Anti-diffusive Methods for Hyperbolic Heat Transfer

Transcription

1 Anti-diffusive Methods for Hperbolic Heat Transfer Wensheng Shen and Leigh Little Department of Computational Science, SUNY Brockport Brockport, NY 44, USA Liangjian Hu Department of Applied Mathematics, Donghua Universit Shanghai 6, P.R. China December 5, 9 Abstract The hperbolic heat transfer equation is a model used to replace the Fourier heat conduction for heat transfer of etremel short time duration or at ver low temperature. Unlike the Fourier heat conduction, in which heat energ is transfered b diffusion, thermal energ is transfered as wave propagation at a finite speed in the hperbolic heat transfer model. Therefore methods accurate for Fourier heat conduction ma not be suitable for hperbolic heat transfer. In this paper, we present two anti-diffusive methods, a second-order TVD-based scheme and a fifth-order WENO-based scheme, to solve the hperbolic heat transfer equation and etend them to two dimension, including a nonlinear application caused b temperaturedependent thermal conductivit. Several numerical eamples are applied to validate the methods. The current solution is compared in one-dimension with the analtical one as well as the one obtained from a high-resolution TVD scheme. Numerical results indicate that the fifthorder anti-diffusive method is more accurate than the high-resolution TVD scheme and the second-order anti-diffusive method in solving the hperbolic heat transfer equation. Ke words: Anti-diffusive method, Hperbolic conservation laws, WENO scheme, Thermal wave, H- Corresponding author. wensheng@csr.uk.edu. URL: shen. llittle@brockport.edu. ljhu@dhu.edu.cn.

2 perbolic heat transfer, Temperature-dependent thermal conductivit Introduction With the development of modern science and technolog, the phenomena of non-fourier heat transfer has been observed in man industrial applications [], such as laser-heating, crogenic engineering and nano-technolog. The Fourier law, which states that heat is transferred b diffusion process alone, cannot account for the transient heat transfer in situations such as etremel short time duration, ver low temperature, and etremel large heat flu. Hence, there has been a growing interest in discovering new heat transfer models [7] as well as appling eisting non- Fourier models to phsical problems [, 3, 4]. Most recent eamples include but not limited to the application of non-fourier heat transfer to stud laser-induced thermal damage in biological tissues with nonhomogeneous inner structures [4], to investigate the critical energ characteristics of cooled composite superconductor [], and to eamine the phsical anomalies during the transient heat transfer process under the dual-phase-lag model [3]. More evidence of non-fourier heat conduction will emerge as scientific discover and manufacturing move to microscopic time and length scales. The hperbolic heat transfer equation is one of the models used to replace the Fourier heat conduction for heat transfer in etreme situations. Under Fourier conduction, the resulting energ equation is a Laplacian equation, which can be solved b standard finite difference, finite volume, or finite element methods. Solutions of higher order accurac in time and space ma also be obtained b higher order temporal and spatial discretization. Unlike Fourier heat conduction, in which heat energ is transfered b diffusion, the hperbolic heat transfer model uses wave propagation at a finite speed to transfer thermal energ. As a result, methods accurate for Fourier heat conduction ma not be suitable for hperbolic heat transfer. Much effort has been taken to find stable and accurate solutions to the hperbolic heat transfer equation. A surve of literature indicates that earl practices [5, 8, ] often encounter fictitious numerical oscillations, consequentl, sharp propagation fronts and reflective boundaries cannot be represented accuratel. Various numerical methods have been proposed aimed at removing artificial oscillations near sharp discontinuities. Chen and Lin [6] developed a hbrid technique based on the Laplace transform and control volume method, which successfull suppressed the oscillations b removing the time-dependent terms using Laplace transform. Tamma and Railkar [9] used a speciall tailored transfinite element formulations for hperbolic heat conduction. Instead of regular polnomials, the applied the general solution of the transformed form of the hperbolic heat transfer equation as the shape function of finite elements, and successfull captured the discontinuit. Their method, however, lacks generalit, since the general solution to other specific problems ma not be guaranteed. The stud of hperbolic conservation laws applied to gas dnamics has shed light on the solution of hperbolic heat conduction. In the past decades, a number of effective schemes have

3 been developed to solve all kinds of problems governed b hperbolic conservation laws, including the wave equation, Burger s equation, Euler s equation, and resonant pipe flow [3]. These advanced schemes are called high resolution schemes and share some common features, i.e. using characteristics and appling special limiter functions. The have been proved to be ver successful b providing solutions that are both high-order in accurac and oscillation free in one-dimensional applications. Tpical of those are the Total Variation Diminishing (TVD) scheme contributed b Roe [, ] and Sweb [8]. Their scheme is first order accurate in time, and second order accurate in space in smooth regions. The scheme is widel celebrated for its abilit to capture sharp discontinuities and its simplicit in terms of implementation. An accurate, robust, and oscillation free numerical solution to the hperbolic heat conduction equation was obtained b Yang [3], who took the characteristic approach, emploed the second order TVD scheme of Roe [, ] and Sweb [8], and obtained ver nice results. Being motivated b Yang s success, Shen and Han [4, 5, 6] continued and etended Yang s solution to hperbolic heat conduction involving irregular geometries, temperature dependent material properties, and composite materials. However, the accurac of the method decreases when it is etended to multi-dimension. Other finite element and finite volume methods have been applied to solve hperbolic heat transfer equation as well. Ai and Li [] solved hperbolic thermal wave problems using discontinuous finite element method. Miller and Haber [] has recentl applied a spacetime discontinuous Galerkin method to the hperbolic heat equation and successfull resolved continuous and discontinuous thermal waves in conducting medium. The main purpose of this paper is to search numerical methods which are both mathematicall accurate and computationall efficient for hperbolic heat transfer. Under Yang s representation [3], the hperbolic heat conduction equation is essentiall a sstem of coupled linear transport equations. Thus all newl developed numerical methods for linear wave equations can be applied to solve it. Recentl Bokanowski and Zidani [4] proposed a TVD-based anti-diffusive scheme for advection and Hamilton-Jacobi-Bellman equations, and Xu and Shu proposed a WENO-based anti-diffusive scheme []. In this paper, we would like to present anti-diffusive solutions to the hperbolic heat transfer equation using Bokanowski and Zidani s second-order method [4] and Xu and Shu s fifth-order method []. In one-dimension, the anti-diffusive solutions are compared with the analtical one as well as the one obtained from Yang s method [3]. We further etend the anti-diffusive methods to two-dimension, and eplore the abilit of the fifth-order scheme to solve nonlinear hperbolic heat transfer equation caused b temperature-dependent thermal conductivit. Hperbolic Heat Transfer Model As an eample, one-dimensional hperbolic heat transfer in a finite slab with constant thermal properties is considered. equations [3], The corresponding mathematical model consists of the following two ρc p T t + q = g () 3

4 τ q t + k T = q () where T is the temperature, ρ the densit, c p the specific heat, q the heat flu, and g the heat source per unit volume, τ the relaation parameter (τ = α c ), α the thermal diffusivit (α = k the thermal conductivit, c the propagation velocit of a thermal wave. Equations. () and () can be written in non-dimensional form as k ρc p ), φ t + F = S (3) where φ, F, and S are the unknown vector, flu vector and source vector respectivel. The ma be eplicitl written as φ = T q F = q T g S = q (4) 4α g c k(t w T ), where the nondimensional variables are defined as T = T T T w T, q αq = ck(t, w T ) g = = c α, t = c α t, where T w is the reference temperature (i.e. the temperature of the boundar surface at = ), T the initial temperature. The reason to choose T w T as the denominator of dimensionless temperature is to limit it in the range of [,], if T = T w at = and T = T at =. Note, for convenience, the nondimensional variables are written without s in the rest of the paper. It should be noticed that Eq. (3) consists of two coupled linear equations. We decouple the linear sstem equation b calculating the Jacobian matri A = F φ and reducing matri A into diagonal form A = RΛR. Thus, we rewrite Eq. (3) in the form φ t + φ RΛR = S (5) where R is the matri of eigenvectors and Λ is the matri of eigenvalues. The matrices A, R, Λ, and R are as follows: A = R = Λ = λ λ R = (6) where the eigenvalues are λ = and λ =. Multipling Eq. (5) b R, we obtain the following equation of characteristic variables, W t + Λ W = H (7) 4

5 where the characteristic based variables can be calculated as W = R φ = (T + q) (T q) H = R S = g 4 q g 4 + q (8) In implementing the model, either Eq. (5) with variables T and q, or Eq. (7) with characteristic variables, can be used. If one choose to use the characteristic variables, heat flu q in the right hand side of Eq. (7) ma be epressed as a function of the characteristic variables, i.e. q = W W, where W = (T + q) and W = (T q). It can be seen without much difficult that Eq. (7) is a linear wave equation with a non-trivial diffusive source term. We would like to introduce two anti-diffusive numerical schemes, which will be presented in Section 3, to solve the equation with desirable accurac. 3 Anti-diffusive Numerical Scheme 3. A Second-order Anti-diffusive Scheme The hperbolic conservation law with a non-zero source term can be written as u t + f(u) = s u(, ) = u () (9) Man numerical methods have been proposed to solve Eq. (9), of which the most effective ones include the high resolution TVD schemes [, 8], and high order WENO schemes [7]. In this paper, we eamine the capabilities of two classes of anti-diffusive schemes, the second-order TVD-based schemes and the fifth-order WENO-based schemes, in solving hperbolic heat transfer equation. The anti-diffusive N-Bee scheme, suggested b Bokanowski and Zidani [4], is a miture of Roe s Super Bee and Ultra-Bee schemes [], and quite simple to implement. The Ultra-Bee scheme is onl of first order of accurac, but it has an interesting propert of eact transport of a large set of piecewise constant functions in the case of linear advection [4, 3]. The N-Bee scheme is second order accurate at smooth regions for linear advection equations, as being numericall checked b Bokanowski and Zidani [4], and has similar properties as the Ultra-Bee scheme at discontinuities. We will appl the N-Bee scheme to hperbolic heat transfer equation, and the construction of the scheme will be reiterated here. Note, since there is a diffusive source term, the propert of eact transport doesn t hold for the Ultra-Bee scheme when it is applied to solve hperbolic heat transfer equation. Let be the spatial increment and t be the time step, and assume t n = n t and i = i. 5

6 Equation (9) can be approimated in the following form, u n+ i = u n i t ( f n,l i+ ) f n,r + s () i where f n,l and f n,r are numerical flues at i+ i i+ and i respectivel. For linear advection, we ma write f(u) as f(u) = au, where a is the velocit of traveling wave, and the local CFL number is defined as Assuming that the CFL number satisfies the condition ν i = a i t. () ν i, () the numerical flues of the anti-diffusive scheme are constructed as [4] f n,r i+ f n,l i+ = f i + ψ(r i, ν i ) ( ν i )(f i+ f i ), if a i > (3) = f i+ + ψ(r i+, ν i+ ) ( ν i+ )(f i f i+ ), if a i+ < (4) f n,l i+ = f n,l i+ = (f i + f i+ ), if a i and a i+ (5) f n,l i+ where ψ(r, ν) is the nonlinear limiter function determined b = f n,r, if a i+ i a i+ > (6) ( ( ) ( )) r ψ(r, ν) = ma, min ν,, min r,, (7) ν and r is defined b if a i >, or r i = u i u i u i+ u i (8) r i+ = u i+ u i+ u i+ u i (9) if a i+ <. 3. A Fifth-order Anti-diffusive Scheme A high order anti-diffusive finite difference scheme has recentl been proposed b Xu and Shu [] to improve the original fifth-order weighted essentiall non-oscillator (WENO) scheme of Jiang and Shu [9] in the resolution of contact discontinuities. Third-order temporal accurac is achieved 6

7 b discretizing Eq. (9) using multi-step Runge-Kutta method [], u () = u n + tl(u n ), u () = u n + 4 tl() (u n ) + 4 tl(u() ), u n+ = u n + 6 tl() (u n ) + 6 tl(u() ) + 3 tl(u() ) (a) (b) (c) where the operator L(u n ) is written as and the anti-diffusive flu is defined b L(u n ) = ( ) f i+ f i + s, () ( u f i+ = f n + φ i+ i minmod i u n ) i + f f, f + f, () ν i i i+ i+ i+ where the minmod function is minmod(, ) =, >,, >, >, (3) and f and f + are the left-biased and right-biased upwind flues based on stencils with one i+ i more point to the left and to the right, respectivel [9, ]. The discontinuit indicator in Eq. () is defined b [], φ i = η i η i + ξ i, (4) and η i = ( αi + α ) i+, ξ i = u ma u min, α i = u n i α i α i+ α un i + ɛ, (5) i where u ma and u min are the maimum and minimum values of u i for all grid points, and ɛ is a small positive number. The operators L () (u n ) and L () (u n ) in Eq. () are defined b L () (u n ) = ( L () (u n ) = f () i+ ) f () i ( f () f () i+ i + s, (6a) ) + s, (6b) 7

8 and the numerical flues f () i+ f () i+ = f i+ + φ i minmod and f () i+ are written as ( u n ) i un i 4ν i + f f, f + f, > and < i i+ i+ i+ f i+, otherwise (7) and f () i+ = f i+ ( u n ) + φ i minmod i u n i 6ν i + f f, f + f, > and < i i+ i+ i+ f i+, otherwise (8) where and are defined b = un i un i + f f ν i i i+ and = f + i+ f. (9) i+ We remark that the fifth-order anti-diffusive scheme presented in this paper is slightl different t from that of Xu and Shu []. Specificall, we use ν i = a i while ν i = t was used in []. in Eqs. (), (7), (8), and (9), 4 Etension to Two-dimension It is assumed that heat conduction is isotropic. Similar to Section, b introducing non-dimensional T T αq variables, T = T w T, q = ck(t, w T ) q = ck(t, w t ) g = c k(t, w T ) = α, = α, t = c αt, the hperbolic heat conduction equation in two-dimension can be written as (without superscript ) φ t + F + G = S (3) where φ, F, G, and S ma be eplicitl written as αq 4α g c c φ = T q F = q T G = q S = g q. (3) q T q We can further write Eq. (3) in the form of matri-vector product, φ t + A φ + B φ = S (3) 8

9 where matrices A and B are analticall calculated as A = B =. (33) Similar to handling one-dimensional problem, we can decouple Eq. (3) into three independent equations b decomposing matrices A and B into diagonal form A = R A Λ A R A and B = R B Λ B R B. We onl list the decomposed results of matri A, R A = Λ A = R A =. (34) Readers can obtain the corresponding results of matri B with a straight forward calculation. We etend the anti-diffusive schemes presented in Section 3 to two-dimension using the dimension-b-dimension approach, which will be presented below. Let φ indicate an component in vector φ, the value of φ at time level n + can be represented as, φ n+ i,j = φ n i,j t ) (f ni+,j f ni,j t ( ) h n i,j+ h n i,j + ts i,j, (35) where f n i+,j, f n i,j hn, and h n i,j+ i,j are numerical flues at i+, i, j+, and j respectivel. The flu f n is constructed as i+,j f n i+,j = 3 k= R k A,i+,jλk A,i+,jW k A,i+,j, (36) where Similarl, flu h n i,j+ can be constructed as W A,i+,j = R A,i+,j φ n i,j. (37) h n i,j+ = 3 k= R k B,i,j+ λ k B,i,j+ W k B,i,j+, (38) where W B,i,j+ = R φ n B,i,j+ i,j. (39) We appl the anti-diffusive schemes to each product of λ A,i+,jW A,i+,j or λ B,i,j+ W B,i,j+. 9

10 5 Results and Discussion Eample : constant temperature or heat flu boundar condition In the first eample, we test thermal wave propagation and reflection in a one-dimensional slab with dimensionless length of., and compare our numerical results with the analtical solutions obtained b Care and Tsai [5]. The dimensionless temperature of the slab is kept at T =. initiall, and the temperature at the left boundar is increased to T =. at time t >. A thermal wave with discontinuous front will propagate from left to right with a constant dimensionless speed of λ =.. Two tpes boundar conditions are considered at the right end of the slab, given temperature (T=) and zero heat flu (q=). The boundar conditions are implemented b characteristics. As can be seen from Eq. (7), for one-dimensional hperbolic heat transfer, there are two characteristics, W and W, which move with the speed of λ = and λ = respectivel, i.e., W travels from left to right (right-traveling), while W travels from right to left (left-traveling). For right-traveling W, we need to provide boundar condition at =. We should bear in mind that we can onl specif either temperature T or heat flu q but not both at boundaries. The boundar condition for W at = can be calculated as W () = T () W () (4) and W () = q () + W (), (4) where Eqs. (4) and (4) appl to given temperature or given heat flu conditions respectivel. To obtain W in Eqs. (4) and (4), we need the boundar information of W, which is obtained b simple etrapolation b letting W () = W (), where () indicates the boundar node at the left end, and () indicates the node net to the boundar node. This could also be approimated b upwind differencing as implemented b Yang [3]. Similarl, the boundar condition at the right end ( =.) ma be epressed as W (n) = T (n) W (n) (4) and W (n) = W (n) q (n), (43) where Eqs. (4) and (43) should be used for given temperature and given heat flu boundar conditions respectivel. Again, the value of W can be obtained b etrapolation, W (n) = W (n ), where (n) and (n ) denotes the boundar note and its left neighbor respectivel. The results of the first test are shown in Fig., where solid lines are the analtical solution, and square, triangle, and circular smbols represent the results of fifth-order WENO, N-Bee and Yang s methods respectivel. The solutions corresponding to given temperature boundar condition at the right end ( =.) at dimensionless time of t =.5 is shown in Fig. (a), where the

11 dimensionless temperature is kept at a constant of T =. at =.. As epected, the thermal wave with a sharp front propagates to the location of =.5 at time t =.5. The wave front is captured with onl two grid points b the anti-diffusive WENO and N-Bee schemes, while it is captured with four grid points b Yang s method. We remark that both the current and Yang s methods have similar results in smooth regions for this particular application. Figs. (b) and (c) show results at t =. and t =.5 with the right boundar condition of T =. respectivel. The fifth-order anti-diffusive scheme is superior in these two cases. It captures the sharp discontinuit with three point at t =., while the second-order N-bee scheme and Yang s method give similar results, representing the discontinuit b four points. The thermal wave is reflected from the right end and its front propagates to =.5 at t =.5. The wave front is represented b two points using the fifth-order method, four points using the N-bee method, and five points using Yang s. Fig. (d) shows the results with an insulated boundar condition (zero heat flu) at the right end at t =.5. Again, the thermal wave is reflected after reaching the right end, and propagates to =.5. The wave front is represent b two points in the fifth-order method, three points in the N-bee method, and four points in Yang s. The insulation boundar condition at the right end can be implemented b letting q =, etrapolating right-traveling characteristics, and calculating the left-traveling characteristics according to Eq. (43). We remark that the difference between the second-order N-bee scheme and Yang s method becomes less significant when simulation time is longer, and more points are needed to represent discontinuities for both two methods. The fifthorder anti-diffusive scheme, on the contrar, can consistentl capture discontinuities with onl two points. Eample : on-off heat flu boundar condition This eample was taken from reference [3]. In Eample, the left boundar is kept at a constant temperature, and the right boundar is subjected to a fied temperature or heat flu condition. Here, we appl periodic on-off heat flu to the left boundar and constant temperature of T =. to the right boundar, and test the effectiveness of numerical schemes. The periodic on-off heat flu is given b [8, 3] f(t) = λ (i )P < t < [(i ) + λ)]p [(i ) + λ]p < t < ip (44) where i is the number of period and P is the period. For comparison purposes, the values of P and λ are taken as P =. and λ =.5. It can be verified that the total energ suppl is the same and equals to one at each period. The periodic on-off heat flu is implemented as given heat flu boundar condition, i.e. q = f(t). The left traveling characteristics is calculated as W () = W () + q (45)

12 The eact solution of the problem was obtained b Glass et al. [8], and the numerical one b Yang [3]. We compare the results represented using circles and triangles from different numerical schemes and plot them in Fig. at dimensionless time of t =., where Fig. (a) shows the solutions from the the fifth-order anti-diffusive scheme (represented b circles) and N-Bee scheme (represented b triangles), and Fig. (b) shows that b Yang s method [3]. The solid line in the figure indicates the eact solution. As can be seen from Fig., there are several sharp discontinuities due to the five periodic flues. Both the anti-diffusive schemes and Yang s method produce ver good results, and anti-diffusive methods are more accurate b representing discontinuities with less number of grid points. We use a mesh size of intervals for all numerical methods. Eample 3: radiative boundar condition This eample tests the accurac of the method in the case of radiation boundar condition on the left. The eistence of radiation boundar condition introduces nonlinear propert to the eample. Numerical solution of this case has been studied b Glass et al. [8] and Yang [3]. In the case of radiation, the heat flu on the boundar ma be written as q = f(t) T 4 (), (46) where f(t) is defined b Eq. (44). As mentioned before, the left traveling characteristics W on the left boundar can be obtained b etrapolation, and it is related to the primitive variables b the epression W () = (T () q). (47) A new equation about T () is obtained b combining Eqs. (46) and (47) T 4 () + T () f(t) W () =. (48) Eq. (48) is nonlinear, which can be solved b Newton s iteration. We use the magnitude of relative error as the stopping criteria for the Newton s iteration. The iteration is regarded as converged if the relative error of T () is less than 6, i.e. T n+ () T () T n () 6. We believe that 6 is small enough to claim convergence. To verif our belief, we tried to change the stopping criteria from 6 to 8. For both stopping criteria, it takes about eight iterations to converge in the case of on-flu, and takes about four iterations in the case of off-flu. Once the temperature on the left boundar is known, we can calculate the corresponding right traveling characteristics as W () = (T () + q) = (T () + f(t) T 4 () ) (49) Numerical results are presented in Fig. 3 at dimensionless time of t =., where solid line is the solution obtained using etremel fine mesh, i.e. intervals. Circular and triangular smbols in Fig. 3(a) represent the results computed b the fifth-order anti-diffusive and the N-Bee schemes, respectivel, while circular smbols in Fig. 3(b) represent the results computed b Yang s method

13 [3], with the mesh size of intervals. It is not difficult to find that once again, anti-diffusive schemes out-perform Yang s method in terms of capturing discontinuities. Eample 4: two-dimension with Gaussian laser-pulse heating In this eample, we discuss numerical results of hperbolic heat transfer in two-dimension, and investigate thermal wave propagation in a rectangular geometr of.. and.5.5. Similar to [, ], a laser pulse incident is applied on the left side of the domain and represented b a dimensionless heat source with Gaussian-tpe profile, g(,, t) = [ ep + ( )] Dt p D t W, (5) t c where D and W are the penetration depth and width of the laser beam respectivel, and t c is the characteristic duration of the laser pulse. We use the same parameters as in [], i.e. D =.5, W =., and t c =.. We implemented both the two anti-diffusive methods, introduced in Section 3, in twodimension. However, the N-bee scheme produces unsatisfactor results for the propagation of the Gaussian pulse, which has a smooth gradient over or, and numerical results deteriorate for long time simulations. This is a tpical shortcoming for low-order TVD schemes to solve hperbolic conservation laws. Therefore, in this paper, we present onl solutions obtained b the fifth-order anti-diffusive WENO-based scheme. We calculated the temperature distribution at t =., t =.5, t =., and t =., and showed the results in Fig. 4, where temperature is visualized b both color mapping and contouring. Red color indicates high temperature while blue color indicates low temperature. As epected, thermal waves are reflected when Gaussian pulse reaches the bottom, top, and right walls. Since our code of the fifth-order anti-diffusive scheme is written onl on regular geometr, we cannot directl compare our results with those of [], in which a converging-diverging channel was used. In the near future, we would like to convert the code from the standard Cartesian coordinates to curvilinear coordinates to handle non-rectangular geometr. Eample 5: two-dimension with temperature dependent thermal conductivit To test the validit of the numerical method to nonlinear cases, we further eplore hperbolic heat conduction with temperature dependent thermal conductivit in two-dimension. For simplicit, it is assumed that the thermal conductivit changes linearl with temperature, i.e. k = k ( + βt ), where β is a constant. Substituting k into Eq. (3) and performing dimensionless analsis, we obtain the following new equations for T, q and q, q t q t T t + q + q = g, + ( + βt ) T = q, + ( + βt ) T = q. (5a) (5b) (5c) 3

14 Accordingl, matrices A and B and their diagonal decompositions should be taken new forms as well. For the convenience of readers, we list corresponding matrices below, A = + βt B = + βt (5) R A = λ λ 3 Λ A = λ λ λ 3 R A = +βt +βt (53) where λ =, λ = + βt, and λ 3 = + βt. Nonlinearit is handled b the above matri decomposition. Nonlinear wave propagation can then be represented b characteristics and its traveling speed. We use the same rectangular geometr as in Eample 4 as the computational domain. Due to temperature dependent thermal conductivit, thermal wave propagates at a varing dimensionless speed of + βt, while it propagates at a constant dimensionless speed of in Eample 4. Numerical results corresponding to β =.5 are presented in Fig. 5 for t =., t =.5, t =., and t =.. A comparison between Fig. 4 and Fig. 5 finds that as epected, thermal wave in Eample 5 propagates at a faster speed than that in Eample 4. Similar to Eample 4, thermal wave is reflected when the Gaussian pulse reaches boundar walls of the geometr, but the wave pattern changes due to nonlinearit. It is not difficult to conclude that the fifth-order scheme can be used to solve nonlinear hperbolic heat transfer equation effectivel. 6 Concluding Remarks Two tpes of anti-diffusive schemes, a second-order TVD-based and a fifth-order WENO-based, are used to find the numerical solution of hperbolic heat transfer equation in one and two-dimension based on characteristics. Several numerical tests are used to demonstrate the effectiveness of the anti-diffusive methods. In one-dimensional applications, three different boundar conditions are considered, given temperature, given heat flu, and radiation. In the case of radiation boundar condition, the nonlinearit is handled b the Newton s method. The anti-diffusive schemes are further etended to two-dimension using the dimension-b-dimension approach. Numerical results indicate that both anti-diffusive schemes provide improved alternatives over Yang s method in terms of accurac, and the sharp discontinuities are represented b less number of grid points. The difference between the second-order N-bee scheme and Yang s method, however, is less significant for long time simulations. Both Yang s method and the N-bee scheme have two major drawbacks. 4

15 Their order of accurac decreases when the are etended to two-dimension. The tend to over anti-diffusive when the are applied to solve problems involving rich smooth structures, such as sine waves and Gaussian pulses. Therefore, the are not recommended for multi-dimensional problems with rich smooth structures. Overall, the fifth-order anti-diffusive scheme performs better than the second-order ones in solving hperbolic heat transfer equation. A more comple model of temperature dependent thermal conductivit is introduced to test the capabilit of the proposed numerical method. Our numerical solution indicates that the fifth-order scheme can solve nonlinear hperbolic heat conduction equation fairl well. Similar to two-dimensional etension, the method can be convenientl etended to three-dimension b adding corresponding temperature and flu information in the z-coordinate, decomposing the relevant matrices, and appling the dimensionb-dimension approach. References [] X. Ai and B.Q. Li, A discontinuous finite element method for hperbolic thermal wave problems, J. Engrg. Comput., (5)(4) [] M.Q. Al-Odat and F.M. Al-Hussien, Analtical prediction of quenc energies of cooled superconductors based on the hperbolic heat conduction, International Journal of Thermophsics, 9, , 8. [3] B. Després and F. Lagoutière, Contact discontinuit capturing schemes for linear advection and compressible gas dnamics, J. Sci. Comput., 6, ,. [4] O. Bokanowski and H. Zidani, Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellman equations, J. of Sci. Comput., 3, -33, 7. [5] G.F. Care and M. Ozisik, Hperbolic heat transfer with reflection, Numerical Heat Transfer, 5, 39-37, 98. [6] Han-Taw Chen and Yae-Yuh Lin, Numerical analsis for hperbolic heat conduction, International Journal of Heat and Mass Transfer, 36, , 993. [7] R. Čiegis, A. Dementèv, and G. Jankevičiūtė, Numerical analsis of the hperbolic twotemperature model, Lithuanian Mathematical Journal, 48, 46-6, 8. [8] D.E. Glass, M.N. Ozisik, and B. Vick, Non-Fourier effect on transient temperature resulting from periodic on-off heat flu, International Journal of Heat and Mass Transfer, 3, 63-63, 987. [9] G. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Phsics, 6, -8,

16 [] S.T. Miller and R.B. Haber, A spacetime discontinuous Galerkin method for hperbolic heat conduction, Computer Methods in Applied Mechanics and Engineering, 98, 94-9, 8. [] P.L. Roe, Generalized formulation of TVD La-Wendroff schemes, ICASE Report, 984. [] P.L. Roe, Some contributions to the modelling of discontinuous flows, Lectures in Applied Mathematics,, 63-93, 985. [3] B. Shen and P. Zhang, Notable phsical anomalies manifested in non-fourier heat conduction under the dual-phase-lag model, International Journal of Heat and Mass Transfer, 5, 73-77, 8. [4] W. Shen and S. Han, An eplicit TVD scheme for hperbolic heat conduction in comple geometr, Numerical Heat Transfer, Part B, 4, ,. [5] W. Shen and S. Han, Hperbolic heat conduction in composite materials, 8th AIAA/ASME Joint Thermophsics and Heat Transfer Conference, 4-6 June, St. Louis, Missouri. [6] W. Shen and S. Han, Two-dimensional hperbolic heat conduction with temperature dependent properties, Journal of Thermophsics and Heat Transfer, 8, 85-87, 4. [7] C.W. Shu, Essentiall non-oscillator and weighted essentiall non-oscillator schemes for hperbolic conservation laws, in Advanced Numerical Approimation of Nonlinear Hperbolic Equations, edited b B. Cockburn, C. Johnson, C. W. Shu, and E. Tadmor, Lect. Notes in Math. (Springer-Verlag, Berlin/New York, 998), 697, 35. [8] P.K. Sweb, High resolution schemes using flu limiter for hperbolic conservation laws, SIAM Journal on Numerical Analsis,, 995-, 984. [9] K.K. Tamma and S.B. Railkar, Speciall tailored transfinite-element formulations for hperbolic heat conduction involving non-fourier effects, Numerical Heat Transfer, Part B, 5, -6, 989. [] J.R. Torcznski, D. Gerthsen, and T. Roesgen, Schlieren photograph of second-sound shock waves in superfluid helium, Phs. Fluids, 7, 48-43, 984. [] D.C. Wiggert, Analsis of earl-time transient heat conduction b method of characteristics, ASME Journal of Heat Transfer, 99, 9-97, 983. [] Z. Xu and C.-W. Shu, Anti-diffusive flu corrections for high order finite difference WENO schemes, Journal of Computational Phsics, 5, , 5. [3] H.Q. Yang, Characteristics-based, high-order accurate and nonoscillator numerical method for hperbolic heat conduction, Numerical Heat Transfer, Part B, 8, -4, 99. [4] J. Zhou and Y. Zhang, Non-Fourier heat conduction effect on laser-induced thermal damage in biological tissues, Numerical Heat Transfer, Part A, 54, -9, 8. 6

17 List of Figures Solution to eample (Squares are results of the fifth-order anti-diffusive method, triangles are results of the second-order anti-diffusive method, circles are results b Yang s method [3], and solid line represents analtical solution). (a) Dimensionless temperature distribution at t =.5 with T =. at the right boundar. (b) Dimensionless temperature distribution at t =. with T =. at the right boundar. (c) Dimensionless temperature distribution at t =.5 with T =. at the right boundar. (d) Dimensionless temperature distribution at t =.5 with q =. at the right boundar Solution to eample. (a) Dimensionless temperature distribution at t =. with periodic on-off heat flu at the left boundar and T =. at the right boundar, circles for the fifth-order anti-diffusive scheme and triangles for the N-Bee scheme. (b) Dimensionless temperature distribution at t =. with periodic on-off heat flu at the left boundar and T =. at the right boundar, circles for Yang s solution [3] Solution to eample 3. (a) Dimensionless temperature distribution at t =. with surface radiation at the left boundar and T =. at the right boundar, circular smbols for the fifth-order scheme and triangular smbols for the N-Bee scheme. (b) Dimensionless temperature distribution at t =. with surface radiation at the left boundar and T =. at the right boundar, circular smbols for Yang s solution [3] Solution to eample 4 at various time (A laser pulse incident of Gaussian profile described b Eq. (??) is supplied near the left boundar). (a) Dimensionless temperature distribution at t =.. (b) Dimensionless temperature distribution at t =.5. (c) Dimensionless temperature distribution at t =.. (d) Dimensionless temperature distribution at t = Solution to eample 5 with temperature-dependent thermal conductivit at various time (A laser pulse incident of Gaussian profile described b Eq. (??) is supplied near the left boundar). (a) Dimensionless temperature distribution at t =.. (b) Dimensionless temperature distribution at t =.5. (c) Dimensionless temperature distribution at t =.. (d) Dimensionless temperature distribution at t =.... 7

18 T (a) (b) (c) (d) Figure : Solution to eample (Squares are results of the fifth-order anti-diffusive method, triangles are results of the second-order anti-diffusive method, circles are results b Yang s method [3], and solid line represents analtical solution). (a) Dimensionless temperature distribution at t =.5 with T =. at the right boundar. (b) Dimensionless temperature distribution at t =. with T =. at the right boundar. (c) Dimensionless temperature distribution at t =.5 with T =. at the right boundar. (d) Dimensionless temperature distribution at t =.5 with q =. at the right boundar. 8

19 (a) (b) Figure : Solution to eample. (a) Dimensionless temperature distribution at t =. with periodic on-off heat flu at the left boundar and T =. at the right boundar, circles for the fifth-order anti-diffusive scheme and triangles for the N-Bee scheme. (b) Dimensionless temperature distribution at t =. with periodic on-off heat flu at the left boundar and T =. at the right boundar, circles for Yang s solution [3] (a) (b) Figure 3: Solution to eample 3. (a) Dimensionless temperature distribution at t =. with surface radiation at the left boundar and T =. at the right boundar, circular smbols for the fifth-order scheme and triangular smbols for the N-Bee scheme. (b) Dimensionless temperature distribution at t =. with surface radiation at the left boundar and T =. at the right boundar, circular smbols for Yang s solution [3]. 9

20 (a) (b) (c) (d) Figure 4: Solution to eample 4 at various time (A laser pulse incident of Gaussian profile described b Eq. (5) is supplied near the left boundar). (a) Dimensionless temperature distribution at t =.. (b) Dimensionless temperature distribution at t =.5. (c) Dimensionless temperature distribution at t =.. (d) Dimensionless temperature distribution at t =..

21 (a) (b) (c) (d) Figure 5: Solution to eample 5 with temperature-dependent thermal conductivit at various time (A laser pulse incident of Gaussian profile described b Eq. (5) is supplied near the left boundar). (a) Dimensionless temperature distribution at t =.. (b) Dimensionless temperature distribution at t =.5. (c) Dimensionless temperature distribution at t =.. (d) Dimensionless temperature distribution at t =..