Fourth-Order Accurate Simpliﬁed Coupled Characteristics ......Sep 12, 2009 · (frozen wave...

Applied Mathematical Sciences, Vol. 3, 2009, no. 10, 461 - 490

Fourth-Order Accurate Simplified

Coupled Characteristics-Based

Schemes for the

Hyperbolic Heat Conduction Equations

Brian J. McCartin

Applied Mathematics, Kettering University1700 West University Avenue, Flint, MI 48504-4898, USA

[email protected]

AbstractRoe and Arora [1] derived a fourth-order accurate coupled characteristics-based scheme for the hyperbolic heat conduction equations (mistak-enly concluding that it was only third-order accurate [2]) by discretiz-ing an analytical expression involving the Riemann function (the OPTscheme). Herein, we first derive two alternative fourth-order accurateschemes which do not require knowledge of the Riemann function. Wethen combine these two schemes to produce a third hybrid scheme whichis not only fourth-order accurate but which further possesses principallocal truncation errors identical to those of the OPT scheme. The dis-sipative and dispersive properties of these new schemes are comparedto those of their continuum counterpart as well as to those of the OPTscheme. Both the OPT scheme and the three new schemes are exercisedon a smooth numerical example for which the exact solution is known.

Mathematics Subject Classification: 35L45, 65M06, 65M25

Keywords: hyperbolic heat conduction equations, numerical method ofcharacteristics, dispersion and dissipation analysis, fourth-order accuracy

1 Introduction

In [1], Roe and Arora studied the application of the numerical method ofcharacteristics [3] with unit Courant number to the Cauchy problem for thesource-free hyperbolic heat conduction equations [4]:

∂θ

∂t+∂q

∂x= 0; −∞ < x <∞, t > 0, (1)

462 B. J. McCartin

τ∂q

∂t+ q +

∂θ

∂x= 0; −∞ < x <∞, t > 0, (2)

subject to the initial conditions

θ(x, 0) = φ(x), q(x, 0) = ψ(x); −∞ < x <∞. (3)

In the above, θ(x, t) is the temperature, q(x, t) is the heat flux, and τ isthe relaxation time. Equation (1) expresses the first law of thermodynamics(conservation of thermal energy) while Equation (2) embodies the Maxwell-Cattaneo law of heat conduction (modified Fourier’s law).

Eliminating q from Equations (1) and (2) results in the (hyperbolic) dampedwave equation

1

a2

∂2θ

∂t2+∂θ

∂t=∂2θ

∂x2; −∞ < x <∞, t > 0, (4)

where a = τ−1/2 is the thermal propagation speed. Observe that when τ = 0(Fourier law of heat conduction), the wave speed is infinite and Equation (4)reduces to the (parabolic) diffusion equation.

In some emerging technologies such as fast-pulsed laser heating of materials,transient heat transfer at low cryogenic temperatures, and microwave heatingwith very high frequencies, the wave nature of thermal phenomena must beproperly taken into account. Wiggert [5] first applied the numerical methodof characteristics to the hyperbolic heat conduction equations.

Equations (1-2) are said to be in nonconservative form. Multiplication ofboth equations by the integrating factor et/τ yields the conservative form ofthe hyperbolic heat conduction equations:

∂θ

∂t+∂q

∂x= 0, (5)

∂[qet/τ ]

∂t+∂[(θ/τ)et/τ ]

∂x= 0. (6)

Equations (5-6) reveal that θ and qet/τ are conserved quantities with fluxes qand (θ/τ)et/τ , respectively.

In the numerical method of characteristics, the hyperbolic heat conduc-tion equations are transformed to characteristic form prior to discretization.Introduction of the characteristic coordinates (see Figure 1),

ζ = t+ τ 1/2x; η = t− τ 1/2x, (7)

transforms Equations (1-2) to the nonconservative characteristic form:

∂θ

∂ζ+ τ 1/2

(∂q

∂ζ+

q

2τ

)= 0, (8)

Fourth-order simplified coupled schemes 463

Figure 1: Characteristic Coordinates

∂θ

∂η− τ 1/2

(∂q

∂η+

q

2τ

)= 0. (9)

Likewise, Equations (5-6) transform to the conservative characteristic form:

eζ/2τ ∂θ

∂ζ+ τ 1/2∂[qe

ζ/2τ ]

∂ζ= 0, (10)

eη/2τ ∂θ

∂η− τ 1/2∂[qe

η/2τ ]

∂η= 0. (11)

In characteristic form, both the nonconservative and conservative forms of thehyperbolic heat conduction equations become ordinary differential equationsalong the characteristic lines with constant ζ and η. Moreover, when theCourant number τ−1/2Δt/Δx equals one, the characteristic lines pass throughthe computational grid and either of the characteristic forms may be numeri-cally integrated to yield nonconservative and conservative numerical schemes.

2 Continuous Dispersion Relation

In [1], Roe and Arora make a convincing case that a comparison of the dis-persive and dissipative properties of the hyperbolic heat conduction equationsto those of their numerical approximations provides an important tool for as-sessing the suitablity of the latter. So that we may perform such a comparisonin later sections, we next summarize their continuous analysis of the dispersiveand dissipative properties of the hyperbolic heat conduction equations.

By seeking Fourier mode solutions of Equations (1-2) of the form of amultiple of eı(ωt−ξx), we immediately arrive at the complex dispersion relation

τω2 − ξ2 − ı · ω = 0, (12)

464 B. J. McCartin

where (real) ξ is a wave number and (complex) ω may be written as ω =ωR + ı · ωI with ωR an angular frequency and ωI a damping parameter.

The complex dispersion relation, Equation (12) may be broken into realand imaginary parts thereby yielding the pair of equations

ωR · (1− 2τωI) = 0, (13)

ω2R = ω2

I +ξ2 − ωI

τ. (14)

If ωR �= 0 then Equation (13) implies that

ωI =1

2τ(15)

and Equation (14) implies that

ωR = ±√ξ2

τ− 1

4τ 2. (16)

Defining the wave speed a(ξ) = ωR/ξ for ξ �= 0, we have

a(ξ) = ±τ−1/2 ·√

1− 1

4τξ2. (17)

For high wave numbers, a(ξ) approaches the characteristic speed τ−1/2

(frozen wave speed). For lower wave numbers, a(ξ) decreases, reaching zerowhen ξ = 1

2τ−1/2 (vanishing equilibrium wave speed). All waves with wave

numbers in the range [12τ−1/2,∞] are damped as e−t/2τ .

For wave numbers less than 12τ−1/2, ωR = 0 and the waves do not propagate.

After the relaxation time t = τ , they are damped by e−ωIτ where

ωIτ =1∓√1− 4ξ2τ

2. (18)

In Figure 2, the above dispersion analysis is summarized graphically. Inthe top graph, the nondimensional wave speed, a = τ 1/2a is plotted against thenondimensional wave number ξ = τ 1/2ξ. In the bottom graph, the dampingrate e−ωIτ is plotted against the nondimensional wave number ξ = τ 1/2ξ.

Beginning at the wave number of zero, we have two stationary modes, onewith a damping rate of 1 and the other with a damping rate of e−1. As the wavenumber increases, these two damping rates begin to monotonically approache−1/2. At the nondimensional wave number of one-half, these two modes coa-lesce and thereafter bifurcate into two waves propagating in opposite directionswith identical speeds and constant damping rate e−1/2. The magnitude of thenondimensional wave speeds increase monotonically, approaching one in thelimit of infinite wave number.


10−2

10−1

100

101

102

−1

−0.5

0

0.5

1

(1/2,0)

WA

VE

SP

EE

D

CONTINUOUS DISPERSION DIAGRAM (nondimensional)

10−2

10−1

100

101

102

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

e−1

(1/2,e−1/2)

DA

MP

ING

RA

TE

WAVE NUMBER

CONTINUOUS DISSIPATION DIAGRAM (nondimensional)

Figure 2: Continuous Dispersion/Dissipation Diagrams

3 Coupled Schemes

Roe and Arora [1] introduced and studied so-called coupled characteristics-based schemes for the numerical solution of the hyperbolic heat conductionequations. The point of embarkation for their investigation is the followinganalytical solution to the Cauchy problem for the hyperbolic heat conductionequations (see Figure 3):

θP = e−k · θA + θB

2+τ 1/2

2·∫ B

A

(Ω

τ− Ωt

)· θ dx− τ 1/2

2·∫ B

AΩ · qx dx, (19)

qP = e−k · qA + qB2

+τ 1/2

2·∫ B

AΩt · q dx− τ−1/2

2·∫ B

AΩ · θx dx, (20)

where Ω is the Riemann function for this initial value problem [6, pp. 108-110]. Since these analytical expressions reveal that the exact solution at pointP depends upon the values of θ and q along the entire interval [A,B], theyintroduced coupled schemes of the following form which include informationat the point M :

466 B. J. McCartin

Figure 3: Stencil for Coupled Schemes

θP = Rθ(k) · θA + θB

2+Sθ(k) · τ

1/2

2· (qA− qB) +Tθ(k) · (θA− 2θM + θB), (21)

qP = Rq(k) · qA + qB2

+Sq(k) · τ−1/2

2· (θA− θB)+Tq(k) · (qA−2qM + qB), (22)

where the stiffness parameter, k, is given by

k =1

2· Δtτ

=1

2· Δx

τ 1/2, (23)

and the coefficientsRθ, Sθ, Tθ, Rq, Sq, Tq determine the particular coupled scheme.Equations (21) and (22) may be recast as

θP = U(k) · θA + θB

2+τ 1/2

2·X(k) · (qA − qB) + V (k) · θM , (24)

qP = Z(k) · qA + qB2

+τ−1/2

2· Y (k) · (θA − θB) +W (k) · qM , (25)

where X = Sθ, Y = Sq, Z = Rq + 2Tq, U = Rθ + 2Tθ, V = −2Tθ,W = −2Tq.

3.1 Discrete Dispersion Relation

Seeking Fourier mode solutions of Equations (24-25) in the form of (θ, q) =eı(ω·nΔt−ξ·jΔx)(θ, q) [7], we require nonzero solutions to the homogeneous system

[eıωΔt − U cos (ξΔx)− V −ıτ1/2X sin (ξΔx)−ıτ−1/2Y sin (ξΔx) eıωΔt − Z cos (ξΔx)−W

] [θq

]=

[00

]. (26)


Setting the determinant of the coefficient matrix to zero, we immediately arriveat the complex dispersion relation for coupled schemes

eıωΔt =1

2·[B ±√B2 − 4C

], (27)

whereB := V +W + (U + Z) · cos (ξΔx), (28)

C := (UZ −XY ) · cos2 (ξΔx) + (UW + V Z) · cos (ξΔx) + (XY + VW ). (29)

In the above, (real) ξ is a wave number and (complex) ω may be written asω = ωR + ı · ωI with ωR an angular frequency and ωI a damping parameter.Denoting the complex amplification factor [8], Equation (27), by c±, we have

ωRΔt = Arg(c) (−π < Arg(·) ≤ π), −ωIΔt = ln |c±|. (30)

Before proceeding, let us make an important observation concerning thecomputation of the nondimensional wave speed τ 1/2 · a. If |ωRΔt| > ξΔx then|ωR/ξ| > Δx/Δt, i.e. the Fourier mode

[e−ωIΔt]n · eı(ωR·nΔt−ξ·jΔx) (31)

travels faster than the grid speed. While acceptable mathematically, this in-terpretation violates physical causality and as such is not viable.

Instead, when |ωRΔt| > π/2, we rewrite Equation (31) as

[−e−ωIΔt]n · eı(n·(ωRΔt±π)−j·ξΔx). (32)

Thus, we reinterpret the Fourier mode as traveling with wave speed

τ 1/2 · a± =

⎧⎨⎩ τ 1/2 · ω±

R

ξ, if |ωRΔt| < π/2,

τ 1/2 · ω±R

ξ∓ π

ξΔx, if |ωRΔt| ≥ π/2,

(33)

and possessing a damping rate

r =

{e−ωIτ , if |ωRΔt| < π/2,−e−ωIτ , if |ωRΔt| ≥ π/2.

(34)

Note that for a negatively damped mode of the discretization, the ampli-tude of the wave not only decays but also oscillates in sign at successive timesteps. There is no corresponding behavior exhibited by the original partial dif-ferential equations. Consequently, a thorough discrete dispersion/dissipationanalysis requires that the distinction be made between positively and nega-tively damped modes.

Furthermore, the use of Equation (34) will produce a discontinuity in ourplots of damping rate. This discontinuity is essential if physical causality is tobe enforced. Using Equation (33) produces no corresponding discontinuity inwave speed. See [2] for an alternative perspective on defining wave speed.

468 B. J. McCartin

3.2 Optimum Scheme (OPT)

Assuming a quadratic variation of θ and q on the interval [A,B] in Equa-tions (19-20), Roe and Arora [1] arrive at their Optimum scheme (OPT):

Rθ(k) = 1, Sθ(k) =1− e−2k

2k, Tθ(k) =

e−2k − 1 + 2k − 2k2

4k2; (35)

Rq(k) = e−2k, Sq(k) =1− e−2k

2k, Tq(k) =

1− e−2k(1 + 2k + 2k2)

4k2. (36)

3.2.1 Order of Accuracy

Roe and Arora [1, p. 480] claim that their OPT scheme is only third-orderaccurate. As we next show, the OPT scheme is, in fact, fourth-order accurate(for sufficiently smooth initial data). See [2] for a less direct proof.

Lemma 1 For Equations (35) and (36):

Rθ(k) = 1,

Sθ(k) = 1− k +2

3k2 − 1

3k3 +

2

15k4 + · · · ,

Tθ(k) = −1

3k +

1

6k2 − 1

15k3 + · · · ,

Rq(k) = 1− 2k + 2k2 − 4

3k3 +

2

3k4 − 4

15k5 + · · · ,

Sq(k) = 1− k +2

3k2 − 1

3k3 +

2

15k4 + · · · ,

Tq(k) =1

3k − 1

2k2 +

2

5k3 + · · · .

These facts imply our main result.

Theorem 1 The local discretization/truncation error (LTE) for the OPT scheme,Equations (35-36), is O(k5).

Proof: Simply apply the Order of Accuracy Theorem (Appendix B). The localdiscretization/truncation error is [9, p. 77]:

LTEθM =

4τ

45· [−τθxxxx + τqxxx]M · k5,

LTEqM =

4τ

45· [τqxxxx + θxxx]M · k5. �

This result is hardly surprising when one considers the Simpson-like parabolicapproximation of the OPT scheme and the error term for Simpson’s rule [10,p. 308]. The fourth-order accuracy of the OPT scheme follows immediately.

Corollary 1 The OPT scheme, Equations (35-36), is fourth-order accurate.

Proof: This follows by applying Theorem 2.4 of [9, p. 80] to Theorem 1. �


0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.50

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.55

0 0.2 0.4 0.6 0.8−1

−0.5

0

0.5

1k = 2.00

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1k = 5.00

Figure 4: Wave Speed - OPT Scheme

0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.50

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.55

0 0.2 0.4 0.6 0.8−1

−0.5

0

0.5

1k = 2.00

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1k = 5.00

Figure 5: Damping Rate - OPT Scheme

470 B. J. McCartin

3.2.2 Dispersive and Dissipative Properties

Figures 4-5 provide a graphical summary of the dispersive and dissipativeproperties of the OPT scheme for k = 1.00, 1.30, 1.50, 1.55, 2.00 and 5.00.These values of k are representative of the four qualitatively distinct ranges:0 < k < 1.27847, 1.27847 < k < 1.48, 1.48 < k < 1.54305, and 1.54305 <k < ∞. See [2] for an exhaustive description. In these plots, the solid curvesrepresent the exact values derived from the continuous dispersion relation,Equation (12), while the dotted curves derive from the discrete dispersionrelation, Equation (27). Figure 4 displays the nondimensional wave speed,τ 1/2a, and Figure 5 displays the damping rate, ±e−ωIτ , for nondimensionalwave numbers in the range 0 < ξ = τ 1/2ξ ≤ ξmax = π

2k. Since the magnitude

of the damping rate is bounded by unity, the OPT scheme is stable.

4 Simplified Coupled Schemes

Figure 6: Stencil for Simplified Coupled Schemes

As first observed by Roe and Arora [1], coupled schemes enjoy certainadvantages over uncoupled schemes (omitting point M) by virtue of the in-teraction of the wave families which they permit in accordance with the ex-act solution given by Equations (19-20). However, the OPT scheme requiresknowledge of the Riemann function, Ω. Motivated by a desire to reap the ben-efits of coupled schemes without requiring knowledge of the Riemann function,they introduced the concept of a simplified coupled scheme.

With reference to Figure 6, an uncoupled scheme is first used to provideapproximations to the values of θ at the points F and G. Then, along the char-acteristics AP and BP , θ is assumed to be either constant (PC(M) scheme),or piecewise linear (PC(T) scheme), or quadratic (PC(S) scheme). Finally,


the conservative characteristic form of the hyperbolic heat conduction equa-tions, Equations (10-11), are integrated along the characteristics to yield thesimplified coupled scheme.

Because they mistakenly believed that the OPT scheme was merely third-order accurate, only second- and third-order accurate simplified coupled schemesare presented in [1]. In the present paper, fourth-order accurate simplified cou-pled characteristics-based schemes for the hyperbolic heat conduction equa-tions will be studied.

In what follows, we place the origin of coordinates at P (0, 0) so that we haveA(−Δx,−Δt), M(0,−Δt, ), B(Δx,−Δt), F (−Δx/2,−Δt/2), G(Δx/2,−Δt/2).We then integrate Equations (10) and (11) by parts along the characteristicsAP and BP , respectively, to obtain

(θP − θAe−2k) + τ 1/2(qP − qAe−2k) =

1

τ

∫AP

θ(t)et/τ dt, (37)

(θP − θBe−2k)− τ 1/2(qP − qBe−2k) =

1

τ

∫BP

θ(t)et/τ dt. (38)

Adding Equation (38) to Equation (37) and solving for θP yields

θP = e−2k[θA + θB

2+τ 1/2

2(qA − qB)] +

1

2[1

τ

∫AP

θ(t)et/τ dt+1

τ

∫BP

θ(t)et/τ dt],

(39)while subtracting Equation (38) from Equation (37) and solving for qp yields

qP = e−2k[qA + qB

2+τ−1/2

2(θA−θB)]+

τ−1/2

2[1

τ

∫AP

θ(t)et/τ dt−1

τ

∫BP

θ(t)et/τ dt].

(40)Equations (39) and (40) are exact. Below, we will obtain fourth-order accurateschemes by numerically approximating the integrals which appear above [10].

4.1 Fractional Step Scheme (FS)

Following the lead of Roe and Arora [1], we commence by assuming thatthe temperature varies quadratically along the characteristic AP

θAP (t) = θP +

(3θP − 4θF + θA

2k

)·(t

τ

)+

(θA − 2θF + θP

2k2

)·(t

τ

)2

, (41)

as well as along the characteristic BP

θBP (t) = θP +

(3θP − 4θG + θB

2k

)·(t

τ

)+

(θB − 2θG + θP

2k2

)·(t

τ

)2

. (42)

472 B. J. McCartin

Thus,

1

τ

∫AP

θ(t)et/τ dt+1

τ

∫BP

θ(t)et/τ dt = S0 · 2θP

+ S1 · 6θP − 4(θF + θG) + θA + θB

2k

+ S2 · θA + θB − 2(θF + θG) + 2θP

2k2,

(43)

1

τ

∫AP

θ(t)et/τ dt− 1

τ

∫BP

θ(t)et/τ dt = S1 · 4(θG − θF ) + θA − θB

2k

+ S2 · θA − θB + 2(θG − θF )

2k2, (44)

where

Sn :=1

τn+1

∫ 0

−Δttnet/τ dt (45)

(see Appendix C).Since Roe and Arora [1] used second-order accurate approximations to θF

and θG, their PC(S) scheme is only third-order accurate. In order for such ascheme to be fourth-order accurate, we require third-order accurate approxi-mations to θF and θG. We obtain such third-order accurate approximationsby taking an intermediate fractional time-step to estimate θF and θG.

By suitably modifying Equation (39) for a time-step of Δt/2, we obtain

θF = e−k[θA + θM

2+τ 1/2

2(qA− qM)] +

ek

2[1

τ

∫AF

θ(t)et/τ dt+1

τ

∫MF

θ(t)et/τ dt],

(46)

θG = e−k[θM + θB

2+τ 1/2

2(qM − qB)]+

ek

2[1

τ

∫MG

θ(t)et/τ dt+1

τ

∫BG

θ(t)et/τ dt].

(47)We next describe the computational procedure for θF and obtain the corre-sponding expression for θG via the substitutions F ← G, A←M , M ← B.

Along the characteristic AF , we make the quadratic approximation

θAF (t) = 4θF − 3θA − 2kτθ′A +

(4θF − 4θA − 3kτθ′A

k

)·(t

τ

)

+

(θF − θA − kτθ′A

k2

)·(t

τ

)2

, (48)

while along the characteristic MF , we make the quadratic approximation

θMF (t) = 4θF − 3θM − 2kτθ′M +

(4θF − 4θM − 3kτθ′M

k

)·(t

τ

)

+

(θF − θM − kτθ′M

k2

)·(t

τ

)2

. (49)


In the above,

θ′A = (−qx+τ−1/2θx)A ≈ −−3qA + 4qM − qB2Δx

+τ−1/2 ·−3θA + 4θM − θB

2Δx, (50)

θ′M = (−qx − τ−1/2θx)M ≈ −qB − qA2Δx

− τ−1/2 · θB − θA

2Δx. (51)

Substitution of Equations (48) and (49) into Equation (46) followed bysolution for θF yields

θF = [k2τ 1/2e−2k · (qA − qM) + (k2e−2k − 3k2S0,1 − 4kS1,1 − S2,1) · (θA + θM)

− (2k3τS0,1 + 3k2τS1,1 + kτS2,1) · (θ′A + θ′M )]

/ [2(k2e−2k − 3k2S0,1 − 4kS1,1 − S2,1)], (52)

where

Sn,1 :=1

τn+1

∫ −Δt/2

−Δttnet/τ dt (53)

(see Appendix C).The corresponding expression for θG, derivable from Equation (47), is

θG = [k2τ 1/2e−2k · (qM − qB) + (k2e−2k − 3k2S0,1 − 4kS1,1 − S2,1) · (θM + θB)

− (2k3τS0,1 + 3k2τS1,1 + kτS2,1) · (θ′M + θ′B)]

/ [2(k2e−2k − 3k2S0,1 − 4kS1,1 − S2,1)], (54)

where

θ′M = (−qx + τ−1/2θx)M ≈ −qB − qA2Δx

+ τ−1/2 · θB − θA

2Δx, (55)

θ′B = (−qx − τ−1/2θx)B ≈ −qA − 4qM + 3qB2Δx

− τ−1/2 · θA − 4θM + 3θB

2Δx. (56)

Substitution of Equations (52) and (54) into Equations (43) and (44) andsubsequent substitution into Equations (39) and (40) yield the Fractional Stepscheme (FS):

Rθ(k) = 1, (57)

Sθ(k) =2k2e−2k[(k − 1) + e−k] + [(k − 1) + (k + 1)e−2k][(k − 2) + (k2 + k + 2)e−k]

[(3k − 2) + (k + 2)e−2k][(k − 1) + e−k],

(58)

Tθ(k) =[(k − 1) + (k + 1)e−2k][(4− 3k)− (k + 4)e−k]

2[(3k − 2) + (k + 2)e−2k][(k − 1) + e−k], (59)

Rq(k) = e−2k, (60)

474 B. J. McCartin

Sq(k) =1− e−2k

2k, (61)

Tq(k) =[(k − 1) + (k + 1)e−2k][(k − 2) + (k2 + k + 2)e−k]

4k2[(k − 1) + e−k]. (62)


Lemma 2 For Equations (57-62):

Rθ(k) = 1,

Sθ(k) = 1− k +2

3k2 − 1

3k3 +

43

270k4 + · · · ,

Tθ(k) = −1

3k +

1

6k2 − 43

540k3 + · · · ,

Rq(k) = 1− 2k + 2k2 − 4

3k3 +

2

3k4 − 4

15k5 + · · · ,

Sq(k) = 1− k +2

3k2 − 1

3k3 +

2

15k4 + · · · ,

Tq(k) =1

3k − 1

2k2 +

19

45k3 + · · · .


Theorem 2 The local discretization/truncation error (LTE) for the FS scheme,Equations (57-62), is O(k5).


LTEθM =

4τ

45· [−τθxxxx + τqxxx +

7

12θxx +

7

12qx]M · k5,

LTEqM =

4τ

45· [τqxxxx + θxxx − qxx]M · k5. �

The fourth-order accuracy of the FS scheme follows immediately.

Corollary 2 The FS scheme, Equations (57-62), is fourth-order accurate.



0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.45

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.50

0 0.2 0.4 0.6 0.8−1

−0.5

0

0.5

1k = 2.00

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1k = 5.00

Figure 7: Wave Speed - FS Scheme

0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.45

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.50

0 0.2 0.4 0.6 0.8−1

−0.5

0

0.5

1k = 2.00

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1k = 5.00

Figure 8: Damping Rate - FS Scheme

476 B. J. McCartin


Figures 7-8 provide a graphical summary of the dispersive and dissipativeproperties of the FS scheme for k = 1.00, 1.30, 1.45, 1.50, 2.00 and 5.00.These values of k are representative of the four qualitatively distinct ranges:0 < k < 1.23375, 1.23375 < k < 1.423, 1.423 < k < 1.476, and 1.476 < k <∞.For k = 1.23375, c+ first passes through the origin after its return to the realaxis. For k = 1.423, c+ = c− at the origin upon their return to the real axis.For k = 1.476, c+ and c− coalesce but never leave the real axis. c−(ξmax) isalways negative and so corresponds to a negatively damped mode.

In these plots, the solid curves represent the exact values derived from thecontinuous dispersion relation, Equation (12), while the dotted curves derivefrom the discrete dispersion relation, Equation (27). Figure 7 displays thenondimensional wave speed, τ 1/2a, and Figure 8 displays the damping rate,±e−ωIτ , for nondimensional wave numbers in the range 0 < ξ = τ 1/2ξ ≤ξmax = π

2k. Since the magnitude of the damping rate is bounded by unity, the

FS scheme is stable. Note the qualitative similarity to the OPT scheme exceptthat the ranges for k have been shifted slightly downward.

4.2 Hermite-Bessel Scheme (HB)

We next develop another fourth-order accurate scheme that does not rely ona fractional time-step. Instead, we employ cubic Hermite-Bessel interpolation[10] to proceed directly from one time level to the next.

Specifically, along the characteristic AP , we make the cubic approximation

θAP (t) = θP +

(−3(θA − θP )− 4kτθ′A − 2k2τ 2θ′′A2k

)·(t

τ

)

+

(−3(θA − θP )− 6kτθ′A − 4k2τ 2θ′′A4k2

)·(t

τ

)2

+

(−(θA − θP )− 2kτθ′A − 2k2τ 2θ′′A8k3

)·(t

τ

)3

, (63)

while, along the characteristic BP , we make the cubic approximation

θBP (t) = θP +

(−3(θB − θP )− 4kτθ′B − 2k2τ 2θ′′B2k

)·(t

τ

)

+

(−3(θB − θP )− 6kτθ′B − 4k2τ 2θ′′B4k2

)·(t

τ

)2

+

(−(θB − θP )− 2kτθ′B − 2k2τ 2θ′′B8k3

)·(t

τ

)3

. (64)


Thus,

1

τ

∫AP

θ(t)et/τ dt +1

τ

∫BP

θ(t)et/τ dt = S0 · 2θP

− S1 · 3(θA − 2θP + θB) + 4kτ(θ′A + θ′B) + 2k2τ 2(θ′′A + θ′′B)

2k

− S2 · 3(θA − 2θP + θB) + 6kτ(θ′A + θ′B) + 4k2τ 2(θ′′A + θ′′B)

4k2

− S3 · (θA − 2θP + θB) + 2kτ(θ′A + θ′B) + 2k2τ 2(θ′′A + θ′′B)

8k3,

(65)

1

τ

∫AP

θ(t)et/τ dt − 1

τ

∫BP

θ(t)et/τ dt =

− S1 · 3(θA − θB) + 4kτ(θ′A − θ′B) + 2k2τ 2(θ′′A − θ′′B)

2k

− S2 · 3(θA − θB) + 6kτ(θ′A − θ′B) + 4k2τ 2(θ′′A − θ′′B)

4k2

− S3 · (θA − θB) + 2kτ(θ′A − θ′B) + 2k2τ 2(θ′′A − θ′′B)

8k3.(66)

Finally,θ′A = (−qx + τ−1/2θx)A, (67)

θ′B = (−qx − τ−1/2θx)B, (68)

θ′′A = (τ−1qx − 2τ−1/2qxx + 2τ−1θxx)A, (69)

θ′′B = (τ−1qx + 2τ−1/2qxx + 2τ−1θxx)B, (70)

which imply

θ′A + θ′B ≈1

2τ 1/2k· (qA − qB)− 1

τk· (θA − 2θM + θB), (71)

θ′A − θ′B ≈ −1

2τk· (θA − θB) +

1

τ 1/2k· (qA − 2qM + qB), (72)

θ′′A + θ′′B ≈ −1

2τ 3/2k· (qA − qB) +

1

τ 2k2· (θA − 2θM + θB), (73)

θ′′A − θ′′B ≈ −(1 + k)

τ 3/2k2· (qA − 2qM + qB). (74)

Substitution of Equations (71-74) into Equations (65-66) and subsequentsubstitution into Equations (39-40) yield the Hermite-Bessel scheme (HB):

Rθ(k) = 1, (75)

478 B. J. McCartin

Sθ(k) =(2k3 − 8k2 + 9k − 3) + (−4k3 − 4k2 − 3k + 3)e−2k

3[(−2k2 + 2k − 1) + e−2k], (76)

Tθ(k) =2k[(k − 1) + (k + 1)e−2k]

3[(−2k2 + 2k − 1) + e−2k], (77)

Rq(k) = e−2k, (78)

Sq(k) =1− e−2k

2k, (79)

Tq(k) =(−2k2 + 6k − 5) + (4k + 5)e−2k

4k2. (80)



Rθ(k) = 1,

Sθ(k) = 1− k +2

3k2 − 1

3k3 +

1

10k4 + · · · ,

Tθ(k) = −1

3k +

1

6k2 − 1

20k3 + · · · ,

Rq(k) = 1− 2k + 2k2 − 4

3k3 +

2

3k4 − 4

15k5 + · · · ,

Sq(k) = 1− k +2

3k2 − 1

3k3 +

2

15k4 + · · · ,

Tq(k) =1

3k − 1

2k2 +

1

3k3 + · · · .


Theorem 3 The local discretization/truncation error (LTE) for the HB scheme,Equations (75-80), is O(k5).


LTEθM =

4τ

45· [−τθxxxx + τqxxx − 3

4θxx − 3

4qx]M · k5,

LTEqM =

4τ

45· [τqxxxx + θxxx + 3qxx]M · k5. �

The fourth-order accuracy of the HB scheme follows immediately.

Corollary 3 The HB scheme, Equations (75-80), is fourth-order accurate.



0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.50

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1k = 1.60

0 0.2 0.4 0.6 0.8−1

−0.5

0

0.5

1k = 2.00

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1k = 5.00

Figure 9: Wave Speed - HB Scheme

0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−0.5

0

0.5

1k = 1.50

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1k = 1.60

0 0.2 0.4 0.6 0.80.2

0.4

0.6

0.8

1k = 2.00

0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1k = 5.00

Figure 10: Damping Rate - HB Scheme

480 B. J. McCartin


Figures 9-10 provide a graphical summary of the dispersive and dissipativeproperties of the HB scheme for k = 1.00, 1.30, 1.50, 1.60, 2.00 and 5.00.These values of k are representative of the five qualitatively distinct ranges:0 < k < 1.44, 1.44 < k < 1.55, 1.55 < k < 1.6378, 1.6378 < k < 2.481285,and 2.481285 < k < ∞. For k = 1.44, c+ first passes through the origin afterits return to the real axis. For k = 1.55, c+ = c− at the origin upon theirreturn to the real axis. For 1.55 < k < 1.6378, c−(ξmax) is negative and socorresponds to a negatively damped mode. For 1.6378, c−(ξ) is zero and so thecorresponding mode is extinguished. For 1.6378 < k < 2.481285, c−(ξmax) ispositive and so corresponds to a positively damped mode. For k = 2.481285,c+ and c− coalesce but never leave the real axis while c−(ξmax) is positive andso corresponds to a positively damped mode.



HB scheme is stable. Note the qualitative dissimilarity to the OPT and FSschemes. Of special note is the much better overall fit for k ≈ 1.50. However,inspection of k = 2.00 reveals an expanding band of falsely propagating modeswith increasing k.

4.3 Hybrid Scheme (HYBRID)

The FS and HB schemes may be averaged to yield a fourth-order accuratescheme with principal local truncation errors coinciding with those of the OPTscheme. We thereby arrive at the Hybrid scheme (HYBRID):

Rθ(k) =9

16RFS

θ (k) +7

16RHB

θ (k), (81)

Sθ(k) =9

16SFS

θ (k) +7

16SHB

θ (k), (82)

Tθ(k) =9

16T FS

θ (k) +7

16THB

θ (k), (83)

Rq(k) =3

4RFS

q (k) +1

4RHB

q (k), (84)

Sq(k) =3

4SFS

q (k) +1

4SHB

q (k), (85)

Tq(k) =3

4T FS

q (k) +1

4THB

q (k). (86)


0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.40

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.55

0 0.2 0.4 0.6 0.8−1

−0.5

0

0.5

1k = 2.00

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1k = 5.00

Figure 11: Wave Speed - HYBRID Scheme

0 0.5 1 1.5 2−1

−0.5

0

0.5

1k = 1.00

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.30

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.40

0 0.5 1 1.5−1

−0.5

0

0.5

1k = 1.55

0 0.2 0.4 0.6 0.8−1

−0.5

0

0.5

1k = 2.00

0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1k = 5.00

Figure 12: Damping Rate - HYBRID Scheme

482 B. J. McCartin



Rθ(k) = 1,

Sθ(k) = 1− k +2

3k2 − 1

3k3 +

2

15k4 + · · · ,

Tθ(k) = −1

3k +

1

6k2 − 1

15k3 + · · · ,

Rq(k) = 1− 2k + 2k2 − 4

3k3 +

2

3k4 − 4

15k5 + · · · ,

Sq(k) = 1− k +2

3k2 − 1

3k3 +

2

15k4 + · · · ,

Tq(k) =1

3k − 1

2k2 +

2

5k3 + · · · .


Theorem 4 The local discretization/truncation error (LTE) for the HYBRIDscheme, Equations (81-86), is O(k5).


LTEθM =

4τ

45· [−τθxxxx + τqxxx]M · k5,

LTEqM =

4τ

45· [τqxxxx + θxxx]M · k5. �

The fourth-order accuracy of the HYBRID scheme follows immediately.

Corollary 4 The HYBRID scheme, Equations (81-86), is fourth-order accu-rate.



Figures 11-12 provide a graphical summary of the dispersive and dissipativeproperties of the HYBRID scheme for k = 1.00, 1.30, 1.40, 1.55, 2.00 and 5.00.These values of k are representative of the five qualitatively distinct ranges:0 < k < 1.3035, 1.3035 < k < 1.517, 1.517 < k < 1.595, 1.595 < k < 3.03,and 3.03 < k < ∞. For k = 1.3035, c+ first passes through the origin afterits return to the real axis. For k = 1.517, c+ = c− at the origin upon theirreturn to the real axis. For 1.517 < k < 1.595, c−(ξmax) is negative and so


corresponds to a negatively damped mode. For 1.595, c+ and c− coalesce butnever leave the real axis while c−(ξmax) is negative and so corresponds to anegatively damped mode. For 1.595 < k < 3.03, c−(ξmax) is negative and socorresponds to a negatively damped mode. For k = 3.03, c−(ξmax) is zero andso the corresponding mode is extinguished. For 3.05 < k < ∞, c−(ξmax) ispositive and so corresponds to a positively damped mode.



HYBRID scheme is stable. For k ≈ 1.50, we observe improvement over theOPT/FS schemes without the falsely propagating modes of the HB scheme.Also, there is no negative damping for large k.

5 Numerical Example

0 1 2 3 4 5 6

01

23

450

2

4

xt

TE

MP

ER

AT

UR

E

0 1 2 3 4 5 6

01

23

45

−20

−10

0

10

xt

HE

AT

FLU

X

Figure 13: Exact Solution - Equations (87-88)

Consider the exact solution to the hyperbolic heat conduction equations[11]:

θ(x, t) = sin(

x

2τ 1/2

)· e−t/(2τ) ·

(α + β · t

2τ

), (87)

484 B. J. McCartin

q(x, t) = γ · e−t/τ − τ−1/2 · cos(

x

2τ 1/2

)· e−t/(2τ) ·

(α− β + β · t

2τ

). (88)

This solution is on display in Figure 13 for α = 1, β = 10, γ = −9, and τ = 1on the interval 0 ≤ x ≤ 2π · τ 1/2. The temperature profile at first rises to amaximum when t = 2(β − ατ 2)/(βτ) before asymptotically approaching zero.

We next apply the FS, HB, HYBRID and OPT schemes to this problem.These numerical approximations with Δx = π/8 at the time when the temper-ature profile is cresting are displayed in Figures 14, 15, 16 and 17, respectively.As might be expected for k = π/16, all four schemes provide superlative accu-racy. However, it should be noted that the results for the HYBRID and OPTschemes are a notch above the others. Most importantly, it should be observedthat the HYBRID scheme is able to achieve a level of accuracy comparable tothat of the OPT scheme without requiring the Riemann function.

6 Conclusion

The paper by Roe and Arora [1] is an important and influential one (despiteits flawed truncation error analysis as well as other difficulties revealed in [2])which pioneered and championed the concept of coupled characteristics-basedschemes for the hyperbolic heat conduction equations. The present paper wasmotivated by a recognition of the importance of their work and a desire to buildupon it. The benefits of fourth-order accurate schemes for wave propagationproblems are clearly presented in [12].

In closing, the preceding analysis has been confined to the pure initial valueproblem for the source-free hyperbolic heat conduction equations. In principle,coupled characteristics-based schemes may be extended to the initial-boundaryvalue problem with a heat source. However, as discovered in [13], this consider-ably complicates the details and exposition, especially for higher-order accuratemethods. Finally, extension of these fourth-order accurate approximations tononlinear hyperbolic heat transfer problems [14] is an open problem.

7 Acknowledgements

The author expresses his sincere gratitude to Mrs. Barbara A. McCartinfor her dedicated assistance in the production of this paper. The author alsothanks Matthew F. Causley who contributed significantly to the initial phaseof this investigation.

A Derivative Formulas

Successive differentiation and substitution of Equations (1) and (2) permits


0 2 4 60

1

2

3

4

5

x

TE

MP

ER

AT

UR

E

0 2 4 6 8−10

−8

−6

−4

−2

0

2x 10

−5

x

TE

MP

ER

AT

UR

E−

erro

r0 2 4 6

−20

−15

−10

−5

0

5

10

x

HE

AT

FLU

X

0 2 4 6 8

−2

0

2

4x 10

−5

xH

EA

T F

LUX

−er

ror

Figure 14: Numerical Approximation - FS Scheme

0 2 4 60

1

2

3

4

5

x

TE

MP

ER

AT

UR

E

0 2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−4

x

TE

MP

ER

AT

UR

E−

erro

r

0 2 4 6−20

−15

−10

−5

0

5

10

x

HE

AT

FLU

X

0 2 4 6 8−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

x

HE

AT

FLU

X−

erro

r

Figure 15: Numerical Approximation - HB Scheme

486 B. J. McCartin

0 2 4 60

1

2

3

4

5

x

TE

MP

ER

AT

UR

E

0 2 4 6 80

2

4x 10

−5

x

TE

MP

ER

AT

UR

E−

erro

r

0 2 4 6−20

−15

−10

−5

0

5

10

x

HE

AT

FLU

X

0 2 4 6 8

−2

0

2

4x 10

−5

xH

EA

T F

LUX

−er

ror

Figure 16: Numerical Approximation - HYBRID Scheme

0 2 4 60

1

2

3

4

5

x

TE

MP

ER

AT

UR

E

0 2 4 6 80

0.5

1

1.5

2

2.5

3x 10

−5

x

TE

MP

ER

AT

UR

E−

erro

r

0 2 4 6−20

−15

−10

−5

0

5

10

x

HE

AT

FLU

X

0 2 4 6 8

−2

0

2

4x 10

−5

x

HE

AT

FLU

X−

erro

r

Figure 17: Numerical Approximation - OPT Scheme


the exchange of temporal derivatives for spatial derivatives (assuming that therequisite derivatives exist and are continuous):

θt = −qxθtt = τ−1 · θxx + τ−1 · qx

θttt = −τ−1 · qxxx − τ−2 · θxx − τ−2 · qxθtttt = τ−2 · θxxxx + 2τ−2 · qxxx + τ−3 · θxx + τ−3 · qx

θttttt = −τ−2 · qxxxxx − 2τ−3 · θxxxx − 3τ−3 · qxxx − τ−4 · θxx − τ−4 · qx

qt = −τ−1 · θx − τ−1 · qqtt = τ−1 · qxx + τ−2 · θx + τ−2 · q

qttt = −τ−2 · θxxx − 2τ−2 · qxx − τ−3 · θx − τ−3 · qqtttt = τ−2 · qxxxx + 2τ−3 · θxxx + 3τ−3 · qxx + τ−4 · θx + τ−4 · q

qttttt = −τ−3 · θxxxxx − 3τ−3 · qxxxx − 3τ−4 · θxxx − 4τ−4 · qxx − τ−5 · θx − τ−5 · q

B Order of Accuracy Conditions

We provide necessary and sufficient conditions for coupled characteristics-based schemes, Equations (21) and (22), to be first-, second-, third- and fourth-order accurate approximations to the hyperbolic heat conduction equations,Equations (1) and (2). For fourth-order accurate schemes, we provide explicitexpressions for their principal local truncation errors. It is shown that thereis an accuracy barrier whereby no such scheme can be fifth-order accurate.

Theorem 5 (Order of Accuracy) With reference to Equations (21) and (22),define

Rθ(k) =∞∑

n=0

Ankn, Sθ(k) =

∞∑n=0

Bnkn, Tθ(k) =

∞∑n=0

Cnkn,

and

Rq(k) =∞∑

n=0

ankn, Sq(k) =

∞∑n=0

bnkn, Tq(k) =

∞∑n=0

cnkn.

Then, Equations (21) and (22) are

1. first-order accurate (consistent) if and only if A0 = 1, A1 = 0, B0 =1; a0 = 1, a1 = −2, b0 = 1,

2. second-order accurate if and only if they are first-order accurate andA2 = 0, B1 = −1, C0 = 0; a2 = 2, b1 = −1, c0 = 0,

488 B. J. McCartin

3. third-order accurate if and only if they are second-order accurate andA3 = 0, B2 = 2

3, C1 = −1

3; a3 = −4

3, b2 = 2

3, c1 = 1

3,

4. fourth-order accurate if and only if they are third-order accurate andA4 = 0, B3 = −1

3, C2 = 1

6; a4 = 2

3, b3 = −1

3, c2 = −1

2.

If fourth-order accurate then the principal local truncation errors of Equations(21) and (22) are

LTEθM =

4τ

45·[−τθxxxx+τqxxx−45(C3+

1

15)θxx+

45

2(B4− 2

15)qx− 45

4τA5θ]M ·k5,

LTEqM =

4τ

45·[τqxxxx+θxxx−45(c3− 2

5)qxx+

45

2τ(b4− 2

15)θx− 45

4τ(a5+

4

15)q]M ·k5,

respectively. Equations (21) and (22) cannot be fifth-order accurate. It is as-sumed that the initial data are sufficiently smooth so that all requisite deriva-tives exist and are continuous.

Proof: By Taylor’s Theorem,

θP = [θ + 2τkθt + 2τ 2k2θtt +4

3τ 3k3θttt +

2

3τ 4k4θtttt +

4

15τ 5k5θttttt]M +O(k6),

qP = [q + 2τkqt + 2τ 2k2qtt +4

3τ 3k3qttt +

2

3τ 4k4qtttt +

4

15τ 5k5qttttt]M +O(k6).

Using the identities of Appendix A, these may be rewritten as

θP = [θ − 2τkqx + 2τ 2k2(τ−1θxx + τ−1qx)

+4

3τ 3k3(−τ−1qxxx − τ−2θxx − τ−2qx)

+2

3τ 4k4(τ−2θxxxx + 2τ−2qxxx + τ−3θxx + τ−3qx)

+4

15τ 5k5(−τ−2qxxxxx − 2τ−3θxxxx − 3τ−3qxxx − τ−4θxx − τ−4qx)]M

+ O(k6),

qP = [q + 2τk(−τ−1θx − τ−1q) + 2τ 2k2(τ−1qxx + τ−2θx + τ−2q)

+4

3τ 3k3(−τ−2θxxx − 2τ−2qxx − τ−3θx − τ−3q)

+2

3τ 4k4(τ−2qxxxx + 2τ−3θxxx + 3τ−3qxx + τ−4θx + τ−4q)

+4

15τ 5k5(−τ−3θxxxxx − 3τ−3qxxxx − 3τ−4θxxx − 4τ−4qxx − τ−5θx − τ−5q)]M

+ O(k6).


Also by Taylor’s Theorem,

θA + θB

2= [θ + 2τk2θxx +

2

3τ 2k4θxxxx]M +O(k6),

τ 1/2

2· (qA − qB) = [−2τkqx − 4

3τ 2k3qxxx − 4

15τ 3k5qxxxxx]M +O(k6),

θA − 2θM + θB = [4τk2θxx +4

3τ 2k4θxxxx]M +O(k6),

qA + qB2

= [q + 2τk2qxx +2

3τ 2k4qxxxx]M +O(k6),

τ−1/2

2· (θA − θB) = [−2kθx − 4

3τk3θxxx − 4

15τ 2k5θxxxxx]M +O(k6),

qA − 2qM + qB = [4τk2qxx +4

3τ 2k4qxxxx]M +O(k6).

Substitution of all of these Taylor series into Equations (21) and (22) andsubsequent comparison of like powers of k directly establishes the theorem. �

C Some Useful Integrals

S0,1 :=1

τ

∫ −Δt/2

−Δtet/τ dt = e−k − e−2k

S1,1 :=1

τ 2

∫ −Δt/2

−Δttet/τ dt = −(1 + k)e−k + (1 + 2k)e−2k

S2,1 :=1

τ 3

∫ −Δt/2

−Δtt2et/τ dt = (2 + 2k + k2)e−k + (−2− 4k − 4k2)e−2k

S0 :=1

τ

∫ 0

−Δtet/τ dt = 1− e−2k

S1 :=1

τ 2

∫ 0

−Δttet/τ dt = −1 + (1 + 2k)e−2k

S2 :=1

τ 3

∫ 0

−Δtt2et/τ dt = 2 + (−2− 4k − 4k2)e−2k

S3 :=1

τ 4

∫ 0

−Δtt3et/τ dt = −6 + (6 + 12k + 12k2 + 8k3)e−2k

490 B. J. McCartin

References

[1] P. L. Roe and M. Arora, Characteristic-based schemes for dispersive wavesI. The method of characteristics for smooth solutions, Numerical Methodsfor Partial Differential Equations, 9 (1993), 459-505.

[2] B. J. McCartin, Remarks on a Paper of Roe and Arora, Applied Mathe-matical Sciences, Vol. 3, 2009, no. 2, 85-114.

[3] M. B. Abbott, An Introduction to the Method of Characteristics, AmericanElsevier, New York, NY, 1966.

[4] A. V. Luikov, Analytical Heat Diffusion Theory, Academic, New York,NY, 1968.

[5] D. C. Wiggert, Analysis of Early-Time Transient Heat Conduction byMethod of Characteristics, ASME Journal of Heat Transfer, 99 (1977),35-40.

[6] P. D. Lax, Partial Differential Equations, Courant Institute of Mathemat-ical Sciences, New York University, 1951.

[7] J. W. Thomas, Numerical Partial Differential Equations: Finite DifferenceMethods, Springer, New York, NY, 1995.

[8] K. W. Morton and D. F. Mayers, Numerical Solution of Partial DifferentialEquations, Cambridge, 1994.

[9] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations,Wiley, New York, NY, 1962.

[10] S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algo-rithmic Approach, Third Edition, McGraw-Hill, New York, NY, 1980.

[11] F. H. Miller, Partial Differential Equations, Wiley, New York, NY, 1941.

[12] B. Gustafsson, High Order Difference Methods for Time Dependent PDE,Springer, New York, 2008.

[13] B. J. McCartin and M. F. Causley, Angled Derivative Approximationof the Hyperbolic Heat Conduction Equations, Applied Mathematics andComputation, 182 (2006), 1581-1607.

[14] H. Q. Yang, Characteristics-Based, High-Order Accurate and Nonoscilla-tory Numerical Method for Hyperbolic Heat Conduction, Numerical HeatTransfer, Part B, 18 (1990), 221-241.

Received: August 26, 2008

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