A Nonstandard Finite Difference Scheme for Nonlinear Heat Transfer ...

Journal of Difference Equations and Applications, Vol. 9, No. 11, November 2003, pp. 1015–1021

A Nonstandard Finite Difference Scheme for Nonlinear Heat Transfer in a Thin Finite Rod P.M. JORDAN* Code 7181, Naval Research Laboratory, Stennis Space Center, MS 39529, USA (Received 4 February 2003; In final form 30 April 2003)

This paper is dedicated to Professor Ronald E. Mickens on the occasion of his 60th birthday and in honor of his many contributions to the fields of Applied Mathematics and Physics. A nonstandard finite difference scheme is constructed to solve an initial-boundary value problem involving a quartic nonlinearity that arises in heat transfer involving conduction with thermal radiation. It is noted that the positivity condition is equivalent to the usual linear stability criteria and it is shown that the representation of the nonlinear term in the finite difference scheme, in addition to the magnitudes of the equation parameters, has a direct bearing on the scheme’s stability. Finally, solution profiles are plotted and avenues of further inquiry are discussed. Keywords: Stefan–Boltzmann radiation law; Nonstandard finite difference scheme; Diffusion equation; Positivity

INTRODUCTION The most fundamental modes of heat transfer are conduction and thermal radiation. In the former, physical contact is required for heat flow to occur and the heat flux is given by Fourier’s heat law [1]. In the latter, a body may lose or gain heat without the need of a transport medium, the transfer of heat taking place by means of electromagnetic waves or photons [2]. If a solid with an absolute surface temperature of T is surrounded by a gas at temperature T1, then heat transfer between the surface of the solid and the surrounding medium will take place primarily by means of thermal radiation if jT 2 T1j is sufficiently large [1]. Mathematically, the rate of heat transfer across the solid – gas interface is given by the Stefan –Boltzmann radiation law [1,2] Kð›T=›nÞs ¼ 2se ðT 4 2 T 41 Þ;

ð1:1Þ

where (›T/›n)s is the thermal gradient at the surface of the solid evaluated in the direction of the outward-pointing normal to the surface, K . 0 is the thermal conductivity of the solid (assumed constant), and the constants e [ [0,1] and s < 5:67 £ 1028 W=ðm2 ·K4 Þ are, respectively, the emissivity of the surface and the Stefan – Boltzmann constant [2]. For a perfect blackbody e ¼ 1; while for a perfect insulator e ¼ 0 [2, 3]. *Secondary Address: Department of Mathematics, University of New Orleans, New Orleans, LA 70148, USA. E-mail: [email protected] ISSN 1023-6198 print/ISSN 1563-5120 online q 2003 Taylor & Francis Ltd DOI: 10.1080/1023619031000146922

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Applications involving thermal radiation with/without conduction occur in many areas of science and engineering and include aerospace engineering/design [2,4], power generation [3], glass manufacturing [2] and astrophysics [5,6]. In this work, we consider the problem of unsteady heat conduction in a thin, onedimensional, finite rod that is radiating heat across its lateral surface into a medium of constant temperature. Since the transfer of heat across the rod’s lateral surface is described by Eq. (1.1), the resulting temperature equation will be nonlinear. Consequently, we must turn to numerical approximation techniques since analytical exact solutions to nonlinear PDEs are generally not possible. Specifically, we employ the highly successful nonstandard finite difference method of Mickens (see e.g. Refs. [7 – 11] and the references therein) to solve the nonlinear initial-boundary value problem (IBVP) formulated in the next section.

PROBLEM FORMULATION AND SCHEME CONSTRUCTION Consider a very thin, homogeneous, thermally conducting solid rod of constant crosssectional area A, perimeter p, length ‘ and constant thermal diffusivity k . 0 that occupies the open interval (0, ‘) along the x-axis of a Cartesian coordinate system. Let T ¼ Tðx; tÞ denote the temperature distribution in the rod, T0 sin [px/‘] the rod’s initial temperature, and let the ends at x ¼ 0; ‘ be maintained at the constant temperatures T1 and T2, respectively. Furthermore, we assume that the rod is surrounded by a medium at constant temperature T1 and that the rod radiates (resp. absorbs) thermal energy across its lateral face into (resp. from) this medium according to the Stefan – Boltzmann law. Requiring that all temperatures be expressed in absolute units (e.g. degrees Kelvin), the mathematical model of the above physical system consists of the following IBVP (see Ref. [1], Chap. IV): 8 T t ¼ kT xx 2 b0 ðT 4 2 T 41 Þ; ðx; tÞ [ ð0; ‘Þ £ ð0; 1Þ; ð2:1aÞ > > < ð2:1bÞ Tð0; tÞ ¼ T 1 ; Tð‘; tÞ ¼ T 2 ; t . 0; ð2:1Þ > > : Tðx; 0Þ ¼ T 0 sin ½px=‘; x [ ð0; ‘Þ; ð2:1cÞ where t is the temporal variable; b0 ¼ ðkse pÞ=ðKAÞ; and based on physical considerations we require that T be nonnegative. Employing the nondimensional quantities u ¼ T=T 0 ;

x ¼ x=‘;

t ¼ tðk=‘2 Þ;

b ¼ ðT 30 ‘2 se pÞ=ðKAÞ;

u1 ¼ T 1 =T 0 ;

where T 0 . 0 is taken as constant, we recast IBVP (2.1) in dimensionless form as 8 ut ¼ uxx 2 bðu 4 2 u41 Þ; ðx; tÞ [ ð0; 1Þ £ ð0; 1Þ; ð2:3aÞ > > < ð2:3bÞ uð0; tÞ ¼ U 1 ; uð1; tÞ ¼ U 2 ; t . 0; > > : uðx; 0Þ ¼ sin ½px; x [ ð0; 1Þ; ð2:3cÞ

ð2:2Þ

ð2:3Þ

where U1,2, respectively, denote the dimensionless forms of T1,2. From Refs. [8,10], we see that a possible nonstandard finite-difference scheme for Eq. (2.3a) is k uk 2 2ukm þ ukm21 ukþ1 4 m 2 um ¼ mþ1 2 bðukm Þ3 ukþ1 m þ bu 1 ; Dt ðDxÞ2

ð2:4Þ

A NONSTANDARD FINITE DIFFERENCE SCHEME

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where x ! xm ¼ mðDxÞ; t ! tk ¼ kðDtÞ; uðxm ; tk Þ < ukm ; and we note the following discrete, nonlocal (in time only) representation for the nonlinear term: u 4 ! ðukm Þ3 ukþ1 m ;

ð2:5Þ

which is the simplest such representation possible. (The nonlocal representation of nonlinear terms is a key aspect of nonstandard finite-differencing methods [7,8,10].) Now, since Eq. (2.4) is linear in the most advanced time-step approximation ukþ1 m ; we can solve for this term to obtain the explicit scheme: ukþ1 ¼ m

ukm ð1 2 2RÞ þ Rðukmþ1 þ ukm21 Þ þ bu41 ðDtÞ ; 1 þ bðDtÞðukm Þ3

ð2:6Þ

where R ¼ Dt=ðDxÞ2 ; the truncation error is of order O[Dt þ (Dx)2], and the two denominator functions f1,2 were taken to be Dt and (Dx)2, respectively [10]. The discrete form of the positivity condition reads [8] ukm $ 0 ) ukþ1 $ 0; m

ð2:7Þ

for k fixed and all relevant values of m. From Eq. (2.6) it is clear that if 1 2 2R $ 0; then the positivity condition is satisfied and, moreover, that we can place the following bound on the temporal step size: Dt # ðDxÞ2 =2;

ð2:8Þ

assuming that the spatial step size has been specified. Here, we note that the bound on Dt given in Eq. (2.8), which is a consequence of the positivity condition, is identical to the von Neumann (i.e. usual) stability criteria for the linear heat equation [4]. In contrast, if we discretize Eq. (2.3a) using the most obvious standard finite difference scheme, we find that n 4 o ukþ1 ¼ ukm ð1 2 2RÞ þ R ukmþ1 þ ukm21 2 bðDtÞ ukm 2u41 ; ð2:9Þ m where the (local) representation u 4 ! ðukm Þ4 was used. From Eq. (2.9) it is evident that we cannot, in general, be assured that our solution will be nonnegative using standard finite differencing methods, even if we require that 1 2 2R $ 0: Finally, we observe that if the lateral surface of the rod is insulated, then b ¼ 0 (i.e. e ¼ 0) and Eq. (2.3a) reduces to the heat equation. Consequently, IBVP Eq. (2.3) simplifies to 8 ut ¼ uxx; ðx; tÞ [ ð0; 1Þ £ ð0; 1Þ; ð2:10aÞ > > < uð0; tÞ ¼ U 1 ; uð1; tÞ ¼ U 2 ; t . 0; ð2:10bÞ ð2:10Þ > > : uðx; 0Þ ¼ sin ½px; x [ ð0; 1Þ: ð2:10cÞ Using the substitution uðx; tÞ ¼ vðx; tÞ þ U 1 þ ðU 2 2 U 1 Þx and then separation of variables, the solution to IBVP Eq. (2.10) is easily found to be 2

uðx; tÞ ¼ U 1 þ e 2p t sin ½px þ ðU 2 2 U 1 Þx 2

ð2:11Þ

1 2X sin ½npx 2 2 : e 2n p t {U 1 2 U 2 ð21Þn } n p n¼1 2

Here, we note that if U 1 ¼ U 2 ¼ 0; then Eq. (2.11) reduces to uðx; tÞ ¼ e 2p t sin ½px:

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NUMERICAL RESULTS In this section, we compute and plot the approximate solution to IBVP Eq. (2.3) by iterating the difference scheme given in Eq. (2.6). The algorithm that we use was implemented through the software package Mathematica 4.2 and the data points were interpolated using Mathematica’s built-in cubic spline routine [12]. For simplicity, we have taken U1 ¼ U2 ¼ 0 in all plots. The sequence given in Fig. 1 shows the temporal evolution of the temperature profiles corresponding to IBVP Eq. (2.3), for b ¼ 2; and IBVP Eq. (2.10), the insulated case, the latter being generated from Eq. (2.11). From it, we see that for small values of t, the solution curves are practically identical. As t is increased, however, it is clear that the b ¼ 0 case rapidly pulls away as it decays to its steady-state solution u ¼ 0; while the profile corresponding to IBVP Eq. (2.3) approaches a nonzero steady-state solution. In Fig. 2, we have plotted temperature profiles corresponding to IBVP Eq. (2.3), again by iterating Eq. (2.6) with t ¼ 1 and b ¼ 2; but now using a larger value of u1. Here, we see that for R ¼ 0:5 ( ) the largest temporal step-size that satisfies the positivity condition for the value of Dx taken) and for R ¼ 0:4999 the scheme is unstable and the solution has begun to oscillate. For R ¼ 0:4998; the scheme gives the (essentially) correct profile. Obviously, there is a problem with our scheme when both b and u1 are “large” and R < 0:5: (Actually, for R ¼ 0:5 and b # 2; Eq. (2.6) is stable for u1 & 1:90:) This type of instability was also encountered by Mickens and Gumel [11] in their study of the Burgers– Fisher equation, these authors having employed the quadratic version of Eq. (2.5). As we will soon demonstrate, such instability is related in part to the representation we have chosen for

FIGURE 1 u vs. x for k ¼ 5; 50; 500; 5000 and U 1 ¼ U 2 ¼ 0: Solid: Eq. (2.6) with b ¼ 2; u1 ¼ 1 and Dx ¼ 0:02; Dt ¼ 2 £ 1024 ð) R ¼ 0:5Þ: Dashing: Eq. (2.11).


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FIGURE 2 u vs. x (generated using Eq. (2.6)) for t ¼ 1; U 1 ¼ U 2 ¼ 0; b ¼ u1 ¼ 2 and Dx ¼ 0:02: Solid. Dt ¼ ð5000Þ21 ð) R ¼ 0:5Þ: Dashing: Dt ¼ ð5001Þ21 ð) R ¼ 0:4999Þ: Dotted: Dt ¼ ð5002Þ21 ð) R ¼ 0:4998Þ:

the nonlinear term. To this end, let us replace the representation of the quartic term given in Eq. (2.5) with the following nonlinear-averaging type nonlocal representation:

k 3 ðumþ1 Þ þ ðukm21 Þ3 kþ1 4 u ! ð3:1Þ um : 2 Here, we note that in contrast to the former, this new representation of the quartic term is nonlocal both in time and space. Now, replacing ðukm Þ3 ukþ1 in Eq. (2.4) with Eq. (3.1) and m then solving for ukþ1 ; we obtain the new scheme m ¼ ukþ1 m

ukm ð1 2 2RÞ þ Rðukmþ1 þ ukm21 Þ þ bu41 ðDtÞ ; {1 þ bðDtÞ½ðukmþ1 Þ3 þ ðukm21 Þ3 =2}

ð3:2Þ

where the truncation error is again of order O[Dt þ (Dx)2]. In Fig. 3, we have plotted the Eq. (2.6) (thin-solid curve) and Eq. (3.2) (bold curve) based schemes for b ¼ u1 ¼ 2 and R ¼ 0:5: It is immediately clear from the profiles shown that, using the largest Dt allowable, the latter (i.e. new) scheme is clearly stable. Moreover, iterating Eq. (2.6) to generate Fig. 3 required approximately 88 s (of CPU time) while the same task was accomplished in the case of Eq. (3.2) in about 84 s. (We should also mention that Eq. (3.2) as been tested up to b ¼ u1 ¼ 6 with no evidence of instability.

DISCUSSION In this work we have constructed, based on the work on Mickens [7 –11], nonstandard finite difference schemes for an IBVP involving a nonlinear diffusion equation. We have shown that the overall stability of the scheme based on the representation of the nonlinear term given in Eq. (2.5), which is nonlocal only in time, is inversely related to the magnitudes of b and u1. We have also shown that for the IBVP studied here, the representation of u 4 given in Eq. (3.1), which is nonlocal both in time and space, results in a scheme that is clearly

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FIGURE 3 u vs. x for t ¼ 1; U 1 ¼ U 2 ¼ 0; b ¼ u1 ¼ 2 and Dx ¼ 0:02; Dt ¼ ð5000Þ21 ð) R ¼ 0:5Þ: Thin-solid: Eq. (2.6). Bold: Eq. (3.2).

superior, both in terms of stability and efficiency, to the one based on Eq. (2.5). (See Refs. [7,8] for the rational behind representing nonlinear terms nonlocally.) Moreover, it must be noted that there are many other aspects of this IBVP that should be investigated. For example, one could apply physically realistic boundary conditions (e.g. T 1;2 . 0) and/or consider other forms of initial data. If jT 2 T 1 j ! T 1 ; then T 4 2 T 41 < 4T 31 ðT 2 T 1 Þ: As a result, Eq. (2.3a) can be approximated by the linear PDE ut ¼ uxx 2 4bu31 ðu 2 u1 Þ:

ð4:1Þ

Note that Eq. (4.1) corresponds to modeling heat transfer across the lateral face of the rod using Newton’s law of cooling [1,3]. Furthermore, a comparative study of the three IBVPs mentioned above could be carried out in an effort to understand the impact of the quartic nonlinearity. For example, one could determine the steady-state development time (see e.g. Refs. [13,14]) and the numerical results presented here could be compared with the analytical results given in other works (see Ref. [1] and the references therein) for similar problems. In addition, the stability, accuracy and efficiency of our scheme (i.e. Eq. (3.2)) could be tested against those based on other finite differencing methods (see e.g. Refs. [15 – 17]) in the context of the present IBVP. Finally, we point out that in the derivation of Eq. (2.1a) [1], one could replace Fourier’s heat conduction law with the more realistic Maxwell – Cattaneo (MC) law (see Refs. [18,19] and the references therein). Under the latter, Eq. (2.1a), the nonlinear parabolic temperature equation, would become

l0 T tt þ {1 þ ð4l0 b0 ÞT 3 }T t ¼ kT xx 2 b0 ðT 4 2 T 41 Þ;

ð4:2Þ

where the constant l0 . 0 is a relaxation time [18,19]. Equation (4.2) is a PDE that is both nonlinear and of the hyperbolic type, i.e. a nonlinear wave equation. Currently, Prof. Mickens and the author are attempting to formulate nonstandard finite difference schemes to solve IBVPs involving Eq. (4.2), as well as other PDEs arising from the MC law [19].


1021

(Note that since Eq. (4.2) is of second order in t, solving it requires that both T and Tt be specified at t ¼ 0:) Acknowledgements This work was supported by ONR/NRL funding (PE 602435N). The author is indebted to Professor Ronald E. Mickens for many helpful discussions.

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