Applied Mathematics and Computation 217 (2010) 2631–2638

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Numerical investigation of the generalized lubrication equation

E. Momoniat *, C. Harley, E. AdlemCentre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics,University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

a r t i c l e i n f o

Keywords:Thin filmGeneralized lubricationCrank–Nicolson

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.08.001

* Corresponding author.E-mail addresses: Ebrahim.Momoniat@wits.ac.za

a b s t r a c t

Explicit, implicit–explicit and Crank–Nicolson implicit–explicit numerical schemes forsolving the generalized lubrication equation are derived. We prove that the implicit–explicit and Crank–Nicolson implicit–explicit numerical schemes are unconditionallystable. Numerical solutions obtained from both schemes are compared. Initial curves withboth zero and finite contact angles are considered.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

In this paper we consider numerical solutions of the generalized lubrication equation [14,20]

@h@t¼ � @

@xhn @

3h@x3

!; h P 0; ð1:1Þ

where the non-negative function h(x, t) denotes the height of the free surface above the solid substrate, x is the horizontalcoordinate and t the time. The generalized lubrication Eq. (1.1) models the spreading of a thin film driven by surface tension.The constant n denotes the kind of flow. The case n = 3 corresponds to the surface tension driven spreading of a thinNewtonian fluid [14]. The case n = 1 models the thickness of a thin film in a Hele-Shaw cell [10]. The fourth-order nonlinearpartial differential Eq. (1.1) is degenerate because the coefficient of the highest derivative tends to zero as h tends to zero.Non-negative solutions admitted by (1.1) have been investigated by Beretta et al. [4]. Smyth and Hill [24] investigate sim-ilarity solutions of (1.1) which lead to waiting time solutions. A generalisation of this similarity solution has been obtained inMomoniat et al. [19] using a Lie group approach and is given by

hðx; tÞ ¼ j xþ a1ð Þ4

t þ a2

!1=3

; ð1:2Þ

where a1 and a2 are constants and j = �81/56. They also investigate asymptotic solutions valid for small 0 < n� 1. Furthersimilarity solutions have been investigated by Witelski [28] and Bernoff and Witelski [6]. Investigations into the behaviour ofanalytical solutions of (1.1) have been undertaken by Bernis et al. [5] and Bowen and King [9]. Oron et al. [22] discuss appli-cations of thin film flow. Myers [20] presents a review of analytical results relating to various investigations of the general-ized lubrication equation. Asymptotic solutions to (1.1) for dip coating, draining flows, contact lens flows and spreadingdrops are discussed.

In this paper we derive an explicit, implicit–explicit and a Crank–Nicolson implicit–explicit numerical scheme to deter-mine numerical solutions of (1.1). The degenerate nature of (1.1) makes imposing boundary conditions at the contact line

. All rights reserved.

, Ebrahim.Momoniat@gmail.com (E. Momoniat).

2632 E. Momoniat et al. / Applied Mathematics and Computation 217 (2010) 2631–2638

h = 0 very challenging. Bertozzi [7,8] focuses specifically on the contact line singularity and other singularities that can de-velop in solutions of (1.1). Existence and the behaviour of solutions of (1.1) based on entropy estimates are investigated byPasso et al. [23]. Numerical schemes based on these entropy estimates in higher spatial dimensions have been considered byGrün [15]. Finite element approximations to (1.1) are investigated by Barrett et al. [2] and An et al. [1]. Zhornitskaya andBertozzi [29] develop finite difference schemes that ensure h P 0. The finite difference schemes satisfy conservation of mass,conservation of surface energy dissipation and a nonlinear energy dissipation. Ha et al. [16] have investigated numericalsolutions of (1.1) coupled with a nonlinear source term @/@x(u2�u3). Ha et al. [16] solve the nonlinear diffusion term givenby (1.1) using a Crank–Nicolson scheme. In this paper we show that an implicit–explicit scheme is unconditionally stableand is more efficient for solving the nonlinear diffusion Eq. (1.1).

In this paper we firstly derive an explicit numerical scheme for solving (1.1). We show that this explicit numerical schemeis conditionally stable. We then derive an implicit–explicit scheme. Implicit–explicit schemes for solving hyperbolic partialdifferential equations have been investigated by McGuire and Morris [17]. Taha and Ablowitz [26] in particular show howthese implicit–explicit schemes can be used to obtain numerical solutions of the nonlinear Schrodinger equation. We provethe implicit–explicit scheme is unconditionally stable. We then derive a Crank–Nicolson [11] implicit–explicit scheme. Weprove that this Crank–Nicolson implicit–explicit scheme is unconditionally stable. Numerical solutions obtained from theimplicit-scheme and the Crank–Nicolson implicit–explicit scheme are applied to initial curves with both zero and finite con-tact angles. The advantage of the approach taken in this paper is that we obtain numerical schemes that are unconditionallystable. The numerical schemes are evaluated by solving a linear system of equations. This makes the numerical schemes wehave derived easy to implement. The accuracy of our numerical solutions depend on the steplengths in the time and spatialdirections.

The paper is divided up as follows: In Section 2 we derive an explicit, implicit–explicit and a Crank–Nicolson implicit–explicit numerical scheme. We show that the explicit numerical scheme is conditionally stable while the implicit–explicitand Crank–Nicolson implicit–explicit numerical schemes are unconditionally stable. In Section 3 we implement the impli-cit–explicit and Crank–Nicolson implicit–explicit numerical schemes by solving a linear system of equations. Concluding re-marks are made in Section 4.

2. Stability analysis

In this section we derive and investigate the stability of numerical schemes to solve (1.1). Derivatives in (1.1) are approx-imated using finite differences. We approximate the time derivative by a forward difference approximation

@hi;j

@t� hi;jþ1 � hi;j

Dtþ O Dtð Þ: ð2:1Þ

A central difference approximation to the first and second derivatives is given by

@hi;j

@x� hiþ1;j � hi�1;j

2Dxþ O Dx2

� �; ð2:2Þ

@2hi;j

@x2 �hiþ1;j � 2hi;j þ hi�1;j

Dx2 þ O Dx2� �; ð2:3Þ

where hi,j = h(xi, tj). The domain x 2 [�a,a] is divided into m + 1 intervals of equal length. We define xi+1 = �a + iDx whereDx = 2a/m. We approximate the third derivative as a first derivative of the second derivative to obtain the approximation

@3hi;j

@x3 �hiþ2;j � 2hiþ1;j þ 2hi�1;j � hi�2;j

2Dx3 þ O Dx2� �: ð2:4Þ

To overcome the contact line singularity an appropriate boundary condition is given by

h �a; tð Þ ¼ �; �� 1; ð2:5Þ

where � is the height of the precursor film [18,13,21,25,8]. To close the problem we include the additional boundaryconditions

@h �a; tð Þ@x

¼ 0: ð2:6Þ

The boundary condition (2.6) implies that the contact angle, the angle made by the film with the substrate, is zero. This im-plies that complete wetting takes place. We impose the normalizing condition [27,12]

hð0;0Þ ¼ 1; ð2:7Þ

i.e. the height of the film at the center is one. An appropriate initial profile that satisfies the boundary conditions (2.5), (2.6)and (2.7) on a large enough finite domain is given by

hðx;0Þ ¼ exp �x2� �: ð2:8Þ

If the domain under consideration is given by [�a,a] then the height of the precursor film is given by � = exp(�a2).

E. Momoniat et al. / Applied Mathematics and Computation 217 (2010) 2631–2638 2633

2.1. Explicit approximation

Approximating the outer spatial derivative in (1.1) first we obtain

@hi;j

@t¼ � 1

2Dxhn

iþ1;j@3hiþ1;j

@x3 � hni�1;j

@3hi�1;j

@x3

!þ O Dx2� �

: ð2:9Þ

Note that both the nonlinear term hn and the linear derivative @3h/@x3 are approximated explicitly. Substituting (2.1) and(2.9) into (1.1) and dropping the truncation error term we obtain the difference equation

hi;jþ1 ¼ hi;j �Dt

4Dx4 hniþ1;j hiþ3;j � 2hiþ2;j þ 2hi;j � hi�1;j� �

� hni�1;j hiþ1;j � 2hi;j þ 2hi�2;j � hi�3;j� �h i

: ð2:10Þ

Setting

ai;j ¼ khni;j; k ¼ Dt

4Dx4 ð2:11Þ

and dropping the truncation errors we write (2.10) as:

hi;jþ1 ¼ hi;j � aiþ1;j hiþ3;j � 2hiþ2;j þ 2hi;j � hi�1;j� �

þ ai�1;j hiþ1;j � 2hi;j þ 2hi�2;j � hi�3;j� �

: ð2:12Þ

To perform a von Neumann stability analysis we set

hi;j ¼ UjeIxiDx ð2:13Þ

in (2.12) where I2 = �1 and x a constant. Substituting (2.13) into (2.12) we obtain

Ujþ1

Uj¼ 1� baiþ1;j þ cai�1;j; ð2:14Þ

where

b ¼ e3IxDx � 2e2IxDx þ 2� e�IxDx ð2:15Þ

and

c ¼ eIxDx � 2þ 2e�2IxDx � e�3IxDx: ð2:16Þ

The stability criteria is usually specified in terms of the amplification factor g where

gj j ¼ Ujþ1

Uj

����������: ð2:17Þ

Taking absolute values on the left and right hand side of (2.14) we get

gj j ¼ Ujþ1

Uj

���������� ¼ 1� baiþ1;j þ cai�1;j

�� ��: ð2:18Þ

Imposing the stability criteria

gj j < 1 ð2:19Þ

on (2.18) we obtain the expression

1� baiþ1;j þ cai�1;j

�� �� < 1: ð2:20Þ

The boundary condition (2.7) at the center of the film places an upper bound on the value of ai�1,j 6 k and ai+1,j 6 k wherehi�1,j = hi+1,j = 1. The assumption that hi�1,j = hi+1,j = 1 is a valid one as there is no source term to increase the height of the film.Since surface tension is the only driving force the height of the film will decrease until a steady profile is reached [27]. Theinequality (2.20) simplifies to

1þ k c� bð Þj j < 1: ð2:21Þ

From definitions (2.15) and (2.16) we find that

b� c ¼ 16 sin2 xDxð Þ sin2 xDx=2ð Þ: ð2:22Þ

The inequality (2.21) simplifies to

1� 16k sin2 xDxð Þ sin2 xDx=2ð Þ��� ��� < 1: ð2:23Þ

2634 E. Momoniat et al. / Applied Mathematics and Computation 217 (2010) 2631–2638

Given the bound j sin2ðxDxÞ sin2ðxDx=2Þj 6 1 on the trigonometric function we simplify (2.23) to

0 < k < 1=8: ð2:24Þ

From (2.11) we express the inequality (2.24) as:

0 < Dt < Dx4=2: ð2:25Þ

We can thus conclude that the explicit scheme (2.12) is conditionally stable as the stability criteria (2.19) holds provided(2.25) is true.

The stability criteria (2.25) for the explicit scheme suggests that it is impractical to use the explicit scheme (2.12) to deter-mine numerical solutions of (1.1) as a large number of iterations are required to obtain solutions for reasonable times. As anexample, if we consider Dx = 0.1 then we require Dt < 0.0005 for stability. To obtain solutions at t = 1.0 more than 20000 iter-ations are required. A spatial step ofDx = 0.1 gives a spatial truncation error ofOð10�2Þwhich is too big to give reliable solutions.

2.2. Implicit–explicit scheme

An implicit–explicit scheme for solving (1.1) can be obtained by writing (2.9) as:

@hi;j

@t¼ � 1

2Dxhn

iþ1;j@3hiþ1;jþ1

@x3 � hni�1;j

@3hi�1;jþ1

@x3

!þ O Dx2

� �: ð2:26Þ

Substituting (2.1) and (2.4) into (2.26), re-arranging terms and dropping the truncation error term we obtain

hi;jþ1 þ aiþ1;j hiþ3;jþ1 � 2hiþ2;jþ1 þ 2hi;jþ1 � hi�1;jþ1� �

� ai�1;j hiþ1;jþ1 � 2hi;jþ1 þ 2hi�2;jþ1 � hi�3;jþ1� �

¼ hi;j: ð2:27Þ

We perform a linear stability analysis by substituting (2.13) into (2.27) to obtain

Ujþ1 1þ baiþ1;j � cai�1;j� �

¼ Uj: ð2:28Þ

We again impose the conditions ai+1,j 6 k and ai�1,j 6 k, where hi+1,j = hi�1,j = 1. Then from (2.28)

Ujþ1

Uj

!1þ k b� cð Þ½ � ¼ 1: ð2:29Þ

The stability criteria (2.19) combined with (2.22) simplifies (2.29) to

1

1þ 16k sin2 xDxð Þ sin2 xDx=2ð Þ

���������� < 1: ð2:30Þ

The condition (2.30) is satisfied for k > 0 and hence the implicit–explicit scheme (2.27) is unconditionally stable as the cri-teria (2.19) is always satisfied.

2.3. Crank–Nicolson implicit–explicit scheme

A Crank–Nicolson [11] improvement on the scheme (2.26) is given by

@hi;j

@t¼ � 1

4Dxhn

iþ1;j@3hiþ1;jþ1

@x3 þ @3hiþ1;j

@x3

!� hn

i�1;j@3hi�1;jþ1

@x3 þ @3hi�1;j

@x3

! !þ O Dx2� �

: ð2:31Þ

Substituting (2.1) and (2.4) into (2.31), re-arranging terms and dropping the truncation error term we obtain

hi;jþ1 þ12aiþ1;j hiþ3;jþ1 � 2hiþ2;jþ1 þ 2hi;jþ1 � hi�1;jþ1

� �� 1

2ai�1;j hiþ1;jþ1 � 2hi;jþ1 þ 2hi�2;jþ1 � hi�3;jþ1

� �¼ hi;j �

12aiþ1;j hiþ3;j � 2hiþ2;j þ 2hi;j � hi�1;j

� �þ 1

2ai�1;j hiþ1;j � 2hi;j þ 2hi�2;j � hi�3;j

� �: ð2:32Þ

We perform a linear stability analysis by substituting (2.13) into (2.32) to obtain

Ujþ1 1þ 12

baiþ1;j � cai�1;j� �� �

¼ Uj 1� 12

baiþ1;j � cai�1;j� �� �

: ð2:33Þ

As before we impose the conditions ai+1,j 6 k and ai�1,j 6 k where hi+1,j = hi�1,j = 1. The stability criteria (2.19) combined with(2.22) simplifies (2.33) to

1� 8k sin2 xDxð Þ sin2 xDx=2ð Þ1þ 8k sin2 xDxð Þ sin2 xDx=2ð Þ

���������� < 1: ð2:34Þ

The condition (2.34) is satisfied for k > 0 and we can therefore conclude that the Crank–Nicolson implicit–explicit scheme(2.32) is unconditionally stable as the criteria (2.19) is always satisfied.

E. Momoniat et al. / Applied Mathematics and Computation 217 (2010) 2631–2638 2635

In the next section we compare numerical solutions to (1.1) obtained from the implicit–explicit scheme (2.27) and theCrank–Nicolson implicit–explicit scheme (2.32).

3. Comparison of numerical schemes

In the previous section we have indicated that the explicit scheme is impractical to implement due to the large number ofiterations that would be required to obtain accurate results. In this section we show how the implicit–explicit scheme (2.27)and the Crank–Nicolson implicit–explicit scheme (2.32) can be implemented using linear iterative schemes.

The implicit–explicit scheme (2.27) is written as the linear system

Ahðjþ1Þ ¼ hðjÞ; ð3:1Þ

where

A¼

1 0 0 0 0 0 0 0 ��� an�5 �2an�5 �an�3 ð1þ2an�5þ2an�4Þ �an�5 �2an�3 an�3 0�a2 ð1þ2a2Þ 0 �2a2 a2 0 0 0 ��� 0 an�4 �2an�4 �an�2 ð1þ2an�4þ2an�2Þ �an�4 �2an�2 an�2

�2a1 a1�a3 ð1þ2a1þ2a3Þ �a1 �2a3 a3 0 0 ��� 0 0 an�3 �2an�3 �an�1 ð1þ2an�3þ2an�1Þ an�1�an�3 �2an�1

a2 �2a2 �a4 ð1þ2a2þ2a4Þ �a2 �2a4 a4 0 ��� 0 0 0 an�2 �2an�2 0 ð1þ2an�2Þ �an�2

0 a3 �2a3 �a5 ð1þ2a3þa5Þ �a3 �2a5 a5 ��� 0 0 0 0 0 0 0 1

26666664

37777775

ð3:2Þ

and

hðjÞ ¼ hðjÞ0 ;hðjÞ1 ; . . . ; hðjÞn�1;h

ðjÞn

h iT: ð3:3Þ

We have imposed the boundary conditions (2.5) and (2.6) as:

hjþ10 ¼ hj

0; hðjþ1Þn ¼ hðjÞn ð3:4Þ

and

hðjÞ�1 ¼ hðjÞ1 ; hðjÞ�2 ¼ hðjÞ2 ; hðjÞnþ1 ¼ hðjÞn�1; hðjÞnþ2 ¼ hðjÞn�2; ð3:5Þ

respectively.The Crank–Nicolson implicit–explicit scheme (2.32) is implemented as:

Ahðjþ1Þ ¼ BhðjÞ; ð3:6Þ

where A is given by (3.2) and

B¼

1 0 0 0 0 0 0 0 �� � �an�5 2an�5 an�3 ð1�2an�5�2an�4Þ an�5 2an�3 �an�3 0a2 ð1�2a2Þ 0 2a2 �a2 0 0 0 �� � 0 �an�4 2an�4 an�2 ð1�2an�4�2an�2Þ an�4 2an�2 �an�2

2a1 �a1þa3 ð1�2a1�2a3Þ a1 2a3 �a3 0 0 �� � 0 0 �an�3 2an�3 an�1 ð1�2an�3�2an�1Þ �an�1þan�3 2an�1

�a2 2a2 a4 ð1�2a2�2a4Þ a2 2a4 �a4 0 �� � 0 0 0 �an�2 2an�2 0 ð1�2an�2Þ an�2

0 �a3 2a3 a5 ð1�2a3�a5Þ a3 2a5 �a5 �� � 0 0 0 0 0 0 0 1

26666664

37777775:

ð3:7Þ

We use MATHEMATICA to solve the systems (3.1) and (3.6). In the case of (3.1) the system is evaluated as:

hðjþ1Þ ¼ A�1hðjÞ; ð3:8Þ

while for (3.6) the system is evaluated as:

hðjþ1Þ ¼ A�1BhðjÞ: ð3:9Þ

Both the implicit–explicit scheme and the Crank–Nicolson implicit–explicit scheme converge to the same solution. To testwhich approach is the most efficient we consider steady solutions to (1.1). For steady solutions we have hi,j+1 = hi,j. Theimplicit–explicit scheme (3.1) is then evaluated as:

A hðjÞ� �zþ1 ¼ hðjÞ

� �z; ð3:10Þ

where z is the iteration number and ðhðjÞÞ0 is an initial guess to the steady profile. Similarly, the Crank–Nicolson implicit–explicit scheme (3.6) is written as:

A hðjþ1Þ� �zþ1 ¼ B hðjÞ� �z

: ð3:11Þ

Note here that both (3.10) and (3.11) need to be iterated until the convergence criteria

max hðjÞ� �zþ1 � hðjÞ

� �z��� ��� < d; ð3:12Þ

where d is the tolerance is satisfied. We use the zero-contact exponential curve (2.8) as our initial guess. Choosing d = 0.01 wetabulate the number of iterations to convergence to the tolerance d in Table 1. From Table 1 we note that the Crank–Nicolsonscheme is more costly as it takes longer to converge.

Table 1Table comparing number of iterations required to reach a tolerance of d = 0.01. We havechosen m = 500, Dt = 0.1, a = 4 and Dx = 2a/m = 0.016. The initial curve is given byh(x, 0) = exp(�x2).

n Implicit–explicit Crank–Nicolson implicit–explicit

2 10 143 8 12

Fig. 1. Numerical solutions admitted by (1.1) obtained from the implicit–explicit scheme (3.1) for n = 1, n = 2 and n = 3 at t = 0.5. We have chosen m = 500,Dt = 0.1, a = 4 and Dx = 2a/m = 0.016. The initial curve is given by h(x,0) = exp(�x2).

2636 E. Momoniat et al. / Applied Mathematics and Computation 217 (2010) 2631–2638

As both the implicit–explicit scheme and the Crank–Nicolson implicit–explicit scheme converge to the same solution it isnot necessary to plot the output of both schemes. In Fig. 1 we plot the numerical solution obtained from the implicit–explicitscheme (3.1) for n = 1, n = 2 and n = 3. We note from Fig. 1 that as n increases the film height increases at a fixed time.

We next consider initial profiles with finite contact angle. For the arbitrary initial profile

hðx;0Þ ¼ f ðxÞ ð3:13Þ

the appropriate boundary conditions are given by [20]

@h �a; tð Þ@x

¼ df �að Þdx

;@3h �a; tð Þ

@x3 ¼ d3f �að Þdx3 : ð3:14Þ

The precursor film height is imposed as:

hð�a; tÞ ¼ f ð�aÞ: ð3:15Þ

Choosing a = 1 we have

f ð�1Þ ¼ �; df ð�1Þdx

¼ � tan /; f ð0Þ ¼ 1; ð3:16Þ

where / is the contact angle. A fourth-order polynomial that satisfies (3.16) is given by

f ðxÞ ¼ 1þ 2 �� 1ð Þx2 � �� 1ð Þx4 � 12

x2 x2 � 1� �

tan /: ð3:17Þ

Imposing the boundary conditions (3.14) as:

hð�1; tÞ ¼ �; @hð�1; tÞ@x

¼ df ð�1Þdx

;@3hð�1; tÞ

@x3 ¼ d3f ð�1Þdx3 ; ð3:18Þ

we end up solving the system

Ahðjþ1Þ ¼ ChðjÞ; ð3:19Þ

Fig. 2. Numerical solutions admitted by (1.1) obtained from the implicit–explicit scheme (3.19) for n = 3 at t = 0.1. We have chosen m = 250, � = 0.01,Dt = 0.1, a = 1 and Dx = 2a/m = 0.008. The initial curve is given by hðx;0Þ ¼ 1þ 2ð�� 1Þx2 � ð�� 1Þx4 � 1

2 x2ðx2 � 1Þ tan /.

E. Momoniat et al. / Applied Mathematics and Computation 217 (2010) 2631–2638 2637

where

C ¼

1 0 0 0 0 0 � � � 0 0 0 0 0 0 0�1 1 0 0 0 0 � � � 0 0 0 0 0 0 0�1 2 0 �2 1 0 � � � 0 0 0 0 0 0 00 0 0 1 0 0 � � � 0 0 0 0 0 0 00 0 0 0 1 0 � � � 0 0 0 0 0 0 00 0 0 0 0 1 � � � 0 0 0 0 0 0 0... ..

. ... ..

. ... ..

.� � � ..

. ... ..

. ... ..

. ...

0 0 0 0 0 0 � � � 1 0 0 0 0 0 00 0 0 0 0 0 � � � 0 1 0 0 0 0 00 0 0 0 0 0 � � � 0 0 1 0 0 0 00 0 0 0 0 0 � � � 0 0 1 2 0 �2 10 0 0 0 0 0 � � � 0 0 0 0 0 �1 10 0 0 0 0 0 � � � 0 0 0 0 0 0 1

26666666666666666666666666664

37777777777777777777777777775

: ð3:20Þ

The system (3.19) is the implicit–explicit scheme for finite contact angle /. In Fig. 2 we plot the numerical solution obtainedfrom (3.19) for different values of /.

Concluding remarks are made in the next section.

4. Concluding remarks

In this paper we have derived unconditionally stable numerical schemes for solving the lubrication Eq. (1.1). The numer-ical schemes are based on implicit–explicit and Crank–Nicolson implicit–explicit finite difference approximations. Thenumerical schemes have a truncation error OðDt þ Dx2Þ. Both numerical schemes are easily implemented using linear iter-ation. The correctness of the results obtained in this paper are verified by the fact that both the implicit–explicit and Crank–

2638 E. Momoniat et al. / Applied Mathematics and Computation 217 (2010) 2631–2638

Nicolson implicit–explicit finite difference schemes give the same solutions. We have shown that the implicit–explicitscheme is more efficient than the Crank–Nicolson scheme. This is an improvement on the work of Ha et al. [16]. The finitedifference schemes presented in this work are easier to implement than the schemes derived by Zhornitskaya and Bertozzi[29]. The accuracy of the numerical schemes can be improved by considering a higher order approximation to the third-orderderivative given by

@3hi;j

@x3 ��hiþ3;j þ 8hiþ2;j � 13hiþ1;j þ 13hi�1;j � 8hi�2;j þ hi�3;j

8Dx3 þ O Dx4� �: ð4:21Þ

This improves the truncation error to OðDt þ Dx4Þ. The resulting matrices are however more complicated and would requiremore computation to solve. Another possible approach that reduces a nonlinear partial differential equation to a linear sys-tem is the quasi-linearisation approach introduced by Bellman and Kalaba [3]. This approach does however need iterationsto take place at each time step.

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