An Alternative Discretization and Solution Procedure for the Dual Phase-Lag Equation

Transcription

1 An Alternative Discretization and Solution Procedure for the Dual Phase-Lag Equation J. M. McDonough, Departents of Mechanical Engineering and Matheatics, University of Kentucky, Lexington, Kentucky , USA I. Kunadian, R. R. Kuar, Departent of Mechanical Engineering, University of Kentucky, Lexington, Kentucky , USA T. Yang Convergent Thinking, LLC, Madison, WI 53719, USA Abstract Alternative discretization and solution procedures are developed for the 1-D dual phase-lag (DPL) equation, a partial differential equation for very short tie, icroscale heat transfer obtained fro a delay partial differential equation that is transfored to the usual non-delay for via Taylor expansions with respect to each of the two tie delays. Then in contrast to the usual practice of decoposing this equation into a syste of two equations, we utilize this forulation directly. Truncation error analysis is perfored to show consistency and first-order teporal accuracy of the discretized 1-D DPL equation, and it is shown by von Neuann stability analysis and nuerical results that the proposed nuerical technique is unconditionally stable. The overall approach is then extended to three diensions via Douglas Gunn tie-splitting, and a siple arguent for stability is given for the tie-split forulation. Based on a straightforward arithetic operation count, qualitative coparisons are ade with explicit and iterative ethods with the expected result that the current approach is generally significantly ore efficient, and this is deonstrated with CPU-tie results. Application of Richardson extrapolation in tie is then investigated to iprove the first-order teporal accuracy, and finally, it is shown that nuerical predictions agree with available experiental data during sub-picosecond laser heating of a thin fil. Key words: icroscale, heat transport equation, phase lags, delay equations, von Neuann stability, truncation error, Douglas Gunn tie splitting Preprint subitted to Elsevier Science 11 August 2005

2 1 Introduction Fourier s law predicts infinite-velocity propagation of theral disturbances, iplying that a theral perturbation applied at any location in a solid ediu can be sensed iediately anywhere else in the ediu (violating precepts of special relativity). Associated with this is the fact that the parabolic character of the heat equation obtained fro Fourier s law iplies that heat flow starts (stops) siultaneously with appearance (disappearance) of a teperature gradient, thus violating the causality principle which states that two causally correlated events cannot happen at the sae tie; rather, the cause ust precede the effect, as noted by Cielli [1]. In order to ensure finite propagation of theral disturbances a hyperbolic heat conduction equation (HHCE) was proposed at least as early as the studies of Luikov [2] and Baueister and Hail [3]. We reark that this equation is of the sae for as that tered the telegraph equation in the atheatics literature (see essentially any interediate PDE text). More recently, it has been shown that in certain situations HHCEs violate the second law of therodynaics resulting in physically unrealistic teperature distributions such as teperature overshoot phenoena observed in a slab subject to a sudden teperature rise on its boundaries (see, e.g., Taitel [4]). Also, since the classical Fourier and hyperbolic odels neglect theralization tie (tie for electrons and lattice to reach theral equilibriu) and relaxation tie of the electrons, their applicability to very short-pulse laser heating becoes questionable, as noted by Qiu and Tien [5,6] and Qiu et al. [7]. Kagnov et al. [8] were aong the first to theoretically evaluate the icroscopic (theral) exchange between electrons and the lattice. Following this, Anisiov et al. [9] proposed a two-step odel to describe electron and lattice teperatures, T e and T l, respectively, during the short-pulse laser heating of etals. Later, Qiu and Tien [5,6] rigorously derived the hyperbolic two-step odel fro the Boltzann transport equation for electrons. They then nuerically solved the odel equations for the case of a 96fs duration laser pulse irradiating a thin fil of thickness 0.1 µ. The predicted teperature change of the electron gas during the picosecond transient agreed well with experiental data, supporting validity of the hyperbolic two-step odel for describing heat transfer echaniss during short-pulse laser heating of etals. Even though this icroscopic odel works quite well at sall scales, when investigating acroscopic effects a different odel ight be desirable. Tzou proposed the dual phase-lag (DPL) odel [10 13] that reduces to diffusion, Corresponding author. Eail addresses: jcd@uky.edu (J. M. McDonough), ikuna0@engr.uky.edu (I. Kunadian), rrkua0@engr.uky.edu (R. R. Kuar), tlyang@c-think.co (T. Yang). 2

3 theral wave, the phonon-electron interaction [5,6], and the pure phonon scattering (Guyer and Kruhansl [14]) odels as values of odel paraeters τ q and τ T are changed, peritting coverage of a wide range of physical responses, fro icroscopic to acroscopic scales, in both space and tie, ostensibly with a single forulation in ters of a single teperature. Tzou has attepted to copare this one-teperature forulation with the two-step teperature forulations, but it has been noted that the DPL oneteperature description of heat conduction in solids cannot be used to explain the icroscale two-teperature physics for pulse-laser heating of etals because these two forulations have different physical bases. Zhou et al. [15] argue that the DPL equation is only a relaxed atheatical representation with no sound physical interpretation attached to it. In particular, the teperature distribution predicted by the DPL odel does not correspond to either the electron teperature T e or the lattice teperature T l. Thus, the objective of the present paper is not to attept validation of the DPL odel but rather to present a ethod for treating equations such as this (involving lagging and ixed derivatives) and provide a procedure to efficiently solve the in the context of a relatively siple (single-teperature) forulation. In recent years, various nuerical ethods have been investigated for solving the DPL equation. Most early nuerical studies involved only the 1-D equation, often using explicit discretization in tie. Recent studies have begun to consider 2-D and 3-D DPL equations, with iplicit discretizations. Dai and Nassar [16,17] have developed an iplicit finite-difference schee in which the DPL equation is split into a syste of two equations; the individual equations are discretized using the Crank Nicolson schee and solved sequentially. These authors use the discrete energy ethod of Lee [18] to show that the approach is unconditionally stable, and that the nuerical solutions are non-oscillatory. The ethod has been generalized to 3-D by Dai and Nassar [19 21] and applied to the case of heating a rectangular thin fil with thickness at the sub-icron scale. Zhang and Zhao [22,23] have eployed the iterative techniques Gauss Seidel, successive overrelaxation (SOR) with optial overrelaxation paraeters, conjugate gradient (CG), and preconditioned conjugate gradient (PCG) to solve the 3-D discrete DPL equation with Dirichlet boundary conditions. In contrast, applying Neuann boundary conditions (as often needed for heat transfer probles) can result in non-syetric seven-band (in 3-D) positive seidefinite atrices that can be unsuitable for treatent with iterative ethods of the nature of CG and PCG, suggesting that other approaches should also be considered. The purpose of the present paper is to provide a forulation based on a single 1-D DPL equation (in contrast to the usual two coupled equations) solved 3

4 via trapezoidal integration. We will deonstrate consistency and first-order teporal accuracy (despite use of trapezoidal integration) of the nuerical schee by perforing a truncation error analysis, and we eploy a siple von Neuann analysis to show stability. The nuerical technique will be extended to three diensions using a Douglas Gunn tie-splitting ethod in δ for, and efficiency of this approach will be assessed via coparisons with other current solution procedures reported in the recent literature. We then briefly study application of Richardson extrapolation in tie for the 1-D case to obtain second-order accurate results in both space and tie. Coparisons of 1-D results with both analytical and experiental results, and 3-D siulations, are provided, and we close the paper with a suary, conclusions and suggestions for further work. 2 Analysis We first outline a derivation of the DPL equation eploying a heat flux forulation containing two phase-lag tie scales that lead to finite theral propagation rates and at the sae tie to a delay partial differential equation containing two delays. We siplify this via Taylor expansion of the teperature with respect to each of these tie scales. Following this we provide detailed analysis of the discretizations being eployed, and in particular deonstrate consistency and stability of the overall forulation in one space diension. 2.1 Brief review of DPL equation origins Matheatically, the dual phase-lag concept can be represented as in [11] with heat flux expressed as q(x, t + τ q ) = λ T (x, t + τ T ), (1) where τ T is the phase lag of the teperature gradient, and τ q is the corresponding lag of the heat flux vector. Here λ represents theral conductivity, and x is a spatial location. There are three characteristic ties involved in the dual phase-lag odel: the instant of tie t + τ T at which the teperature gradient is established across a aterial volue, the tie t + τ q for the onset of heat flow, and tie t for the transient heat transport. The heat flux and teperature gradient shown in the Eq. (1) represent local responses within the solid ediu and should not necessarily be associated with the global quantities specified in boundary conditions. Application of heat flux at the boundary does not iply precedence of the heat flux vector to the teperature gradient throughout the aterial doain; the teperature gradient established 4

5 at a aterial point within the solid ediu can still precede the heat flux vector. Whether heat flux precedes the teperature gradient depends on the cobined effects of type of theral loading, geoetry of the specien, and theral properties of the aterial. For the case of τ T > τ q, the teperature gradient established across a aterial volue is a result of the heat flow, iplying that the heat flux vector is the cause (e.g., due to localized internal heating), and the teperature gradient is the effect. For τ T < τ q, on the other hand, heat flow is induced by the teperature gradient established at an earlier tie t, iplying that the teperature gradient is the cause, while heat flux is the effect the view of heat transfer fro classical therodynaics. Application of the first law of therodynaics in the usual way, but now with the tie lags described above, leads to T t (x, t + τ q) = α T (x, t + τ T ), (2) which is a delay partial differential equation with two separate delays. Here, α is theral diffusivity, and is the Laplace operator in an appropriate coordinate syste. In this for the atheatical proble is nearly intractable, both analytically and nuerically. But if we assue T is sufficiently sooth in tie we can expand it in Taylor series about t, separately, with respect to each of τ q and τ T : T (x, t + τ q ) = T (x, t) + T t (x, t)τ q +, and T (x, t + τ T ) = T (x, t) + T t (x, t)τ +, T where we have neglected all higher-order ters. Clearly, if higher-order derivatives reain bounded we should expect this approxiation to be at least qualitatively accurate because τ T and τ q are expected to be sall. Substitution of these expressions into (2) leads to τ q 2 T t 2 + T t τ α ( T ) T t = α T. (3) It should be clear that analysis of (3) is far ore straightforward than that of Eq. (2), despite presence of the third-order ixed derivative ter, because there are now no delays. In particular, coplete well-posed probles for the HHCE are also well posed for (3). Furtherore, we see that when τ T = 0 the HHCE is recovered, and if both τ T and τ q are zero the usual parabolic heat equation results. As was hinted above, it has been custoary to decopose (3) as a syste of two equations containing no ixed derivative. There are at least a couple 5

6 versions of this, and we refer the reader to [21] for one exaple. But such decopositions rely on constancy of τ q and τ T, and while this is not an unreasonable expectation it is at least of interest to seek a solution approach not depending on this requireent. Loss of this required constancy could occur (spatially) for nonhoogeneous aterials, and, in general, if τ q and τ T depend on T, leading to nonlinearities. Directly solving Eq. (3), as we will describe in the sequel, provides a way to to handle these situations although this is not the focus of the present work. By introducing diensionless quantities defined by Chiu in [24], we can express Eq. (3) as θ(x, t) = T (x, t) T 0 T W T 0, β = t τ q, δ = x ατq, (4) T t + 2 T t 2 Z t ( 2 ) T x 2 = 2 T x 2, (5) where Z is the ratio of τ T to τ q, and τ q is equated to the diffusive tie scale. We reark that it has been necessary to use constant values of these paraeters to achieve this scaling, which is convenient for the reainder of the analyses; but it is not unduly difficult to work with the diensional for of equation (3) to perit treatent of nonconstant paraeter values as entioned above. Indeed, this for is eployed in our later siulations of physical probles, results of which will be presented in Sec Discretization and analysis of the 1-D DPL equation We begin by applying trapezoidal integration to Eq. (5) to obtain T n+1 T n + ( ) n+1 ( ) n T T Z t t ( 2 ) n+1 ( T 2 ) n T x 2 x 2 = k ( 2 ) n+1 ( T 2 ) n T +, (6) 2 x 2 x 2 with k denoting the tie step size ( t); subscripts provide discrete spatial indexing, and superscripts represent tie levels. It is of interest to note that backward-euler integration results in the sae set of ters shown in (6), but with different signs and coefficients for soe ters. 6

7 We apply a second-order backward difference to discretize the tie derivative at tie level n + 1 and a centered difference for the tie derivative at n. The second-order derivatives in space are approxiated using a centered-difference schee: thus, ( T t ( T t ( 2 T x 2 ) n+1 ) n ) n 1 [ = 3T n+1 2k 1 [ = T n+1 2k 4T n + T ] n 1, (7a) ] T n 1, (7b) = 1 h 2 [T +1 2T + T 1 ], (7c) where we have written these without their associated truncation errors, and we note that the last of these holds for both tie levels included in Eq. (6). In Eq. (7c) h (b a)/(n x 1) is the spatial step size for a 1-D spatial doain Ω [a, b] discretized with N x uniforly-spaced grid points. The approxiations shown in Eqs. (7) render the nuerical schee globally first-order accurate in tie and second-order accurate in space, as we now show. Proposition 1 Suppose for all x Ω R and t (t 0, t f ] that T (x, t) is sufficiently sooth to possess bounded ixed derivatives through at least order six. Then the cobination of trapezoidal integration and discretizations (7) applied to Eq. (5) yields a consistent approxiation that is first order in tie and second order in space for all 0 Z <. Proof. After substituting Eqs. (7) into Eq. (6) we obtain ( 1 [ T n+1 T n + 3T n+1 2k ] ) ( 1 [ 4T n + T n 1 T n+1 2k ] ) T n 1 Z ([ T n+1 n+1 h T + T [ 1] n+1 T n +1 2T n + T 1 n = k ([ T n+1 n+1 2h T + T [ 1] n+1 + T n +1 2T n + T 1]) n, (8) and siplification and rearrangeent leads to ]) T 1 n+1 + C 1T n+1 + T +1 n+1 = C ( ) 2 T n 1 + T+1 n + C3 T n C 4T n 1 (9) 7

8 for = 1, 2,..., N x (odulo boundary condition treatent); and C 1 = h2 (1 + 1/k) + 2Z + k Z + k/2 C 2 = Z k/2 Z + k/2, C 3 = h2 (1 + 2/k) + 2Z k Z + k/2 C 4 = h2 /k Z + k/2., (10a) (10b), (10c) (10d) Expansion of T n+1 1 (= T (x 1, t n+1 )), etc., in (9) about an arbitrary spacetie point (x, t n ) Ω (t 0, t f ], and using Eqs. (10) followed by straightforward but tedious algebra, results in T t + 2 T t 2 Z 3 T t x 2 T 2 x = k ( 3 T 2 2 h 2 12 ) t x 2 T 2 t + Z 4 T + 2 t 2 x 2 ( 4 ) ( T x + Z 5 T k2 1 3 T 4 t x t T 6 t h 2 ( k 5 T 24 k T t 2 x Z 5 ) T + (11) 2 3 t 3 x 2 ) 6 T t x + Z 4 t 2 x 4 ( 5 T t 3 x 1 4 T 2 2 t + Z ) T + t 4 x 2 at (x, t n ), where we have suppressed index notation since this holds at all points other than those corresponding to Dirichlet boundaries. Clearly, the right-hand side approaches zero as h, k 0 independent of the order of these liits, and the leading error is O(k + h 2 ) independent of Z. This copletes the proof. Notice that Eqs. (8) and (9) are three-level in tie. Moreover, the right-hand side of Eq. (9) consists of known values, and the iplicit part on the left-hand side is tridiagonal, so the collection of all such equations at each tie level can be easily solved using failiar tridiagonal LU decoposition, or possibly cyclic reduction. 8

9 2.3 Stability analysis We analyze the stability of the above schee via a von Neuann analysis. Because Eq. (9) is a three-level difference approxiation, the von Neuann condition supplies only a necessary (but not sufficient) stability criterion in general. Thus, we also present soe nuerical results that at least suggest that the ethod is unconditionally stable. Proposition 2 The discretization of the DPL equation given by Eq. (9) satisfies a von Neuann necessary condition for stability h, k 0, and 0 Z <. Proof. Since the DPL equation in the for treated here is linear and constant coefficient, it is clear that the discrete evolution equation for the error {e n } is e n C 1e n+1 n + e n+1 +1 = C 2(e n 1 + en +1 ) + C 3e n C 4e n 1, (12) with the C i s given in (10). As usual, let e n = ξ n e iβh (13) with β arbitrary. Then substitution of (13) into (12) results in which we express as for ease of presentation. Then ξ 2 C 3 + 2C 2 cos βh C cos βh ξ C 4 C cos βh = 0, (14) ξ 2 C 1 ξ + C 2 = 0 (15) ξ ± = C 1 2 ± C C 2, (16) and we ust show that ξ ± 1, independent of h, k and Z, to coplete the proof. We will do this indirectly by appealing to a lea that appears in Young [25] (pg. 171). This states that the agnitude of both roots of a quadratic polynoial of the for (15) is less than unity C 1, C 2 R satisfying C 2 < 1, and C 1 < 1 + C 2. (17) Hence, all that is needed is to deonstrate satisfaction of these inequalities. We reark that the lea is used in [25] in the context of studying convergence of linear iterative ethods for solving sparse algebraic systes. Thus, 9

10 strict inequalities are required in (17) to guarantee spectral radii strictly less than unity. In the present case of stability analysis, non-strict inequalities are perissible. Substitution of Eqs. (10a) and (10d) into the definition of C2 (Eq. (14)) results in C2 h 2 /k = h 2 (1 + 1/k) + (2Z + k)(1 cos βh). Clearly, C2 0; but the denoinator is nonnegative, and we need only check that h 2 k h2 (1 + 1/k) + (2Z + k)(1 cos βh), or k + k (2Z + k)(1 cos βh) 0. h2 But this holds trivially h, k, Z 0. Next, we construct C 1 in a siilar anner to obtain C 1 = h2 (1 + 2/k) + (2Z k)(1 cos βh) h 2 (1 + 1/k) + (2Z + k)(1 cos βh). The denoinator of this expression is the sae as that in C2 and is nonnegative; but this does not necessarily hold for the nuerator if Z k/2 and h is sufficiently sall. Thus, we ust consider C1 1 + C 2 written in the for h 2 (1+2/k)+(2Z k)(1 cos βh) h 2 (1+1/k)+(2Z +k)(1 cos βh)+h 2 /k. Rearrangeent of the right-hand side of this inequality leads to h 2 (1 + 2/k) + (2Z k)(1 cos βh) h 2 (1 + 2/k) + (2Z + k)(1 cos βh), which clearly holds h, k, Z 0 since the right-hand side is nonnegative. Hence, both inequalities in (17) are satisfied (nonstrictly), thus guaranteeing that ξ ± 1 holds independent of the cobination of space and tie step sizes and the chosen value of Z, and proof of the proposition is coplete. But as already noted, this alone does not iply unconditional stability because the difference equation (9) is three level. We have perfored nuerous coputations with a wide range of values of h, k and Z and have been unable to find any cobinations that lead to long-tie blowup of solutions. Thus, unconditional stability appears to be likely, but it cannot be proven with the siple von Neuann analysis provided here. Figure 1 displays tie series at the idpoint of the spatial doain [0, 2] for several cases of step sizes and values of Z including both rather noinal and soewhat extree sets of these paraeters. Part (a) of the figure corresponds to the case Z = 0, which at early ties exhibits oscillations due to incopatibility of initial and boundary 10

11 1.2 (a) k/h = k/h = 5 2 k/h = (b) 1.0 Scaled Teperature k/h = k/h = 5 2 k/h = (c) k/h = k/h = 5 2 k/h = Scaled Tie Fig. 1. Nuerical confiration of stability; (a) Z = 0, (b) Z = 1, (c) Z = 10. conditions and the hyperbolicity of Eq. (5). In particular, T is initially identically zero (as is T/ t), but a non-zero boundary teperature is iposed on the left boundary. Part (b) presents results for Z = 1, and part (c) for 11

12 Z = 10. In each of these results are presented for cobinations of k and h such that k/h 2 = 1 2, 5 and 104. Except for the first, these could lead to instability for typical explicit schees for soe values of Z. The figure shows that the long tie behavior corresponds to a steady, bounded solution independent of cobination of space and tie step, and the insets provide an indication of convergence as tie step k is decreased, except in the pure hyperbolic case. Furtherore, any other runs were coputed, soe with k/h 2 as high as 10 6 ; but all show the sae qualitative behavior, so we are reasonably confident that the ethod is unconditionally stable. 3 Efficient Solution of 3-D DPL Equation In this section we extend the approach analyzed above to the physicallyrelevant 3-D case. Recent work in both 2D and 3D has often eployed iterative ethods to solve the algebraic systes arising at each discrete tie step; this can be quite inefficient, especially when very fine spatial grids are being used. Here, we will utilize a very old approach that has been widely eployed in coputational fluid dynaics to efficiently treat the 3-D proble, but we note that this technique is basically not applicable when unstructured eshes are used. We begin with presentation of the diensional, non-hogeneous 3-D DPL equation, apply discretizations analogous to those used earlier in 1D, and then derive the Douglas Gunn [26] tie-split forulas that result in only O(N) arithetic operations per tie step, where N is the nuber of points in the coputational grid. By coparison, even fairly efficient iterative ethods often require as uch as O(N 1.5 ) operations per tie step as is suggested by CPU-tie coparisons with an often-used such ethod presented in a final subsection of the current section. 3.1 Discretization of a 3-D nonhoogeneous DPL equation The governing DPL equation used to describe the theral response of icrostructures subjected to laser heating is expressed in diensional for in [10] as τ q 2 T α t T α t τ ( T ) T t = T + 1 λ ( S + τ q S t ). (18) 12

13 In one diension the voluetric source ter S describing laser heating of the electron-phonon syste fro a theralization state is given by [ ] ( 1 R S(x, t) = 0.94J exp x t p δ δ t 2t ) p, (19) t p where laser fluence J = 13.7J/ 2, and t p = 96fs; penetration depth is δ = 15.3 n, and reflectivity is R = 0.93, as presented in [24]. We have extended this to 3D as [ ] ( 1 R Lx (x 2 S(r, t) = 0.94J exp )2 +(y Ly t p δ 2r 2 o 2 )2 ) z δ t 2t p t p, (20) where L x and L y are the length and width of the etal fil respectively, and r o is radius of the laser bea oriented in the z direction. Applying trapezoidal integration to Eq. (18) between tie levels n and n + 1 yields ( ) n+1 T n+1 T n + τ q T t = α k 2 ( ) n T τ t T α ( T n+1 T n) ( T n+1 + T n) + α k ( S n+1 + S n) + α λ 2 λ τ ( q S n+1 S n) (21) at any grid point (x i, y j, z k ). As in the earlier 1-D case, we apply a second-order backward difference for the tie derivative at tie level n + 1 and a centered difference at tie level n. Second-order derivatives in space are approxiated using the usual centered-difference schee. We first consider contributions fro the left-hand side of (21) plus the tielevel n + 1 Laplacian fro the right-hand side that results fro these approxiations. These ters can be expressed as Ti,j,k n+1 ατ + k/2 [ T 1 ( ) 1 Ti 1,j,k 2T 1 + τ q /k h 2 i,j,k + T i+1,j,k + x h 2 y ) 1 ( ) ] n+1 2T i,j,k + T i,j+1,k + Ti,j,k 1 2T h 2 i,j,k + T i,j,k+1 z 2τ q/k 1 + τ q /k T i,j,k n + τ q/k 1 + τ q /k T i,j,k n 1. ( Ti,j 1,k (22) Siilarly, the reaining ters fro the right-hand side differential operators of (21), and including the tie-level n ter fro the ixed derivative on the left-hand side, are 13

14 α τ k/2 [ T 1 ( ) 1 ( ) Ti 1,j,k 2T 1 + τ q /k h 2 i,j,k + T i+1,j,k + Ti,j 1,k 2T x h 2 i,j,k + T i,j+1,k y + 1 ( ) ] n Ti,j,k 1 2T h 2 i,j,k + T i,j,k+1. (23) z Finally, the nonhoogeneous ter can be written as s n = α/λ [( τq + k/2 ) S n+1 ( τ q k/2 ) S n]. (24) 1 + τ q /k If we now cobine the results fro expressions (22) (24) and ove all ters to the left-hand side except the nonhoogeneous ter, we can express the result in the standard for of a M +2-level difference equation to which the Douglas Gunn foralis [26] can be directly applied: ( ) M I + A n+1 T n+1 + B n T n = s n, (25) =0 with M = 1 in the case of the three-level schee being considered here. Bold sybols denote N N atrices with N N x N y N z, the product of the nuber of grid points in each coordinate direction, and I is the identity atrix. This equation holds at all points of the discrete solution doain except at boundaries, where soe odifications are necessary. We also reark that in the cases being treated herein the teporal indexing of atrices is erely foral. The atrix A n+1 consists of the ters in (22) arising fro spatial discretization at tie level n + 1; viz, N s A n+1 = l=1 A n+1 l, (26) where N s is in general the nuber of split steps (which in the present case is the nuber of spatial diensions of the solution doain). Also, the B atrices are constructed as follows: B n 0 = An 2τ q/k 1 + τ q /k I, Bn 1 = τ q/k 1 + τ q /k I. (27) In the first of these the atrix A n arises fro spatial discretization of the Laplacian at tie level n appearing in (23). 14

15 3.2 Douglas Gunn tie splitting Foral tie splitting of this equation is based on the following l th split step forula fro [26]: (I + A n+1 l )T (l) l 1 + A n+1 r T (r) + and r=1 N s r=l+1 A n+1 r T + 1 B n T n = s n, (28) =0 l = 1,..., N s, T n+1 T (Ns). (29) It is clear fro the for of (28) that all ters are evaluated explicitly except the first. Indeed, all those in the second ter (first suation) have already been coputed during earlier split steps while those in the second suation are usually evaluated with data fro the preceding tie level, or via extrapolation if higher accuracy is sought. In our case, the latter will not be necessary since the basic unsplit schee is only first-order accurate in tie. Finally, all inforation needed for the third suation coes fro earlier tie levels. Thus, we can express (28) as l 1 (I + A l )T (l) = s n A r T (r) r=1 N s 1 A r T B T n, (30) r=l+1 =0 with all quantities on the right-hand side known, and where we have now suppressed tie-step indexing of atrices. Although the first suation in (30) is null for l = 1 and the second for l = N s, there is nevertheless considerable arithetic required for evaluation of these equations for all l = 1,..., N s. But by first writing the above for the l 1 th step and subtracting this fro the l th -step forula for all but the first step, and then adding and subtracting (I + A l )T (l) for each of the resulting N s equations (including the first), one obtains the so-called δ for of Douglas Gunn tie splitting: 1 (I + A 1 )δt (1) = s n (I + A)T n B T n, =0 (I + A l )δt (l) = δt (l 1), l = 2,... N s. where δt (l) T (l) T n T n+1 = T n + δt (Ns). In this for we recognize that the right-hand side of the first equation is the original discrete equation evaluated at tie level n, so the aount of required arithetic is O(N) essentially that required for an explicit ethod. Then there are N s atrix-vector solves to be perfored. In the present case, each of 15

16 these involves a tridiagonal atrix, which also leads to O(N) arithetic per solve. For the l th split step this takes the for with the coefficients defined as C (l) 1 δt (l) i 1 + C(l) 2 δt (l) i + C (l) 1 δt (l) (l 1) i+1 = δt i C (l) 1 τ T + k/2 h 2 l (1 + τ q/k, C(l) (τ T + k/2) h 2 l (1 + τ q/k, and i denotes a generic ulti index for (3-D) gridpoint notation with shifts only in the l th slot. Furtherore, we associate h x with l = 1, i.e., h 1 = h x, etc. It is worthwhile at this point to recall soe of the features of Douglas Gunn tie splitting presented in [26]. First, it is proven in that work that up through second-order teporal accuracy, the split schee retains the accuracy of the unsplit schee. Second, it is also shown that stability of the unsplit schee is inherited by the split schee. While we have not carried out a detailed stability analysis in the 3D case, the equation is of the sae for as that studied in 1D where unconditional stability is suggested, and our nuerical results in 3D have provided no counterexaples. Moreover, it is generally true that split schees are ore stable than unsplit ones (even for explicit ultidiensional ethods) for a rather basic reason. Naely, the aplification factor of a split ethod is usually, up to a sall perturbation, the product of the aplification factors of the individual split steps. So, if each of these is stable, then the overall split schee is even ore strongly stable. Indeed, one sees fro this that it is even possible for one (or ore) steps of a split schee to be unstable while the ethod, as a whole, retains stability. Finally, we reark that these favorable properties of the Douglas Gunn split schee are proven in [26] under quite stringent conditions associated with solution regularity and coutativity of the various atrices appearing in Eqs. (26) and (27). But it is usually found in practice that these requireents can be relaxed significantly without affecting behavior of the ethod. 3.3 CPU tie coparisons with other solution ethods As we have already ephasized, Douglas Gunn tie splittings exhibit very favorable stability and accuracy characteristics; but beyond this is their inherent coputational efficiency. We quantify this with data presented in Table 1 which contains a sapling of results given by Kunadian et al. [27]. This table shows CPU tie in seconds required to coplete an entire siulation to a fixed final tie for explicit, conjugate gradient and δ-for Douglas Gunn tie-splitting ethods using different values of nuber of grid points 16

17 N = N x N y N z. The spatial doain consists of a thin square slab described in ore detail in Sec. 5 with unifor grid spacing, and equal nuber of grid points, in each of the three separate directions. Table 1. Perforance coparison of different nuerical ethods for solving the discretized 3-D DPL equation. Nuerical techniques Total required CPU tie (seconds) N = 21 3 N = 41 3 N = 51 3 N = Explicit Schee Conjugate Gradient δ-for Douglas Gunn Fro the table we can observe that for extreely coarse grids the explicit ethod consues less CPU tie than that required by the other nuerical techniques, but as the spatial resolution is refined the iplicit ethods perfor better than the siple forward-euler/centered-difference explicit ethod eployed in this research due to the sall tie steps required for stability of the latter. The δ-for Douglas Gunn tie-splitting used in the present study consues the least CPU tie of the three ethods considered; oreover, its degree of superiority increases with spatial resolution, and it is highly parallelizable. Further coparisons of this sort over a wide range of currently-used ethods are forthcoing in a paper by Kuar et al. [28] where it is shown that for high-resolution calculations on grids having greater than a illion grid points, tie splitting is significantly ore efficient for tie-dependent probles than any iterative technique. 4 Application of Richardson Extrapolation in Tie In an earlier subsection we showed that our basic ethod is only first-order accurate in tie. A straightforward reedy is Richardson extrapolation, as described in essentially any eleentary nuerical analysis text, e.g., Isaacson and Keller [29]. Here we present the forula to be used to ipleent this technique and provide soe 1-D results to deonstrate its effectiveness. Because we are extrapolating only in tie, the algorith needed for spatially ulti-diensional probles is the sae. It is well known that Richardson extrapolation akes use of two (or ore) solutions to the sae set of discrete equations calculated with different discrete step sizes, and cobined in such a way as to cancel truncation error at lowest order. In the case of first-order ethods and discretization step sizes differing by a factor of two, the forula used is identical to siple linear extrapolation, 17

18 naely T = 2T (k/2) T (k), (31) where T is the extrapolated result having accuracy corresponding to the next higher order in the truncation error expansion; T (k) is the result coputed with tie step size k, and T (k/2) is coputed with a tie step size k/2. Of course, different cobinations of tie step sizes can be used, leading to soewhat different forulas. Clearly, the process can be repeated to successively reove higher-order errors, but here we will apply it only once using Eq. (31). It is also iportant to note that teporal extrapolation of approxiate solutions to differential equations ust be done globally to avoid destabilization of the underlying tie integration technique. Figure 2 presents results deonstrating the effects on grid-function convergence of applying Eq. (31). Part (a) of this figure shows tie series of teperature coputed with Z = 1.5 in the 1-D DPL equation (5) at the center point of the spatial interval [0, 2] with fixed spatial step size h = 0.01 and the sequence of tie step sizes k = 0.04, 0.02, 0.01 along with a result of higher accuracy obtained by Richardson extrapolation of the k = 0.01 solution. The first-order convergence rate is obvious: the error is decreased by a factor of two each tie the tie step size is reduced by the sae factor. In part (b) of the figure we show results for the sae grid spacing and tie step sizes, but now with Richardson extrapolation. (Of course, to accoplish extrapolation for the sallest tie step it was necessary to also copute with a k = step.) It is clear that the three extrapolated tie series lie very close to one another, and in fact are all ore accurate than the unextrapolated k = 0.01 result. Several rearks are needed for a better understanding of these extrapolated results. First, the (rather short) interval chosen for displaying results is an optial one (for visualizing the extrapolation effects) with respect to the overall tie series coputed for t (0, 10]. That is, the various curves are farthest apart on this subinterval, thus peritting easy qualitative assessent of convergence rate. Otherwise, there is nothing special regarding this interval. Second, the value Z = 1.5, while not the only one providing an easily visualized depiction of results, is a reasonable and practical one; oreover, curves coputed with the various tie step sizes shown here are significantly closer together (and thus harder to interpret graphically) when either very large (say Z > 10) or very sall (Z < 0.1) values are used. Convergence rates are not changed in these cases, but they are ore difficult to present clearly in plots such as contained in Fig. 2. Third, the results of this figure were coputed with the spatial step size held fixed. In general, one ust decrease both h and k to obtain a converged solution. But the basic ethod is already second-order accurate in space, and 18

19 0.26 (a) k = 0.01, extrapolated 0.22 k = 0.01 Scaled Teperature (b) k = 0.01, extrapolated k = 0.04 k = k = 0.02, extrapolated k = 0.04, extrapolated Scaled Tie Fig. 2. Richardson extrapolation of 1-D DPL equation results; (a) deonstation of first-order convergence of basic ethod, (b) results obtained with extrapolation. it is easy to see that the for of extrapolation used in tie does not alter this foral order of accuracy. In particular, consider construction of a forula such as (31), but now with spatial discretization also taken into account. Fro the for of Eq. (11) it is clear that T n (k, h) = T (x, t n ) + τ 1 k + σt 1 h 2 + O(k 2,...), with τ 1 and σ 1 representing evaluations of appropriate derivatives as found in (11). Then if the space and tie steps are both halved siultaneously, T n (k/2, h/2) = T (x, t n k ) + τ σ h O(k2,...). To eliinate the O(k) teporal error we ultiply the second of these by two and subtract the first: T = 2T n (k/2, h/2) T n (k, h) 1 2 σ 1h 2 + O(k 2, h 2 k + k 3,...), (32) 19

20 which is of the for (31) but now displays the new (after extrapolation) doinant global truncation errors. It is worth noting that if h is not changed during extrapolation, the doinant spatial truncation error will be twice as large, and of opposite sign, as that in Eq. (32); but it will still be of the sae order. On the other hand, the O(h 2 k) ter(s) will be copletely eliinated for fixed h but only ade saller if h is reduced by a factor two at the sae tie this is done with k. The net effect of changing both h and k yields results that are qualitatively indistinguishable fro those of the figure for the cases we have tested. We next coent that the foral order of accuracy of extrapolated results indicated by (32) was not observed uniforly in tie or space in the coputed results. There are two ain contributions to this discrepancy. First, the truncation error expansions are sufficiently coplicated, as is clear fro (11), to adit local (at least partial) error cancellations. When (and where) this occurs, although the foral order of accuracy ay (or ay not) be altered, the truncation error expansions will assue soewhat different fors than that of (32). Furtherore, if a contribution goes through zero, resulting in a sign change, extrapolation in a neighborhood of such an occurence is often ineffective. On the other hand, one expects that for sufficiently sall h and k such sign changes should not occur. But with sall h and k a second effect can be experienced. In the present calculations, despite the fact that the values eployed for h and k appear to be relatively large, coputed results are quite accurate. One sees in Fig. 2(a) that even the k = 0.04 solution contains errors of only 5%, and as already indicated, the errors shown in this subinterval are aong the largest of the whole calculation. Furtherore, it is clear fro Fig. 2(b) that errors in the extrapolated solutions are nearly insignificant for all values of k. The consequence of this is that when coparing extrapolated solutions in an effort to quantitatively deduce observed order of teporal accuracy, it was often the case that the ain differences between solutions coputed with different values of tie step arose fro rounding errors. This was confired for soe, but not all, cases by repeating runs utilizing quadruple-precision arithetic. In general, as close exaination of Fig. 2(b) suggests, the Richardson extrapolated results exhibited teporal convergence rates between first and second order (rather than the theoretical second order), depending on the specific space-tie point selected for analysis. But as is also evident fro coparing the two parts of Fig. 2, the extrapolated results are far ore accurate in an absolute sense than are the basic first-order results. Finally, it is iportant to note that even though global solutions ust be used in the Richardson extrapolation process to aintain stability of the underlying tie-stepping procedure, this does not iply that two copies of the coplete (for all spatial points and for all tie) solution to the PDE ust be first 20

21 generated (and stored) before extrapolation can be accoplished. In fact, with only a couple extra arrays of size N the Richardson procedure can be ipleented to perfor extrapolation after each new tie step and to write the results to files, as desired (as would be done without extrapolation). The key to aintaining stability is siply to not evolve the extrapolated results. The additional code needed to ipleent such a procedure is only a few lines. 5 Coputed Results for Specific Probles In this section we provide soe representative results associated with siulating physical probles of the sort for which the DPL equation was originally intended. In the first subsection calculations corresponding to a 1-D odel of laser heating of a thin gold fil are presented and copared with analogous analytical and experiental results, and in a second subsection siulations fro a 3-D odel proble are presented D laser heating of gold fil The equation solved in this case is Eq. (18) with the Laplacian replaced by 2 / x 2 and the right-hand side forcing constructed fro Eq. (19). The initial conditions are T 0, and T/ t 0 on Ω; no-flux boundary conditions are eployed at both ends of the interval. We note that differentiation of S as required in (18) poses a ild difficulty due to the for of (19), but this does not create a ajor proble since the Heaviside function is in L 1 (Ω) for bounded Ω. In this proble the spatial doain Ω is the fil thickness (0.1 µ). The proble is discretized with N x = 1001 points, and a tie step k = seconds (= t) was used. Our earlier grid-function convergence tests, although conducted with a diensionless for of the DPL equation, suggest that this should provide sufficient resolution. Figure 3 presents a coparison between the nuerical, analytical [24] and experiental results of Brorson et al. [30] and Qiu and Tien [5,6] corresponding to the front surface transient response for a 0.1µ thick gold fil. Theral properties of the aterial (α = s 1, λ = 315 W 1 K 1, τ T = 90 ps, τ q = 8.5 ps) are assued to be constant. The teperature change has been noralized by the axiu value that occurs during the short-tie transient. Results fro the present nuerical schee copare nearly perfectly with analytical results (confiring adequacy of the discretization step sizes) and reasonably well with experiental results, while the HHCE and parabolic od- 21

22 1.0 Analytical, Chiu Experiental, Qui & Tien Noralized Teperature Change Nuerical, HHCE Experiental, Brorson et al. Nuerical, Parabolic Nuerical, DPL Tie (ps) Fig. 3. Front surface transient response for a 0.1 µ gold fil. Coparison aong nuerical, analytical and experiental results. els, which neglect the icrostructural interaction effects during the short-tie transient, overestiate teperature during ost of the transient response, as shown in the figure D pulsed-laser heating of gold fil For this proble the full 3-D for of Eq. (18) is eployed with a source ter constructed fro S given in Eq. (20); initial and boundary conditions are the sae as in the 1-D case but with the forer iposed on the whole volue Ω of the proble doain and the latter now applied over the entire surface Ω. This doain is again a gold fil of thickness 0.1µ but now with lateral extent 0.5µ 0.5µ. The spatial discretization was constructed on a grid of uniforly-spaced points in all three directions and utilizing ties steps of seconds. As noted in Sec. 2, Eq. (18) acquires DPL, HHCE and parabolic character according to the values selected for τ T and τ q. Here, we have used τ T = 90 ps, τ q = 8.5 ps DPL odel; τ T = τ q = 0 parabolic odel and τ q = 8.5 ps, τ T = 0 hyperbolic odel. Thus, we are able to eploy the sae code for all calculations. 22

23 Figures 4 display a coparison (at two different ties) of transient teperature distribution caused by a pulsating laser bea of 200 n diaeter heating the top surface of the gold fil at various locations near its corners (with oveent fro one corner to the next successive one in a counter-clockwise direction) every 0.3 ps, as predicted by DPL, hyperbolic and parabolic heat conduction odels. (Bright red represents the highest teperatures, and deep blue corresponds to the lowest ones.) The figures show that HHCE and parabolic diffusion odels predict a higher teperature over a wider area near the fil s surface in the heat-affected zone than does the DPL odel, but the penetration depth is uch shallower. The heat-affected zone is significantly deeper for the DPL odel than for the other odels due to the icrostructural interaction effect incorporated in the forulation of this odel. Furtherore, discrepancies between DPL and the other two odels grow significantly with tie, as is evident fro part (b) of the figure. We have been unable to acquire 3-D experiental data with which to ake quantitative coparisons with these siulations. (a) (b) DPL HHCE Parabolic Fig. 4. Teperature distribution in gold fil predicted by DPL, hyperbolic and parabolic heat conduction odels; (a) at 0.31 ps, (b) at 2.18 ps. 23

24 6 Suary and Conclusions We began by deriving the DPL equation by siple Taylor expansion applied to a uch ore (atheatically-) coplicated delay PDE. We then developed a strongly stable iplicit finite-difference schee of Crank Nicolson type for solving the one-diensional DPL equation, without appealing to the standard decoposition typically eployed, and analyzed its accuracy and stability via standard Taylor expansion and von Neuann stability analysis, respectively. The ethod was shown to be only first-order accurate in tie (despite its trapezoidal integration origins) but second order in space; it appears (based on nuerical experients) to be unconditionally stable, but as we earlier noted this cannot be guaranteed by only a von Neuann analysis. The forulation was extended to three-diensional geoetry where the discretized 3-D DPL equation was solved using a δ-for Douglas Gunn tiesplitting ethod. This approach was shown to outperfor both explicit tie stepping and a conjugate gradient iterative procedure in ters of coputation tie required to coplete a siulation. We then deonstrated application of Richardson extrapolation to attain foral second-order accuracy in both space and tie. Finally, we presented coputed results for realistic physical probles to highlight DPL equation capabilities. The treatent we have eployed to arrive at the DPL equation is quite siple and ay have applicability ore generally for other delay PDEs; but as we noted earlier, use of Taylor expansions iposes rather severe regularity requireents and at the sae tie restricts this approach to relatively short delays. The un-decoposed solution foralis eployed here has the advantage of direct application to variable-coefficient PDEs. Its only disadvantage is the need to nuerically treat a third-order ixed (space and tie) derivative and a second-order tie derivative. The teporal part of the first of these is only first order, so it can be directly integrated as we have done. But this introduces a proble in treating the second-order tie derivative ter, resulting in a three-level ethod that is only first order in tie. We showed, however, that this is easily iproved to foral second order via Richardson extrapolation. Use of Douglas Gunn tie splitting in the solution of the ulti-diensional DPL equation appears to be nearly optial, especially as proble size (nuber of spatial grid points and/or required nuber of tie steps) becoes large. This, of course, is not a new result; but it sees to have been widely ignored in recent years, probably in part due to the extree difficulty in ipleenting tie-splitting procedures in the context of unstructured eshes, and possibly also because of concerns associated with splitting or factorization errors. 24

25 Finally, we should ephasize that in the context of applications of the sort considered here (laser heating of aterials), the DPL equation is probably not the best choice of physical odel. In particular, the good agreeent with experiental results displayed in Fig. 3 is widely recognized to require choices of paraeter values that are theselves not physically realistic. Thus, we prefer to view the DPL equation as a siple exaple fro a class of ore general odels of siilar atheatical structure that ight also be treated via the techniques presented herein. Application of the algorith studied here to siilar equations representing ore realistic physical situations would be a fruitful direction for future efforts, and treatent of variable-coefficient fors of Eq. (3) would also be interesting and useful. References [1] V. A. Cielli, Boundary conditions for diffusive hyperbolic systes in non equilibriu therodynaics, Technische Mechanik, Band 22, Heft 3, 181 (2002). [2] A. V. Luikov, Application of irreversible therodynaics ethods to investigation of heat and ass transfer, International Journal of Heat and Mass Transfer 9, 139 (1966). [3] K. J. Baueister and T. D. Hail, Hyperbolic heat conduction equation: a solution for the sei-infinite body proble, ASME Journal of Heat Transfer 91, 543 (1969). [4] Y. Taitel, On the parabolic, hyperbolic and discrete forulation of the heat conduction equation, International Journal of Heat and Mass Transfer 15, 369 (1972). [5] T. Q. Qiu and C. L. Tien, Short-pulse laser heating of etals, International Journal of Heat and Mass Transfer 35, 719 (1992). [6] T. Q. Qiu and C. L. Tien, Heat transfer echaniss during short-pulse laser heating of etals, ASME Journal of Heat Transfer 115, 835 (1993). [7] T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron and C. L. Tien, Fetosecond laser heating of ulti-layered etals II. Experients, International Journal of Heat and Mass Transfer 37, 2799 (1994). [8] M. I. Kagnov, I. M. Lifshitz and M. V. Tanatarov, Relaxation between electrons and crystalline lattices, Soviet Physics JETP 4, 173 (1957). [9] S. I. Anisiov, B. L. Kapeliovich and T. L. Perel an, Electron eission fro etal surfaces exposed to ultrashort laser pulses, Soviet Physics JETP 39, 375 (1974). [10] D. Y. Tzou, Macro-to-Microscale Heat Transfer: The Lagging Behavior, Taylor and Francis, Washington, DC (1996). 25