde{k}(t)}\theta^p,\quad t\in\mathbb{R},\\ \theta(0) = \theta(T). \end{cases} \end{equation*}

    Finally, we prove that \theta(t) > 0 in t\in[0, T] . We define \theta^{\varepsilon}_{\mu}(x, t) as the unique positive periodic solution of Eq (4.1) for small \varepsilon > 0 and large \mu . Similarly to the previous argument, it can be shown that

    \begin{equation} \lim\limits_{\mu\to\infty}\theta^{\varepsilon}_\mu(x,t) = \omega^{\varepsilon}(t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} (4.5)

    where \omega^{\varepsilon}(t) satisfies

    \begin{equation} \begin{cases} \omega^{\varepsilon}_t = \frac{M_*(t)}{k_*(t)} (\omega^{\varepsilon}+\varepsilon) -\frac{c_*(t)}{k_*(t)}(\omega^{\varepsilon})^p,\quad t\in\mathbb{R},\\ \omega^{\varepsilon}(0) = \omega^{\varepsilon}(T). \end{cases} \end{equation} (4.6)

    Since

    \begin{equation*} \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx > 0\,\,\text{ in }\,\,[0,T], \end{equation*}

    we can obtain that Eq (1.5) admits a unique periodic positive solution \omega(t) . Note that \omega(t) is a lower solution of Eq (4.6). Then there exists a unique positive periodic solution \omega^{\varepsilon}(t) to Eq (4.6). In addition, \omega^{\varepsilon}(t) is monotonically increasing about \varepsilon , and \omega_{\varepsilon}(t)\geq\omega(t) > 0 . We obtain that there exists a positive continuous function \omega_{0}(t) such that

    \begin{equation*} \lim\limits_{\varepsilon\to0+}\omega^{\varepsilon}(t) = \omega_{0}(t)\,\,\text{ in }\,\,[0,T]. \end{equation*}

    The uniqueness of the positive solution of Eq (4.6) implies that

    \begin{equation*} \lim\limits_{\varepsilon\to0+}\omega^{\varepsilon}(t) = \omega(t)\,\,\text{ in }\,\,[0,T]. \end{equation*}

    It follows from Lemma 4.1 that

    \begin{equation*} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \theta_\mu(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T). \end{equation*}

    This means that \theta(t) is positive, together with (4.4)–(4.6). Thus, we must have

    \begin{equation*} \theta(t)\equiv\omega(t)\,\,\text{ in }\,\,[0,T]. \end{equation*}

    This also implies that

    \begin{equation*} \lim\limits_{\mu\to\infty}\theta_\mu(x,t) = \omega(t) \,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation*}

    holds for the entire sequence.

    We consider the positive solutions of the periodic-parabolic logistic equation with indefinite weight function and nonhomogeneous diffusion coefficient. When the dispersal rate is small, we can obtain a similar result as in the homogeneous diffusion coefficient. Here we are interested in the case of large diffusion coefficient with nonhomogeneous diffusion coefficient.

    In Theorem 1.1, we obtain the condition of m(x, t) to guarantee a positive periodic solution for all \mu > 0 . Then we investigate the effect of large \mu on the positive solution and establish that the limiting profile is determined by the positive solution of Eq (1.5). More precisely, we prove that the positive periodic solution tends to the unique positive solution of the corresponding non-autonomous logistic equation when the diffusion rate is large.

    M. Fan was responsible for writing the original draft. J. Sun handled the review and supervision.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank the anonymous reviewers for the careful reading and several valuable comments to revise the paper. Fan was supported by Gansu postgraduate scientifc research (20230XZX-055) and Sun was supported by NSF of China (12371170) and NSF of Gansu Province of China (21JR7RA535, 21JR7RA537).

    The authors declare there is no conflict of interest.



    [1] Farmer JT, Howell JR (1998) Comparison of monte carlo strategies for radiative transfer in participating media. Adv Heat Transfer 31: 333-429. doi: 10.1016/S0065-2717(08)70243-0
    [2] Evans TM, Urbatsch TJ, Lichtenstein H, et al. (2003) A residual monte carlo method for discrete thermal radiative diffusion. J Comput Phys 189: 539-556. doi: 10.1016/S0021-9991(03)00233-X
    [3] Zhou HC, Chen DL, Cheng Q (2004) A new way to calculate radiative intensity and solve radiative transfer equation through using the monte carlo method. J Quant Spectrosc Radiat Transfer 83: 459-481. doi: 10.1016/S0022-4073(03)00031-1
    [4] Xia XL, Ren DP, Tan HP (2006) A curve monte carlo method for radiative heat transfer in absorbing and scattering gradient-Index medium. Numer Heat Transfer, Part B 50: 181-192. doi: 10.1080/10407790500459387
    [5] Wang A, Modest MF (2007) Spectral monte carlo models for nongray radiation analyses in inhomogeneous participating media. Int J Heat Mass Transfer 50: 3877-3889. doi: 10.1016/j.ijheatmasstransfer.2007.02.018
    [6] Mazumder S (2019) Application of a variance reduction technique to surface-to-surface monte carlo radiation exchange calculations. Int J Heat Mass Transfer 131: 424-431. doi: 10.1016/j.ijheatmasstransfer.2018.11.050
    [7] Hsu PF, Farmer JT (1997) Benchmark solutions of radiative heat transfer within nonhomogeneous participating media using the monte carlo and YIX method. J Heat Transfer 119: 185-188. doi: 10.1115/1.2824087
    [8] Ruan LM, Tan HP (2002) Solutions of radiative heat transfer in three-dimensional inhomogeneous, scattering media. J Heat Transfer 124: 985-988. doi: 10.1115/1.1495519
    [9] Hohn RH (1998) The monte carlo method in tadiative heat transfer. J Heat Transfer 120: 547. doi: 10.1115/1.2824310
    [10] Lu Z, Zhang D (2003) On importance sampling monte carlo approach to uncertainty analysis for flow and transport in porous media. Adv Water Resour 26: 1177-1188. doi: 10.1016/S0309-1708(03)00106-4
    [11] Wang X (2000) Improving the rejection sampling method in quasi-monte carlo methods. J Comput Appl Math 114: 231-246. doi: 10.1016/S0377-0427(99)00194-6
    [12] Peplow DE, Verghese K (2000) Differential sampling for the monte carlo practitioner. Prog Nucl Energy 36: 39-75. doi: 10.1016/S0149-1970(99)00024-4
    [13] Xia XL, Ren DP, Dong SK, et al. (2004) Radiative heat flux characteristics of coupled heat transfer in tubes and comparison of random sampling modes. J Eng Thermophys 25: 287-289.
    [14] Walters DV, Buckius RO (1992) Rigorous development for radiation heat transfer in nonhomogeneous absorbing, emitting and scattering media. Int J Heat Mass Transfer 35: 3323-3333. doi: 10.1016/0017-9310(92)90219-I
    [15] Cherkaoui M, Dufresne JL, Fournier R, et al. (1996) Monte carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation. J Heat Transfer 118: 401-407. doi: 10.1115/1.2825858
    [16] Cherkaoui M, Dufresne JL, Fournier R, et al. (1998) Radiative net exchange formulation within one-dimensional gas enclosures with reflective surfaces. Trans Am Soc Mech Eng 120: 275-278.
    [17] De Lataillade A, Dufresne JL, El Hafi M, et al. (2002) A net-exchange monte carlo approach to radiation in optically thick systems. J Quant Spectrosc Radiat Transfer 74: 563-584. doi: 10.1016/S0022-4073(01)00272-2
    [18] Eymet V, Fournier R, Blanco S, et al. (2005) A boundary-based net-exchange monte carlo method for absorbing and scattering thick media. J Quant Spectrosc Radiat Transfer 19: 27-46. doi: 10.1016/j.jqsrt.2004.05.049
    [19] Lionel Tessé, Francis Dupoirieux, Bernard Zamuner, et al. (2002) Radiative transfer in real gases using reciprocal and forward monte carlo methods and a correlated-k approach. Int J Heat Mass Transfer 45: 2797-2814. doi: 10.1016/S0017-9310(02)00009-1
    [20] Modest MF (2003) Backward monte carlo simulations in radiative heat transfer. J Heat Transfer 125: 57-62. doi: 10.1115/1.1518491
    [21] Lu XD, Hsu PF (2004) Reverse monte carlo method for transient radiative transfer in participating media. J Heat Transfer 126: 621-627. doi: 10.1115/1.1773587
    [22] Tan HP, Shuai Y, Dong SK (2005) Analysis of rocket plume base heating by using backward monte-carlo method. J Thermophysics Heat Transfer 19: 125-127. doi: 10.2514/1.10519
    [23] Shuai Y, Dong SK, Tan HP (2005) Simulation of the infrared radiation characteristics of high-temperature exhaust plume including particles using the backward monte carlo method. J Quant Spectrosc Radiat Transfer 195: 231-240. doi: 10.1016/j.jqsrt.2004.11.001
    [24] Lu X, Hsu PF (2005) Reverse monte carlo simulations of light pulse propagation in nonhomogeneous media. J Quant Spectrosc Radiat Transfer 93: 349-367. doi: 10.1016/j.jqsrt.2004.08.029
    [25] Kovtanyuk AE, Botkin ND, Hoffmann KH (2012) Numerical simulations of a coupled radiative-conductive heat transfer model using a modified monte carlo method. Int J Heat Mass Transfer 55: 649-654. doi: 10.1016/j.ijheatmasstransfer.2011.10.045
    [26] Soucasse L, Rivière P, Soufiani A (2013) Monte carlo methods for radiative transfer in quasi-isothermal participating media. Eurotherm Semin Comput Thermal Radiat Participating Media IV 128: 34-42.
    [27] Ruan LM, Qi H, Liu LH, et al. (2004) The radiative transfer in cylindrical medium and partition allocation method by overlap regions. J Quant Spectrosc Radiat Transfer 86: 343-352. doi: 10.1016/j.jqsrt.2003.08.011
    [28] Shuai Y, Zhang HC, Tan HP (2008) Radiation symmetry test and uncertainty analysis of monte carlo method based on radiative exchange factor. J Quant Spectrosc Radiat Transfer 109: 1281-1296. doi: 10.1016/j.jqsrt.2007.10.001
    [29] Yang WJ (1995) Radiative heat transfer by the monte carlo method. Adv Heat Transf 27: 45-91. doi: 10.1016/S0065-2717(08)70315-0
    [30] Dupoirieux F, Tessé L, Avila S, et al. (2006) An optimized reciprocity monte carlo method for the calculation of radiative transfer in media of various optical thicknesses. Int J Heat Mass Transfer 49: 1310-1319. doi: 10.1016/j.ijheatmasstransfer.2005.10.009
    [31] Ruan LM, Tan HP, Yan YY (2002) A monte carlo method applied to the medium with nongray absorbing-emitting-anisotropic scattering particles and gray approximation. Numer Heat Transfer, Part A 42: 253-268. doi: 10.1080/10407780290059530
    [32] Howell JR, Mengüç MP, Daun K, et al. (2020) Thermal radiation heat transfer. Boca Raton: CRC Press. doi: 10.1201/9780429327308
  • This article has been cited by:

    1. Peng E, Tingting Xu, Linhua Deng, Yulin Shan, Miao Wan, Weihong Zhou, Solutions of a class of higher order variable coefficient homogeneous differential equations, 2025, 20, 1556-1801, 213, 10.3934/nhm.2025011
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2816) PDF downloads(76) Cited by(1)

Figures and Tables

Figures(7)  /  Tables(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog