Exponential Fitting of the Damped Wave EquationApplied Mathematical Sciences, Vol. 4, 2010, no. 39, 1899 - 1930 Exponential Fitting of the Damped Wave Equation Brian J. McCartin Applied

Applied Mathematical Sciences, Vol. 4, 2010, no. 39, 1899 - 1930

Exponential Fitting of the

Damped Wave Equation

Brian J. McCartin

Applied Mathematics, Kettering University1700 University Avenue, Flint, MI 48504-6214 USA

[email protected]

Abstract

Herein, the Control Region Approximation [1, 2] is applied to the dampedwave equation both with and without exponential fitting [3]. It is shownthat exponential fitting is advantageous as regards the accuracy, asymp-totics, stability and dispersive/dissipative properties of the resultingnumerical approximation.

Mathematics Subject Classification: 35L15, 65M06, 65M12

Keywords: damped wave equation, exponential fitting, stability analysis,dispersion and dissipation analysis, control region approximation

1 Introduction

In [4], the advantages of Control Region Approximation and exponentialfitting were extolled. In this paper, we study their application to the Cauchyproblem for the damped wave equation:

α · ∂2u

∂t2+ β · ∂u

∂t=∂2u

∂x2; −∞ < x <∞, t > 0, (1)

subject to the initial conditions

u(x, 0) = φ(x),∂u

∂t(x, 0) = ψ(x); −∞ < x <∞. (2)

This initial value problem arises in the motion of a vibrating string underthe influence of air resistance [5] where α � β as well as in hyperbolic heatconduction [6] where α � β. We seek a numerical approximation that performswell in both parameter regimes.

1900 B. J. McCartin

2 Continuous Dispersion Relation

Dimensional analysis of Equation (1) reveals that [α] = L−2T 2 and [β] =L−2T [7]. Hence, we may define a characteristic length, time and (conse-

quently) speed as x∗ = α1/2

β, t∗ = α

βand a∗ = α−1/2 , respectively. We have

nondimensionalized u thereby absorbing the coefficient of ∂2u∂x2 . However, we

refrain from further nondimensionalization as we desire to study limiting be-havior on a fixed spatiotemporal domain.

A comparison of the dispersive and dissipative properties of the dampedwave equation to those of its numerical approximations provides an importanttool for assessing the suitability of the latter [8]. We next present a continu-ous analysis of the dispersive and dissipative properties of the damped waveequation modeled after that for the hyperbolic heat conduction equations [9].

By seeking Fourier mode solutions of Equation (1) of the form of a multipleof eı(ωt−ξx), we immediately arrive at the complex dispersion relation

αω2 − ξ2 − ı · βω = 0, (3)

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 (3) may be broken into real andimaginary parts thereby yielding the pair of equations

ωR · (β − 2αωI) = 0, (4)

ω2R = ω2

I +ξ2 − βωI

α. (5)

If ωR �= 0 then Equations (4) and (5) imply that

ωI =β

2α, (6)

ωR = ±√ξ2

α− β2

4α2. (7)

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

a(ξ) = ±α−1/2 ·√√√√1 − 1

4[α1/2

β· ξ]2 . (8)

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

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

2· β

α1/2 (vanishing equilibrium wave speed). All waves with wave

numbers in the range [12· β

α1/2 ,∞] are damped as e−βt/(2α).

Damped wave equation 1901

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 1: Continuous Dispersion/Dissipation Diagrams

For wave numbers less than 12· βα1/2 , ωR = 0 and the waves do not propagate.

After the relaxation time t = αβ, they are damped by e−ωI ·α

β where

ωI · αβ

=1 ∓

√1 − 4[α1/2

β· ξ]2

2. (9)

In Figure 1, the above dispersion analysis is summarized graphically. In thetop graph, the nondimensional wave speed, a = α1/2 · a is plotted against thenondimensional wave number ξ = α1/2

β· ξ. In the bottom graph, the damping

rate e−ωI ·αβ is plotted against the nondimensional wave number ξ = α1/2

β· ξ.

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.

1902 B. J. McCartin

3 Control Region Approximation

Figure 2: Computational Stencil

With reference to Figure 2, the Control Region Approximation [4] firstrecasts the damped wave equation, Equation (1), in conservation form:

∇ ·[ −∂u

∂x

α∂u∂t

+ βu

]= 0; ∇ :=

[∂∂x∂∂t

]. (10)

Subsequent integration over the the control region D (the dashed rectangle ofFigure 2) and application of the Divergence Theorem yields

∮∂D

[ −∂u∂x

α∂u∂t

+ βu

]· n dl = 0, (11)

where n is the outward pointing unit normal to ∂D and dl is the correspondingelement of arc length. Denoting the temporal flux by J := α∂u

∂t+βu, Equation

(11) may be decomposed as

−∫ (xj+1/2,tn+1/2)

(xj+1/2,tn−1/2)

∂u

∂xdt+

∫ (xj+1/2,tn+1/2)

(xj−1/2,tn+1/2)J dt

+∫ (xj−1/2,tn+1/2)

(xj−1/2,tn−1/2)

∂u

∂xdt−

∫ (xj+1/2,tn−1/2)

(xj−1/2,tn−1/2)J dt = 0. (12)


As it stands, Equation (12) is exact. However, we now approximate each ofits four component integrals by the midpoint rule [10, p. 305]. Furthermore,we approximate the spatial flux −∂u

∂xby the central difference formula [10, p.

298]. The net result of these approximations is the discrete equation

Jj,n+1/2 = Jj,n−1/2 +Δt

(Δx)2· (uj+1,n − 2uj,n + uj−1,n). (13)

Each particular variant of the Control Region Approximation is now deter-mined by the specific approximations used for the temporal fluxes Jj,n±1/2.

In any event, the Control Region Approximation may be rearranged as

uj,n+1 = X(ν) · uj,n + Y (ν) · uj,n−1 + μ2 · Z(ν) · (uj+1,n − 2uj,n + uj−1,n), (14)

where the Courant number and stiffness parameter are given, respectively, by

μ :=1

α1/2· Δt

Δx; ν :=

β

α· Δt = μ · β

α1/2· Δx, (15)

and the coefficients X, Y, Z determine the particular variant.

3.1 Discrete Dispersion Relation

Seeking Fourier mode solutions of Equation (14) in the form of uj,n =u · eı(ω·nΔt−ξ·jΔx) [8], we require nonzero solutions to the quadratic equation

z2 +Bz + C = 0, (16)

wherez = eıωΔt, B = −[X + 2μ2Z · (cos (ξΔx) − 1)], C = −Y. (17)

Thus, we immediately arrive at the complex discrete dispersion relation

eıωΔt =1

2·[−B ±√

B2 − 4C]. (18)

In the above, (real) ξ is a wave number and (complex) ω may be writtenas ω = ωR + ı ·ωI with ωR an angular frequency and ωI a damping parameter.Denoting the complex amplification factors [11], Equation (18), by z±, we have

ω±RΔt = Arg(z±) (−π < Arg(·) ≤ π), −ω±

I Δt = ln |z±|. (19)

For stability, we require that both roots of Equation (16) lie within the closedunit disc. By the Cohn-Schur criteria [12, p. 67], this will occur if and only if

|C| ≤ 1; |B(1 − C)| ≤ |1 − C2|. (20)

1904 B. J. McCartin

Before proceeding, let us make an important observation concerning thecomputation of the nondimensional wave speed α1/2 · a. If |ωRΔt| > ξΔx then|ωR/ξ| > Δx/Δt, i.e. the Fourier mode

[e−ωIΔt]n · eı(ωR·nΔt−ξ·jΔx) (21)

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

Instead, when |ωRΔt| > ξΔx, we may rewrite Equation (21) as

[−e−ωIΔt]n · eı(n·(ωRΔt±π)−j·ξΔx). (22)

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

α1/2 · a± =

⎧⎨⎩ α1/2 · ω±

R

ξ, if |ωRΔt| ≤ ξΔx,

α1/2 · ω±R

ξ∓ π

ξΔx, if |ωRΔt| > ξΔx,

(23)

and possessing a damping rate

δ =

{e−ωI ·α

β , if |ωRΔt| ≤ ξΔx,

−e−ωI ·αβ , if |ωRΔt| > ξΔx.

(24)

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 equation. Consequently, a thorough discrete dispersion/dissipationanalysis requires that a distinction be made between positively and negativelydamped modes.

Furthermore, the use of Equations (23) and (24) will produce discontinu-ities in wave speed and damping rate, respectively. These discontinuites areessential if physical causality is to be enforced. However, we instead prefer con-tinuity of our plots of wave speed and damping rate and, consequently, leaveit to the reader to properly interpret the modes as being negatively dampedwhen the nondimensional wave speed exceeds unity.

4 Temporal Flux Approximations

As previously noted, individual Control Region Approximation schemesfor the damped wave equation are determined by how they approximate thetemporal fluxes:

J(x, t) :=

[α∂u

∂t+ βu

](x, t). (25)

We next consider specific instances of such temporal flux approximation.


4.1 Exponentially-Fitted Scheme (EF)

We first consider exponentially-fitted temporal fluxes [13, pp. 77-79]. Aswill be shown, this results in a Control Region Approximation scheme whichdisplays desirable limiting behavior for fixed Δx and Δt as ν → 0 and asν → ∞. Indeed, it is optimal in the sense that it is second-order accurate forμ �= 1 and is fourth-order accurate for μ = 1 (see Appendix A). Furthermorewhen compared to straightforward central differencing, it possesses an enlargedregion of stability and superior dispersive and dissipative properties.

From Equation (25), we have that

eβα·t · J(x, t) = α · [(e β

α·t · u)t](x, t). (26)

At xj , integrate Equation (26) with respect to time from tn to tn+1:∫ tn+1

tne

βα·t · J(xj , t) dt = α ·

∫ tn+1

tn[(e

βα·t · u)t](xj , t) dt. (27)

Next, in Equation (27), make an exponential midpoint approximation on theleft-hand side while evaluating the right-hand side exactly:

Jj,n+1/2 ·∫ tn+1

tne

βα·t dt = α · [e β

α·tn+1 · uj,n+1 − e

βα·tn · uj,n]. (28)

Finally, exact evaluation of the integral on the left-hand side of Equation(28) leads directly to the exponential approximation for the temporal flux:

Jj,n+1/2 = β · eβα·tn+1 · uj,n+1 − e

βα·tn · uj,n

eβα·tn+1 − e

βα·tn

= β · uj,n+1 − e−ν · uj,n

1 − e−ν. (29)

An entirely analogous derivation leads directly to the exponential approxima-tion for the temporal flux:

Jj,n−1/2 = β · uj,n − e−ν · uj,n−1

1 − e−ν=

α

Δt· [B(−ν) · uj,n − B(ν) · uj,n−1] , (30)

where B(ν) is the Bernoulli function [13, p. 77].Substitution of Equations (29-30) into Equation (13) and subsequent solu-

tion for uj,n+1 yield the Exponentially-Fitted Control Region Approximationscheme (EF), Equation (14), with:

X(ν) = 1 + e−ν = 1 − Y (ν), (31)

Y (ν) = −e−ν = − Z(ν)

Z(−ν) , (32)

Z(ν) =1 − e−ν

ν=

1

B(−ν) . (33)

1906 B. J. McCartin

4.1.1 Order of Accuracy

Lemma 1 For Equations (31-33):

X(ν) = 2 − ν +1

2ν2 − 1

6ν3 +

1

24ν4 − 1

120ν5 +

1

720ν6 + · · · ,

Y (ν) = −1 + ν − 1

2ν2 +

1

6ν3 − 1

24ν4 +

1

120ν5 − 1

720ν6 + · · · ,

Z(ν) = 1 − 1

2ν +

1

6ν2 − 1

24ν3 +

1

120ν4 + · · · .

These facts imply our main result.

Theorem 1 The local discretization/truncation error (LTE) for the EF scheme,Equations (31-33), is O(ν4) if μ �= 1 and is O(ν6) if μ = 1.

Proof: Simply apply the Order of Accuracy Theorem (Appendix A). ByLemma 1, the local discretization/truncation error is:

LTEj,n = [1

12· (1 − 1

μ2) · α

2

β4· uxxxx]j,n · ν4

+ [1

24· ( 1

μ2− 1) · α

4

β4· utttt +

1

12· ( 1

μ2− 1) · α

3

β3· uttt]j,n · ν5

− 1

720· [2 · ( 1

μ4− 1) · α

6

β6· utttttt + 6 · ( 1

μ4− 1) · α

5

β5· uttttt

+(−15 +10

μ2+

6

μ4) · α

4

β4· utttt

+2 · (−10 +10

μ2+

1

μ4) · α

3

β3· uttt

+(−9 +10

μ2) · α

2

β2· utt]j,n · ν6.

which, for μ = 1, reduces to:

LTEj,n = [− 1

720· α

2

β4· uxxxx]j,n · ν6.

The order of accuracy of the EF scheme follows immediately.

Corollary 1 The EF scheme, Equations (31-33), is second-order accurate forμ �= 1 and is fourth-order accurate for μ = 1.

Proof: This follows directly from Theorem 1 since the formal accuracy is twoorders of magnitude less than that of the LTE. �


4.1.2 Asymptotic Behavior

If we let β → 0 with α, Δx and Δt fixed, then ν → 0 and Equation (14)becomes

uj,n+1 = 2uj,n − uj,n−1 +1

α· (Δt)2

(Δx)2· (uj+1,n − 2uj,n + uj−1,n), (34)

which is the centered time–centered space (CTCS) approximation to the waveequation [14, p. 94].

If we let α → 0 with β, Δx and Δt fixed, then ν → ∞ and Equation (14)becomes

uj,n+1 = uj,n +1

β· Δt

(Δx)2· (uj+1,n − 2uj,n + uj−1,n), (35)

which is the forward time–centered space (FTCS) approximation to the heat(diffusion) equation [14, p. 12].

4.1.3 Stability Region

Since 0 < C = e−ν < 1 for 0 < ν < ∞, the Cohn-Schur criteria, Equation(20), reduces to the simple inequality |B| < |1 + C|. Furthermore, by virtue

of B = −[1 + e−ν + 2μ2 · 1−e−ν

ν· (cos (ξΔx) − 1)

], this leads directly to the

following necessary and sufficient condition for the stability of the EF scheme:

μ ≤√ν

2· 1 + e−ν

1 − e−ν. (36)

If ν << 1 then the right-hand side of the stability inequality, Equation (36),

is approximately√

2−ν2

. Thus, as β → 0 with α fixed, the stability restrictionbecomes μ ≤ 1 which is the familiar one for the CTCS approximation to thewave equation.

On the other hand, if ν >> 1 then the right-hand side of the stabilityinequality, Equation (36), is approximately

√ν2. Introducing r := Δt

β(Δx)2, we

note that r = μ2

ν. Thus, as α → 0 with β fixed, the stability restriction

becomes r ≤ 1/2 which is the familiar one for the FTCS approximation to theheat (diffusion) equation.

These stability results are summarized in Figure 3. Of particular note arethe asymptotic behaviors of the EF scheme as ν → 0 (CTCS scheme for thewave equation) and as ν → ∞ (FTCS scheme for the heat equation). Inparticular, we observe that the stability region for the EF scheme completelyencloses those for the CTCS scheme for the wave equation and the FTCSscheme for the heat (diffusion) equation.

For fixed α and β, we can select any pair (Δx,Δt) such that the cor-responding pair (ν, μ) lies in the stability region of Figure 3. However, as

1908 B. J. McCartin

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ν

μ

EF Scheme

FTCS Scheme

CTCS Scheme

Figure 3: Stability Region

|Δx| + |Δt| → 0, ν → 0 so that μ must be correspondingly reduced so asto stay within the stability limits. In particular, we require a limiting valueof μ ≤ 1, thereby adhering to the CFL necessary condition for convergence[11, pp. 87-89]. Moreover, by the Lax equivalence theorem [11, p. 139], theconsistency and stability of the EF scheme guarantee its convergence.

4.1.4 Dispersive and Dissipative Properties

Setting ξΔx = 0 in Equation (17) and utilizing Equations (31) and (33),we find from Equation (18) that eıω+Δt = 1 and eıω−Δt = e−ν which coincidewith those for the continuous counterpart Equation (9). Setting ξΔx = πin Equation (17) and utilizing Equations (31-33), we find from Equation (18)

that |z±| = e−ıω±I

Δt = e−ν/2 when B2 < 4C which coincide with those for thecontinuous counterpart Equation (6). Thus, for the lowest and highest wavenumbers resolvable on the computational grid, the EF scheme provides exactvalues for the damping rate.

Figures 4-5 provide a graphical summary of the dispersive and dissipativeproperties of the EF scheme with ν = 1.00 for μ = 0.25, 0.50, 0.75, 1.00, 1.01


0.2 0.4 0.6

−0.5

0

0.5

nu = 1.00, mu = 0.25

0.5 1 1.5

−0.5

0

0.5

nu = 1.00, mu = 0.50

0.5 1 1.5 2

−0.5

0

0.5

nu = 1.00, mu = 0.75

0.5 1 1.5 2 2.5 3

−0.5

0

0.5

nu = 1.00, mu = 1.00

0.5 1 1.5 2 2.5 3

−0.5

0

0.5

nu = 1.00, mu = 1.01

0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

nu = 1.00, mu = 1.04

Figure 4: Wave Speed - EF Scheme (ν = 1)

0.2 0.4 0.60.4

0.6

0.8

nu = 1.00, mu = 0.25

0.5 1 1.50.4

0.6

0.8

nu = 1.00, mu = 0.50

0.5 1 1.5 20.4

0.6

0.8

nu = 1.00, mu = 0.75

0.5 1 1.5 2 2.5 30.4

0.6

0.8

nu = 1.00, mu = 1.00

0.5 1 1.5 2 2.5 30.4

0.6

0.8

nu = 1.00, mu = 1.01

0.5 1 1.5 2 2.5 30.4

0.6

0.8

nu = 1.00, mu = 1.04

Figure 5: Damping Rate - EF Scheme (ν = 1)

1910 B. J. McCartin

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ν

CRITICAL WAVE NUMBER (nondimensional)

ξ

Figure 6: Bifurcation Wave Number: μopt = ρ(ν)

and 1.04. There, the solid curves represent the exact values derived from thecontinuous dispersion relation, Equation (3), while the dotted curves derivefrom the discrete dispersion relation, Equation (18). Figure 4 displays thenondimensional wave speed, α1/2 · a, and Figure 5 displays the damping rate,e−ωI ·α

β , for nondimensional wave numbers 0 < ξ = α1/2

β· ξ ≤ ξmax = μπ

ν.

For μ = 0.25, there is a range of stationary discrete modes, one under-damped and the other overdamped, which should be propagating. Moreover,when they do begin to propagate, it is at a diminished wave speed. The sit-uation progressively improves for μ = 0.50, μ = 0.75 and μ = 1.00. However,for μ = 1.04, which is close to the stability boundary of μmax = 1.04018,negatively damped oscillatory modes emerge.

Such negatively damped modes are possible only if z± of Equation (17)return to the real axis as ξ → ξmax. As such, an optimal value of μ may befound by setting to zero the discriminant of Equation (18), D := B2 − 4C,when ξ = ξmax. This results in

μopt = ρ(ν) =1

2·√ν(1 + e−ν + 2e−ν/2)/(1 − e−ν), (37)

which appears as the dashed curve in Figure 3. For ν = 1, this yields μopt =1.01032. As is seen from Figures 4 and 5, μ = 1.01 yields extremely closeagreement for both wave speed and damping rate. In general, for given αand β, the stiffness parameter, ν, may be selected to produce the desireddegree of accuracy and then the Courant number, μ, may be selected usingEquation (37) to produce an optimal match between discrete and continuousdispersive/dissipative properties.


When choosing μopt = ρ(ν), we may compute the nondimensional bifurca-tion wave number, where the modes transition from stationary to propagating,by again setting D = 0 but now finding the solution where ξ �= ξmax. Thisnondimensional bifurcation wave number,

ξb =μopt

ν· cos−1

(6e−ν/2 − e−ν − 1

2e−ν/2 + e−ν + 1

), (38)

is shown if Figure 6. Observe the close agreement with its continuum counter-part ξb = 1/2 (see Figure 1) for a broad range of ν.

Figures 7-8 provide a graphical summary of the dispersive and dissipativeproperties of the EF scheme with ν = 2.00 for μ = 0.25, 0.50, 0.75, 1.00, 1.04and 1.14. For μ = 0.25, there are no propagating modes since ν > 2π·μ. At theupper wave numbers, the less heavily damped mode is underdamped and themore heavily damped mode is underdamped. For μ = 0.50, there is a rangeof stationary discrete modes, one underdamped and the other overdamped,which should be propagating. Moreover, when they do begin to propagate it isat a diminished wave speed. The situation improves as μ is increased to 0.75and further improves when μ = 1.00. However, for μ = 1.14, which is close tothe stability boundary of μmax = 1.1459, negatively damped oscillatory modesemerge. For ν = 2, we have μopt = 1.04018. As is seen from Figures 7 and 8,μ = 1.04 yields extremely close agreement for both wave speed and dampingrate.

Figures 9-10 provide a graphical summary of the dispersive and dissipativeproperties of the EF scheme with ν = 4.00 for μ = 0.25, 0.50, 0.75, 1.00, 1.145and 1.44. For μ = 0.25 and μ = 0.50 , there are no propagating modes sinceν > 2π · μ. At the upper wave numbers, the less heavily damped mode is un-derdamped and the more heavily damped mode is underdamped. For μ = 0.75and μ = 1.00, there is a range of stationary discrete modes, one underdampedand the other overdamped, which should be propagating. Moreover, whenthey do begin to propagate as for μ = 1.00 it is at a diminished wave speed.For μ = 1.44, which is close to the stability boundary of μmax = 1.440357, neg-atively damped oscillatory modes emerge. For ν = 4, we have μopt = 1.145876.As is seen from Figures 9 and 10, μ = 1.145 yields extremely close agreementfor both wave speed and damping rate.

In order to fully appreciate the truly remarkable agreement between thedispersive/dissipative properties of the EF scheme for μ = ρ(ν) and those ofits continuum counterpart, the reader is encouraged to compare it to the OPTscheme of Roe and Arora [9] for the hyperbolic heat conduction equations.However, the reader is duly cautioned that Roe and Arora [9] make bothmisleading and incorrect statements in their analyses [15].

1912 B. J. McCartin

0.1 0.2 0.3−1

−0.5

0

0.5

1nu = 2.00, mu = 0.25

0.2 0.4 0.6

−0.5

0

0.5

nu = 2.00, mu = 0.50

0.2 0.4 0.6 0.8 1

−0.5

0

0.5

nu = 2.00, mu = 0.75

0.5 1 1.5

−0.5

0

0.5

nu = 2.00, mu = 1.00

0.5 1 1.5

−0.5

0

0.5

nu = 2.00, mu = 1.04

0.5 1 1.5

−1

−0.5

0

0.5

1

nu = 2.00, mu = 1.14


0.5 1 1.50.4

0.6

0.8

nu = 2.00, mu = 1.14

0.5 1 1.50.4

0.6

0.8

nu = 2.00, mu = 1.040.5 1 1.5

0.4

0.6

0.8

nu = 2.00, mu = 1.00

0.2 0.4 0.6 0.8 10.4

0.6

0.8

nu = 2.00, mu = 0.750.2 0.4 0.6

0.4

0.6

0.8

nu = 2.00, mu = 0.50

0.1 0.2 0.30.4

0.6

0.8

nu = 2.00, mu = 0.25



0.05 0.1 0.15−1

−0.5

0

0.5

1nu = 4.00, mu = 0.25

0.1 0.2 0.3−1

−0.5

0

0.5

1nu = 4.00, mu = 0.50

0.1 0.2 0.3 0.4 0.5−0.5

0

0.5nu = 4.00, mu = 0.75

0.2 0.4 0.6

−0.5

0

0.5

nu = 4.00, mu = 1.00

0.2 0.4 0.6 0.8

−0.5

0

0.5

nu = 4.00, mu = 1.145

0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

nu = 4.00, mu = 1.44


0.05 0.1 0.150.4

0.6

0.8

nu = 4.00, mu = 0.25

0.1 0.2 0.30.4

0.6

0.8

nu = 4.00, mu = 0.50

0.1 0.2 0.3 0.4 0.50.4

0.6

0.8

nu = 4.00, mu = 0.75

0.2 0.4 0.60.4

0.6

0.8

nu = 4.00, mu = 1.00

0.2 0.4 0.6 0.80.4

0.6

0.8

nu = 4.00, mu = 1.145

0.2 0.4 0.6 0.8 10.4

0.6

0.8

nu = 4.00, mu = 1.44


1914 B. J. McCartin

4.2 Central Difference Schemes (CD[r/s])

Replacing the exponential approximation for the temporal fluxes, Equa-tions (29-30), by their order [r/s] Pade approximants [16, p. 8], produces afamily of central difference approximations. Specifically, the approximation

eν ≈ N(ν)

D(ν)=N0 +N1 · ν + · · ·Nr · νr

D0 +D1 · ν + · · ·Ds · νs, (39)

yields the order [r/s] Central Difference scheme (CD[r/s]), Equation (14), with:

X(ν) = 1 − Y (ν) = 1 +Z(ν)

Z(−ν) , (40)

Y (ν) =D(ν)

D(−ν) · D(−ν) −N(−ν)D(ν) −N(ν)

= − Z(ν)

Z(−ν) , (41)

Z(ν) =1

ν· D(−ν) −N(−ν)

D(−ν) . (42)

These schemes have the advantage that they do not require exponential func-tion evaluations. But what of their accuracy, asymptotic behavior, stabilityand dispersive/dissipative properties?

4.2.1 CD[1/1] Scheme

If in place of the exponential approximations to the temporal fluxes, Equa-tions (29) and (30), we simply employ central difference approximations for ∂u

∂t

and arithmetic mean approximations for u in Equation (25), we arrive at thestraightforward temporal flux approximations:

Jj,n+1/2 =α

Δt· (uj,n+1 − uj,n) +

β

2· (uj,n+1 + uj,n), (43)

Jj,n−1/2 =α

Δt· (uj,n − uj,n−1) +

β

2· (uj,n + uj,n−1). (44)

Observe that this is equivalent to utilizing the [1/1] Pade approximant

eν ≈ 1 + ν/2

1 − ν/2(45)

for the exponential functions appearing in Equations (29) and (30).Substitution of Equations (43-44) into Equation (13) and subsequent solu-

tion for uj,n+1 yield the order [1/1] Central Difference scheme (CD[1/1]), Equa-tion (14), with:

X(ν) =2

1 + ν/2, (46)


Y (ν) =ν/2 − 1

ν/2 + 1, (47)

Z(ν) =1

1 + ν/2. (48)


X(ν) = 2 − ν +1

2ν2 − 1

4ν3 +

1

8ν4 − 1

16ν5 +

1

32ν6 + · · · ,

Y (ν) = −1 + ν − 1

2ν2 +

1

4ν3 − 1

8ν4 +

1

16ν5 − 1

32ν6 + · · · ,

Z(ν) = 1 − 1

2ν +

1

4ν2 − 1

8ν3 +

1

16ν4 + · · · .


Theorem 2 The local discretization/truncation error (LTE) for the CD[1/1]

scheme, Equations (46-48), is O(ν4).


LTEj,n = [1

12· (1 − 1

μ2) · α

4

β4· utttt +

1

6· (1 − 1

μ2) · α

3

β3· uttt

− 1

12μ2· α

2

β2· utt]j,n · ν4,


LTEj,n = [− 1

12μ2· α

2


The order of accuracy of the CD[1/1] scheme follows immediately.

Corollary 2 The CD[1/1] scheme, Equations (46-48), is second-order accu-rate.



uj,n+1 = 2uj,n − uj,n−1 +1

α· (Δt)2

(Δx)2· (uj+1,n − 2uj,n + uj−1,n), (49)

1916 B. J. McCartin



uj,n+1 = uj,n−1 +1

2β· Δt

(Δx)2· (uj+1,n − 2uj,n + uj−1,n), (50)

which is the centered time–centered space (CTCS) approximation to the heat(diffusion) equation. This the notorious Richardson scheme [17, p. 125] whichis unconditionally unstable [14, p. 129]! This is a harbinger of impendingdifficulties for all central difference schemes when α << β (i.e. the singularlyperturbed case).

Since 0 < |C| = |1−ν/21+ν/2

| < 1 for 0 < ν < ∞, the Cohn-Schur criteria,

Equation (20), reduces to the simple inequality |B| < |1+C|. Furthermore, by

virtue of |B| = |2+2μ2·(cos (ξΔx)−1)1+ν/2

|, this leads directly to the following necessaryand sufficient condition for the stability of the CD[1/1] scheme: μ ≤ 1. Thiscoincides with that of the CTCS scheme for the wave equation so there is notan expanded stability domain as ν increases as there was for the EF scheme.

Figures 11-12 provide a graphical summary of the dispersive and dissipativeproperties of the CD[1/1] scheme with ν = 1.00 for μ = 0.25, 0.50, 0.75, 0.96,0.97 and 1.00. In these plots, the solid curves represent the exact values derivedfrom the continuous dispersion relation, Equation (3), while the dotted curvesderive from the discrete dispersion relation, Equation (18). Figure 11 displaysthe nondimensional wave speed, α1/2 · a, and Figure 12 displays the dampingrate, e−ωI ·α



ν.

For all values of μ, the more heavily damped mode has incorrect dampingfor all wave numbers. For μ = 0.25, there is a range of stationary discretemodes, one underdamped and the other overdamped, which should be prop-agating. Moreover, when they do begin to propagate, it is at a diminishedwave speed. The situation progressively improves for μ = 0.50, μ = 0.75 andμ = 0.96. However, for μ = 0.97 negatively damped oscillatory modes firstemerge and persist as we approach the stability limit μ = 1.

Figures 13-14 provide a graphical summary of the dispersive and dissipativeproperties of the CD[1/1] scheme with ν = 2.00 for μ = 0.25, 0.50, 0.75 and 1.00.The utter inadequacy of the CD[1/1] scheme for ν = 2 (and larger) is abundantlyclear. Since in this case X = 1, Y = 0 and Z = 1/2, Equation (16) has a singlenonzero solution which is always real. Thus, there are only stationary modeswhich become negatively damped, and hence oscillatory, when z is negative.

Note that the CD[0/2] and CD[2/0] schemes also provide second-order ac-curacy. However, as neither of these alternative schemes is likely to improveupon the CD[1/1] scheme, we next pursue higher-order accurate central differ-ence schemes.


0.2 0.4 0.6

−0.5

0

0.5

nu = 1.00, mu = 0.25

0.5 1 1.5

−0.5

0

0.5

nu = 1.00, mu = 0.50

0.5 1 1.5 2

−0.5

0

0.5

nu = 1.00, mu = 0.75

0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1nu = 1.00, mu = 0.96

0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

nu = 1.00, mu = 0.97

0.5 1 1.5 2 2.5 3

−1

−0.5

0

0.5

1

nu = 1.00, mu = 1.00

Figure 11: Wave Speed - CD[1/1] Scheme (ν = 1)

0.2 0.4 0.6

0.4

0.6

0.8

nu = 1.00, mu = 0.25

0.5 1 1.5

0.4

0.6

0.8

nu = 1.00, mu = 0.50

0.5 1 1.5 2

0.4

0.6

0.8

nu = 1.00, mu = 0.75

0.5 1 1.5 2 2.5 3

0.4

0.6

0.8

nu = 1.00, mu = 0.96

0.5 1 1.5 2 2.5 3

0.4

0.6

0.8

nu = 1.00, mu = 0.97

0.5 1 1.5 2 2.5 3

0.4

0.6

0.8

1nu = 1.00, mu = 1.00

Figure 12: Damping Rate - CD[1/1] Scheme (ν = 1)

1918 B. J. McCartin

0.1 0.2 0.3−1

−0.5

0

0.5

1nu = 2.00, mu = 0.25

0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

nu = 2.00, mu = 0.50

0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

nu = 2.00, mu = 0.75

0.5 1 1.5

−1.5

−1

−0.5

0

0.5

nu = 2.00, mu = 1.00


0.5 1 1.5

0.2

0.4

0.6

0.8

1nu = 2.00, mu = 1.00

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

nu = 2.00, mu = 0.75

0.2 0.4 0.6

0.4

0.5

0.6

0.7

0.8

0.9

nu = 2.00, mu = 0.50

0.1 0.2 0.3

0.4

0.5

0.6

0.7

0.8

0.9

nu = 2.00, mu = 0.25



4.2.2 CD[2/2] Scheme

Observe that, for μ = 1, the principal local truncation error of the CD[1/1]

scheme, [− 112μ2 · α2

β2 · utt]j,n · ν4, may be approximated by central differenceswithout expanding the computational stencil of Figure 2. As such, we maydevelop a Hermitian method [18, p. 291] which is fourth-order accurate whenμ = 1.

Specifically, we modify the temporal flux approximations, Equations (43-44) as follows:

Jj,n+1/2 =α

Δt·⎡⎣1 +

1

12·(β

α· Δt

)2⎤⎦ · (uj,n+1 −uj,n)+

β

2· (uj,n+1 +uj,n), (51)

Jj,n−1/2 =α

Δt·⎡⎣1 +

1

12·(β

α· Δt

)2⎤⎦ · (uj,n −uj,n−1)+

β

2· (uj,n +uj,n−1). (52)

Observe that this is equivalent to utilizing the [2/2] Pade approximant

eν ≈ 1 + ν/2 + ν2/12

1 − ν/2 + ν2/12(53)

for the exponential functions appearing in Equations (29) and (30).Substitution of Equations (51-52) into Equation (13) and subsequent solu-

tion for uj,n+1 yield the order [2/2] Central Difference scheme (CD[2/2]), Equa-tion (14), with:

X(ν) =2 + ν2/6

1 + ν/2 + ν2/12, (54)

Y (ν) =−1 + ν/2 − ν2/12

1 + ν/2 + ν2/12, (55)

Z(ν) =1

1 + ν/2 + ν2/12. (56)


X(ν) = 2 − ν +1

2ν2 − 1

6ν3 +

1

24ν4 − 1

144ν5 + 0ν6 + · · · ,

Y (ν) = −1 + ν − 1

2ν2 +

1

6ν3 − 1

24ν4 +

1

144ν5 − 0ν6 + · · · ,

Z(ν) = 1 − 1

2ν +

1

6ν2 − 1

24ν3 +

1

144ν4 + · · · .


1920 B. J. McCartin

Theorem 3 The local discretization/truncation error (LTE) for the CD[2/2]

scheme, Equations (54-56) is O(ν4) if ν �= 1 and is O(ν6) if μ = 1.


LTEj,n = [1

12· (1 − 1

μ2) · α

2

β4· uxxxx]j,n · ν4

+ [1

24· ( 1

μ2− 1) · α

4

β4· utttt +

1

12· ( 1

μ2− 1) · α

3

β3· uttt]j,n · ν5

− 1

720· [2 · ( 1

μ4− 1) · α

6

β6· utttttt + 6 · ( 1

μ4− 1) · α

5

β5· uttttt

+(−15 +10

μ2+

6

μ4) · α

4

β4· utttt

+2 · (−10 +10

μ2+

1

μ4) · α

3

β3· uttt

+(−10 +10

μ2) · α

2



LTEj,n = [− 1

720· α

4

β4· utttt − 1

360· α

3

β3· uttt]j,n · ν6.

The order of accuracy of the CD[2/2] scheme follows immediately.

Corollary 3 The CD[2/2] scheme, Equations (54-56), is second-order accuratefor μ �= 1 and is fourth-order accurate for μ = 1.



uj,n+1 = 2uj,n − uj,n−1 +1

α· (Δt)2

(Δx)2· (uj+1,n − 2uj,n + uj−1,n), (57)



uj,n+1 = 2uj,n − uj,n−1, (58)


0.2 0.4 0.6

−0.5

0

0.5

nu = 1.00, mu = 0.25

0.5 1 1.5

−0.5

0

0.5

nu = 1.00, mu = 0.50

0.5 1 1.5 2

−0.5

0

0.5

nu = 1.00, mu = 0.75

0.5 1 1.5 2 2.5 3

−0.5

0

0.5

nu = 1.00, mu = 1.00

0.5 1 1.5 2 2.5 3

−0.5

0

0.5

nu = 1.00, mu = 1.01

0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

nu = 1.00, mu = 1.04


0.2 0.4 0.60.4

0.6

0.8

nu = 1.00, mu = 0.25

0.5 1 1.50.4

0.6

0.8

nu = 1.00, mu = 0.50

0.5 1 1.5 20.4

0.6

0.8

nu = 1.00, mu = 0.75

0.5 1 1.5 2 2.5 30.4

0.6

0.8

nu = 1.00, mu = 1.00

0.5 1 1.5 2 2.5 30.4

0.6

0.8

nu = 1.00, mu = 1.01

0.5 1 1.5 2 2.5 30.4

0.6

0.8

nu = 1.00, mu = 1.04


1922 B. J. McCartin

0.1 0.2 0.3−1

−0.5

0

0.5

1nu = 2.00, mu = 0.25

0.2 0.4 0.6

−0.5

0

0.5

nu = 2.00, mu = 0.50

0.2 0.4 0.6 0.8 1

−0.5

0

0.5

nu = 2.00, mu = 0.75

0.5 1 1.5

−0.5

0

0.5

nu = 2.00, mu = 1.00

0.5 1 1.5

−0.5

0

0.5

nu = 2.00, mu = 1.04

0.5 1 1.5

−1

−0.5

0

0.5

1

nu = 2.00, mu = 1.14


0.1 0.2 0.30.4

0.6

0.8

nu = 2.00, mu = 0.25

0.2 0.4 0.60.4

0.6

0.8

nu = 2.00, mu = 0.50

0.2 0.4 0.6 0.8 10.4

0.6

0.8

nu = 2.00, mu = 0.75

0.5 1 1.50.4

0.6

0.8

nu = 2.00, mu = 1.00

0.5 1 1.50.4

0.6

0.8

nu = 2.00, mu = 1.04

0.5 1 1.50.4

0.6

0.8

nu = 2.00, mu = 1.14



0.05 0.1 0.15−1

−0.5

0

0.5

1nu = 4.00, mu = 0.25

0.1 0.2 0.3−1

−0.5

0

0.5

1nu = 4.00, mu = 0.50

0.1 0.2 0.3 0.4 0.5−0.5

0

0.5nu = 4.00, mu = 0.75

0.2 0.4 0.6

−0.5

0

0.5

nu = 4.00, mu = 1.00

0.2 0.4 0.6 0.8

−0.5

0

0.5

nu = 4.00, mu = 1.145

0.2 0.4 0.6 0.8 1

−0.5

0

0.5

nu = 4.00, mu = 1.44


0.05 0.1 0.150.4

0.6

0.8

nu = 4.00, mu = 0.25

0.1 0.2 0.30.4

0.6

0.8

nu = 4.00, mu = 0.50

0.1 0.2 0.3 0.4 0.50.4

0.6

0.8

nu = 4.00, mu = 0.75

0.2 0.4 0.60.4

0.6

0.8

nu = 4.00, mu = 1.00

0.2 0.4 0.6 0.80.4

0.6

0.8

nu = 4.00, mu = 1.145

0.2 0.4 0.6 0.8 10.4

0.6

0.8

nu = 4.00, mu = 1.44


1924 B. J. McCartin

which is the centered time approximation to the equation ∂2u∂t2

= 0! Thus, theCD[2/2] scheme cannot be expected to produce reliable results when ν >> 1.

Since 0 < |C| = |1−ν/2+ν2/121+ν/2+ν2/12

| < 1 for 0 < ν < ∞, the Cohn-Schur criteria,

Equation (20), reduces to the simple inequality |B| < |1 + C|. Furthermore,

by virtue of |B| = |2+ν2/6+2μ2·(cos (ξΔx)−1)1+ν/2+ν2/12

|, this leads directly to the followingnecessary and sufficient condition for the stability of the CD[2/2] scheme: μ ≤√

1 + ν2/12. Hence, there is an expanded stability domain as ν increases asthere was for the EF scheme. Yet, we must bear in mind that the CD[2/2]

scheme is inconsistent with the damped wave equation in the limit ν → ∞, sothat this enhanced stability is of little practical significance.

Figures 15-16 provide a graphical summary of the dispersive and dissipativeproperties of the CD[2/2] scheme with ν = 1.00 for μ = 0.25, 0.50, 0.75, 1.00,1.01 and 1.04. There, the solid curves represent the exact values derived fromthe continuous dispersion relation, Equation (3), while the dotted curves derivefrom the discrete dispersion relation, Equation (18). Figure 15 displays thenondimensional wave speed, α1/2 · a, and Figure 16 displays the damping rate,e−ωI ·α



ν.

For μ = 0.25, there is a range of stationary discrete modes, one under-damped and the other overdamped, which should be propagating. Moreover,when they do begin to propagate, it is at a diminished wave speed. The sit-uation progressively improves for μ = 0.50, μ = 0.75 and μ = 1.00. Veryclose agreement with the continuum counterpart, comparable to that for theEF scheme, is observed for μ = 1.01. However, for μ = 1.04, which is close tothe stability boundary of μmax = 1.0408, negatively damped oscillatory modesemerge.

Figures 17-18 provide a graphical summary of the dispersive and dissipativeproperties of the CD[2/2] scheme with ν = 2.00 for μ = 0.25, 0.50, 0.75, 1.00,1.04 and 1.14. For μ = 0.25, there are no propagating modes since ν > 2π · μ.At the upper wave numbers, the less heavily damped mode is underdamped andthe more heavily damped mode is underdamped. For μ = 0.50, there is a rangeof stationary discrete modes, one underdamped and the other overdamped,which should be propagating. Moreover, when they do begin to propagate it isat a diminished wave speed. The situation improves as μ is increased to 0.75and further improves when μ = 1.00. However, for μ = 1.14, which is close tothe stability boundary of μmax = 1.1547, negatively damped oscillatory modesemerge. As is seen from Figures 17 and 18, μ = 1.04 yields close agreementwith that of the EF scheme for both wave speed and damping rate except atthe highest wave numbers.

Figures 19-20 provide a graphical summary of the dispersive and dissipativeproperties of the CD[2/2] scheme with ν = 4.00 for μ = 0.25, 0.50, 0.75, 1.00,1.145 and 1.44. For all values of μ, the more heavily damped mode has incorrect


damping for all wave numbers. For μ = 0.25 and μ = 0.50, there are nopropagating modes since ν > 2π ·μ. For μ = 0.75, there is a range of stationarydiscrete modes, one underdamped and the other overdamped, which should bepropagating. For μ = 1.00 and μ = 1.145 there is a band of falsely propagatingmodes. For μ = 1.44, which is close to the stability boundary of μmax =1.527525, negatively damped oscillatory modes emerge. As is seen from Figures19 and 20, there is very poor agreement with the continuum counterpart forboth wave speed and damping rate.

Note that the CD[0/4], CD[1/3], CD[3/1] and CD[4/0] schemes would also pro-vide second-order accuracy for μ �= 1 and fourth-order accuracy for μ = 1. Asnone of these alternative schemes is likely to improve upon the CD[2/2] scheme,we here conclude our examination of higher-order accurate central differenceschemes.

−10 0 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1α = 0.01; β = 1.00

−10 0 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1α = 1.00; β = 1.00

−10 0 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1α = 1.00; β = 0.01

Figure 21: Numerical Approximation - EF Scheme

1926 B. J. McCartin

5 Numerical Example

We now present numerical simulations of the damped wave equation usingthe EF scheme. The initial condition is taken to be u(x, 0) = e−x2

and thenumerical results are on display in Figure 21. The leftmost graph concernsα = 1.00 and β = 0.01 with Δx = .2 and Δt = .2 so that μ = 1. and ν = .002.As such, it displays the primarily propagative behavior of the wave equation.The rightmost graph concerns α = 0.01 and β = 1.00 with Δx = .02 andΔt = .002 so that μ = 1. and ν = .2. As such, it displays the primarilydiffusive behavior of the heat equation. The middle graph concerns α = 1.00and β = 1.00 with Δx = .2 and Δt = .2 so that μ = 1. and ν = .2. As such,it displays a behavior that is a hybrid of that of the wave and heat equations.

6 Conclusion

In this paper, an Exponentially-Fitted Control Region Approximationscheme has been presented for the damped wave equation. Its stability anddispersive/dissipative properties have been thoroughly analyzed and comparedto Control Region Approximation schemes based upon central differences. Theproposed scheme has been shown to be optimal in the sense that it is second-order accurate for μ �= 1 and fourth-order accurate for μ = 1. The benefits offourth-order accurate schemes for wave propagation problems are clearly pre-sented in [19]. The EF scheme has also been shown to perform well regardlessof the relative magnitudes of α and β.

Observe that all of the preceding results are directly applicable to theTelegrapher’s Equation for the voltage along a transmission line [20, p. 74]:

CL · vtt + (RC +GL) · vt +RG · v = vxx, (59)

where R is the resistance, L is the inductance, C is the capacitance and G isthe leakage, all per unit length, of the cable. This follows directly from the factthat v = u · e−R

Lt for RC ≤ GL transforms Equation (59) into Equation (1)

with α = LC and β = GL − RC while v = u · e−GC

t for RC ≥ GL transformsEquation (59) into Equation (1) with α = LC and β = RC −GL.

In closing, the preceding analysis has been confined to the pure initialvalue problem for the source-free damped wave equation. In principle, ControlRegion Approximation schemes may be extended to the initial-boundary valueproblem with a source-term. However, this would considerably complicate thedetails and exposition, especially for higher-order accurate methods.


7 Acknowledgement

The author expresses his sincere gratitude to Mrs. Barbara A. McCartinfor her dedicated assistance in the production of this paper. This paper is ded-icated to the fond memory of Dr. Ronald G. Mosier: Thespian, Mathematicianand Friend.

A Order of Accuracy Conditions

We provide necessary and sufficient conditions for Control Region Approx-imation schemes, Equation (14), to be first-, second-, third- and fourth-orderaccurate approximations to the damped wave equation, Equation (1). Forsecond- and fourth-order accurate schemes, we provide explicit expressions fortheir principal local truncation errors. It is shown that there exist accuracybarriers whereby, for μ �= 1, no such scheme can be third-order accurate and,for μ = 1, no such scheme can be fifth-order accurate. In this sense, the CD[2/2]

and EF schemes developed above are optimal.

Theorem 4 (Order of Accuracy) With reference to Equation (14), define

X(ν) =∞∑

n=0

Anνn, Y (ν) =

∞∑n=0

Bnνn, Z(ν) =

∞∑n=0

Cnνn.

Then, Equation (14) is

1. first-order accurate (consistent) if and only if A0 = 2, A1 = −1, A2 =−B2, B0 = −1, B1 = 1, C0 = 1,

2. second-order accurate if and only if it is first-order accurate and A2 =12, A3 = −B3, B2 = −1

2, C1 = −1

2,

3. third-order accurate if and only if it is second-order accurate and μ =1, A3 = −1

6, A4 = −B4, B3 = 1

6, C2 = 1

6,

4. fourth-order accurate if and only if it is third-order accurate and A4 =124, A5 = −B5, B4 = − 1

24, C3 = − 1

24.

If second-order accurate then the principal local truncation error of Equa-tion (14) is

LTEj,n = [1

12· (1 − 1

μ2) · α

4

β4· utttt +

1

6· (1 − 1

μ2) · α

3

β3· uttt

+1

12· (3 − 12C2 − 1

μ2) · α

2

β2· utt + (B3 − C2) · α

β· ut

1928 B. J. McCartin

−(A4 +B4) · u]j,n · ν4

+ [1

24· ( 1

μ2− 1) · α

4

β4· utttt +

1

12· ( 1

μ2− 1) · α

3

β3· uttt

+1

24· (1 − 12B3 − 24C3) · α

2

β2· utt + (B4 − C3) · α

β· ut

−(A5 +B5) · u]j,n · ν5

+ [1

360· (1 − 1

μ4) · α

6

β6· utttttt +

1

120· (1 − 1

μ4) · α

5

β5· uttttt

− 1

120· (5B2 +

10C2

μ2+

1

μ4) · α

4

β4· utttt

+1

360· (60B3 − 60C2

μ2− 1

μ4) · α

3

β3· uttt

− 1

12· (6B4 +

C2

μ2+ 12C4) · α

2

β2· utt + (B5 − C4) · α

β· ut

−(A6 +B6) · u]j,n · ν6.

If μ �= 1 then Equation (14) cannot be third-order accurate.If fourth-order accurate then the principal local truncation error of Equation

(14) is

LTEj,n = [− 1

720· α

4

β4· utttt − 1

360· α

3

β3· uttt + (

1

144− C2) · α

2

β2· utt

+(B5 − C4) · αβ· ut − (A6 +B6) · u]j,n · ν6.

Equation (14) cannot be fifth-order accurate. It is assumed that the initial dataare sufficiently smooth so that all requisite derivatives exist and are continuous.

Proof: We define the local truncation error (LTE) to be the difference be-tween the left and right hand sides of Equation (14) for an exact solution u ofEquation (1). As such, the formal accuracy of the scheme is two orders lowerthan that of the principal (leading) term of the LTE [19].

By Taylor’s Theorem,

uj,n+1 = [u+α

β· ut · ν +

1

2· α

2

β2· utt · ν2 +

1

6· α

3

β3· uttt · ν3

+1

24· α

4

β4· utttt · ν4 +

1

120· α

5

β5· uttttt · ν5

+1

720· α

6

β6· utttttt · ν6]j,n +O(ν7),

uj,n−1 = [u− α

β· ut · ν +

1

2· α

2

β2· utt · ν2 − 1

6· α

3

β3· uttt · ν3


+1

24· α

4

β4· utttt · ν4 − 1

120· α

5

β5· uttttt · ν5

+1

720· α

6

β6· utttttt · ν6]j,n +O(ν7),

uj+1,n − 2uj,n + uj−1,n = [1

μ2· αβ2

· uxx · ν2 +1

12μ4· α

2

β4· uxxxx · ν4

+1

360μ6· α

3

β6· uxxxxxx · ν6]j,n +O(ν8).

Successive differentiation of Equation (1), yields the identities

uxxxx = α2 · utttt + 2αβ · uttt + β2 · utt,

uxxxxxx = α3 · utttttt + 3α2β · uttttt + 3αβ2 · utttt + β3 · uttt.

Substitution of all of these Taylor series into Equation (14) while utilizing theseidentities and subsequent comparison of like powers of ν directly establishesthe theorem. �

References

[1] B. J. McCartin and R. E. LaBarre, Control Region Approximation, Amer-ican Mathematical Society Abstracts, Vol. 6, No. 3 (1985), 266.

[2] B. J. McCartin and R. E. LaBarre, Numerical Computation on ArbitraryLattices of Points, Abstracts of the International Congress of Mathemati-cians, Berkeley (1986), 297.

[3] B. J. McCartin, Exponential Fitting of the Delayed Recruitment/RenewalEquation, Journal of Computational and Applied Mathematics, Vol. 136(2001), 343-356.

[4] B. J. McCartin, Seven Deadly Sins of Numerical Computation, AmericanMathematical Monthly, Vol. 105, No. 10, (1998), 929-941.

[5] H. F. Weinberger, A First Course in Partial Differential Equations, Wiley,New York, NY, 1965.

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

[7] P. W. Bridgman, Dimensional Analysis, Yale University Press, New Haven,CT, 1931.

1930 B. J. McCartin

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

[9] 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.

[10] S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algo-rithmic Approach, 3rd Edition, McGraw-Hill, New York, NY, 1980.

[11] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differen-tial Equations, Cambridge, 1994.

[12] A. Iserles, A First Course in the Numerical Analysis of Differential Equa-tions, Cambridge, 1996.

[13] B. J. McCartin, Discretization of the Semiconductor Device Equations,in New Problems and New Solutions for Device and Process Modelling, J.J. H. Miller (Editor), Boole Press, Dublin, 1985, 72-82.

[14] G. D. Smith, Numerical Solution of Partial Differential Equations: FiniteDifference Methods, 2nd Edition, Oxford, 1978.

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

[16] G. A. Baker, Jr. and P. Graves-Morris, Pade Approximants, 2nd Edition,Cambridge, 1996.

[17] G. E. Forsythe and W. R. Wasow, Finite-Difference Methods for PartialDifferential Equations, Wiley, New York, NY, 1960.

[18] L. Collatz, The Numerical Treatment of Differential Equations, Springer-Verlag, New York, NY, 1966.

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

[20] H. Bateman, Partial Differential Equations of Mathematical Physics,Dover, New York, NY, 1944.

Received: January 2010

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