MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2017/1650380 1650380 Research Article Unconditional Stability of a Numerical Method for the Dual-Phase-Lag Equation Castro M. A. 1 http://orcid.org/0000-0002-4702-278X Martín J. A. 1 http://orcid.org/0000-0002-0753-7826 Rodríguez F. 1 de Monte Filippo Department of Applied Mathematics University of Alicante Apdo. 99 03080 Alicante Spain ua.es 2017 3032017 2017 01 02 2017 26 03 2017 3032017 2017 Copyright © 2017 M. A. Castro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The stability properties of a numerical method for the dual-phase-lag (DPL) equation are analyzed. The DPL equation has been increasingly used to model micro- and nanoscale heat conduction in engineering and bioheat transfer problems. A discretization method for the DPL equation that could yield efficient numerical solutions of 3D problems has been previously proposed, but its stability properties were only suggested by numerical experiments. In this work, the amplification matrix of the method is analyzed, and it is shown that its powers are uniformly bounded. As a result, the unconditional stability of the method is established.

University of Alicante
1. Introduction

Non-Fourier heat conduction models have been increasingly used in recent years to model a variety of engineering and biological heat transfer problems (see, e.g.,  and references therein). The dual-phase-lag model (DPL) proposed by Tzou [4, 5] considers two phase-lags, τq, corresponding to the heat flux vector q, and τT, corresponding to the temperature gradient T, resulting in the non-Fourier heat conduction law(1)qr,t+τq=-kTr,t+τT,where t and r are the time and spatial coordinates and k is the thermal conductivity.

Considering first-order approximations in the phase-lags in (1), in the absence of heat sources and in one dimension (1D), the combination of (1) with the conservation of energy principle leads to the DPL equation(2)Tt+τq2Tt2-τTαΔTt=αΔT,where α is the thermal diffusivity and Δ is the Laplace operator.

Analytical and numerical solutions for the DPL equation and other related DPL models derived from (1) for different specific problems and domains have been developed (e.g., ; see also references in [2, 3, 16]).

As pointed out in , in the case of numerical finite difference methods conditions on the stability of the proposed methods are not always completely established. McDonough et al.  proposed a numerical solution procedure for the DPL equation that could provide the base for an efficient numerical solver for 3D problems. However, only a partial analysis of the stability of the method was provided. As indicated by the authors, a von Neumann analysis had been conducted, though not reported in their paper, but it only provided necessary, but not sufficient, criteria for stability. Also, the authors reported that no stability problems were observed in a wide set of numerical experiments, but, as they acknowledged, the unconditional stability of their method was only suggested, but not established.

The aim of this note was to prove the unconditional stability of the finite difference method proposed in , which will be accomplished by analyzing its amplification matrix (e.g., [19, 20]). This type of von Neumann or Fourier stability analysis is directly valid for pure initial-value problems and for initial-boundary-value problems with periodic boundary conditions, and it can also be applied to general Dirichlet or Neumann boundary conditions by using appropriate periodic extensions, provided that consistent discretizations are used for the boundary conditions [21, Ch. 3].

In the next section we recall the method proposed in  and write the scheme in a form suitable for the posterior stability analysis, in the case where τq0, letting the case τq=0 for a particular simplified analysis. Next, we will obtain the amplification matrix corresponding to the scheme, showing that their powers are uniformly bounded, thus establishing the unconditional stability of the method. Finally, the main conclusions are summarized.

2. Finite Difference Scheme

The numerical solution procedure for the DPL equation proposed by McDonough et al.  consists in applying first trapezoidal integration to (2) and then using the following finite difference approximations: (3)Ttmn+1=12k3Tmn+1-4Tmn+Tmn-1+Ok2,Ttmn=12kTmn+1-Tmn-1+Ok2,2Tx2mn=1h2Tm+1n-2Tmn+Tm-1n+Oh2,where k=Δt>0 and considering the spatial domain xa,b, with Nx uniformly spaced grid points, so that Δx=h=b-a/Nx-1 and m=1,2,,Nx, and superscripts indicate time levels.

The resulting finite difference scheme is globally first-order accurate in time and second-order accurate in space. Assuming τq0, the scheme can be written in the form(4)1+kτqTmn+1-rατqτT+k2Tm+1n+1-2Tmn+1+Tm-1n+1=2+kτqTmn-rατqτT-k2Tm+1n-2Tmn+Tm-1n-Tmn-1,where r=k/h2. The case τq=0 will be examined apart.

3. Unconditional Stability

L 2 stability of the three-level scheme (4) follows from the uniform boundedness in ξ, for every 0<k<δ for a fixed δ, of the powers of the amplification matrix E~ξ ([19, p. 69], [20, p. 65]):(5)E~ξ=a1ξbξa2ξbξ10,where(6)bξ=b-1e-iξ+b0+b1eiξ,a1ξ=a1,-1e-iξ+a1,0+a1,1eiξ,a2ξ=a2,0.For scheme (4), the coefficients bi and ai are as follows: (7)b-1=-αrτqτT+k2,b0=1+kτq+2αrτqτT+k2,b1=-αrτqτT+k2,a1,-1=-αrτqτT-k2,a1,0=2+kτq+2αrτqτT-k2,a1,1=-αrτqτT-k2,a2,0=-1.Therefore,(8)bξ=-αrτqτT+k2e-iξ+1+kτq+2αrτqτT+k2-αrτqτT+k2eiξ=1-2αrτTτqcosξ-1+1-αrcosξ-1kτq,a1ξ=a1,-1e-iξ+a1,0+a1,1eiξ=-αrτqτT-k2e-iξ+21+k2τq+2αrτqτT-k2-αrτqτT-k2eiξ=2-2αrτTτqcosξ-1+1+αrcosξ-1kτq.Introducing the parameter β=cosξ-1, (8) take the form(9)bξ=1-2αrτTτqβ+1-αrβkτq,a1ξ=2-2αrτTτqβ+1+αrβkτq.Thus, writing(10)d=2αrτTτqβ-1-αrβkτqone gets(11)bξ=1-d,a1ξ=2-d+2αrβτqk.Finally, writing(12)a=11-d,b=2kαrβτq11-d,the amplification matrix E~ξ can be written in the simple form(13)E~ξ=1+a+b-a10.

We note the following bounds, which will be of use to obtain bounds on the eigenvalues of the amplification matrix. Since -1cosξ1, it follows that -2β0. Also, from (10), it follows that(14)-4αrτTτq-1-αrβkτqd-1-αrβkτq<0,and, hence,(15)1-d1+1-αrβkτq>1. Therefore, the following bounds for the parameters defined in (12) hold:(16)0<a<1,b<0. Also, since b=O(k), there is a fixed δ>0 such that 1+a+b>0 for every 0<k<δ.

Next we compute the eigenvalues of the amplification matrix, as written in (13), whose characteristic equation is (17)μ2-1+a+bμ+a=0.Thus, the eigenvalues of E~ξ are given by(18)121+a+b±1+a+b2-4a,writing μ1 for + sign and μ2 for − sign. In the case where there are two complex conjugate eigenvalues and also when there is a double root, from (17) one has that μ1μ2=a<1, and, therefore,(19)μ1=μ2<1. When there are two real different eigenvalues, that is, when 1+a+b2-4a>0, one has μ1>μ2>0, and also it follows that μ11, since b<0 and(20)121+a+b+1+a+b2-4a1-4b0.

Therefore, the following bound on the spectral radius of the amplification matrix holds:(21)ρE~ξ1,which shows that the scheme satisfies the von Neumann necessary condition for stability, as stated in . Moreover, it also holds that E~ξ has at least one eigenvalue with modulus strictly lower than 1, which will be the complementary result used to prove the unconditional stability of the scheme.

We will next follow similar arguments to those used in [20, pp. 66–68]. First we note that the sum of the absolute values of the elements of the amplification matrix is bounded:(22)j,k=12E~ξjk=1+a+b+a+1=21+a+b<21+a=22-d1-d4.Then, since E~ξ is a 2×2 matrix, there is a unitary matrix U such that T=UE~ξU is the triangular matrix(23)T=μ1m0μ2,where(24)mj,k=12E~ξjk4. Hence, L2 stability of scheme (4) is equivalent to the uniform boundedness of the powers of the triangular matrix T:(25)Tn=μ1nmnmu1,mu20μ2n,n1,where(26)mnmu1,mu2mj=0n-1μ1n-1-jμ2j. Since maxμ1,μ21, it is only necessary to prove the boundedness of mnmu1,mu2. When the eigenvalues are complex conjugate or real double, there is γ<1 such that μ1=μ2γ, so that(27)mnmu1,mu2mnγn-1C,ξR.

Also, when there are two different real eigenvalues, it holds that μ11, and there is γ<1 such that μ2γ, so that(28)mnmu1,mu2mj=0n-1γjm1-γξR.As a consequence, the unconditional L2 stability of the scheme is proved.

Remark 1.

There are two especial cases of the DPL model that could be singled out. When τT=0, the DPL model reduces to the single-phase-lag (SPL) model [2, 22], and (2) reduces to the Cattaneo-Vernotte (CV) model [23, 24]. In the case where the two phase-lags are exactly the same, τq=τT, there would be an instantaneous response between the temperature gradient and the heat flux, and (1) would be equivalent to the classical Fourier law, except for a shift in time. However, we note that, regarding the numerical scheme (4) and its stability properties, both especial cases are included in the general stability analysis presented above.

3.1. Particular Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M88"><mml:msub><mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula>

In this special case the method is a two-level scheme: (29)kTmn+1-rατT+k2Tm+1n+1-2Tmn+1+Tm-1n+1=kTmn-rατT-k2Tm+1n-2Tmn+Tm-1n,so that the amplification matrix reduces to the amplification factor:(30)E~ξ=a1ξbξ, where(31)bξ=b-1e-iξ+b0+b1eiξ,a1ξ=a1,-1e-iξ+a1,0+a1,1eiξ,with(32)b-1=-rατT+k2,b0=k+2rατT+k2,b1=-rατT+k2.Thus, introducing the parameter β=1-cosξ0, it is not difficult to get to the expressions(33)bξ=2rατTβ+1+rαβk,a1ξ=2rατTβ+1-rαβk,and from this(34)E~ξ=2rατTβ+1-rαβk2rατTβ+1+rαβk1.Therefore, the unconditional L2 stability of the scheme in this special case is also proved.

4. Conclusions

The use of non-Fourier models of heat conduction in engineering problems requires the use of efficient methods to compute numerical solutions, and a basic condition for these numerical methods to be reliably employed is to be confident that they present good stability properties.

The numerical solution procedure for the DPL equation proposed by McDonough et al.  provided the base for an efficient numerical solver for higher dimension problems, but its stability properties were not firmly established.

In this note, by analyzing the amplification matrix of the scheme, we have provided a detailed proof showing that the method is indeed unconditionally stable, including its possible application to the particular case when τq=0.

As stressed in , the use of appropriate boundary conditions is a key issue in the modelling of non-Fourier heat conduction problems. The type of stability analysis presented in this work is valid for different type of boundary conditions, when they are consistently incorporated into the numerical scheme, and it can also provide correct results in a wide range of situations for nonperiodic boundary conditions [25, Ch. 10]. In micro- and nanoscale problems, mixed-type Robin boundary conditions are needed to account for temperature jump on boundaries [26, 27], and the incorporation of these types of boundary conditions might require a particular or complementary stability analysis.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was partially funded by Grant GRE12-08 from University of Alicante.

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