IL NUOVO CIMENTO

VOL. 110 B, N. 12

Dicembre 1995

The Boundary-Forced Regularised Long-Wave Equation (*). L. R. T. GARDNER(I) and I. DAG(2) (1) School of Mathematics, University of Wales - Bangor, Gwynedd LL57 1UT, UK (2) Gazi Osman Pasa Universitesi, Matematik Bolumu - Eskisehir, Turkey (ricevuto il 22 Maggio 1995; approvato fl 17 Luglio 1995)

Summary. - - The regularised long-wave equation is solved numerically by a B-spline finite-element method involving a Galerkin approach with cubic B-spline finite-elements so that the dependent variable and its first derivative are continuous throughout the solution range. Time integration of the resulting system of ordinary differential equations is effected using a Crank-Nicolson approximation. The numerical scheme is validated by studying the motion of a single solitary wave. The amplitude, velocity and position of the wave are well represented and the method shows good conservation. The effect of inhomogeneous boundary conditions on the numerical solution is explored, and found to result in the establishment of a source of solitary waves. PACS 02.60.Cb - Numerical simulation; solution of equations. PACS 47.35 - Hydrodynamic waves.

1. -

Introduction.

A numerical solution of the regularised long-wave (RLW) equation in the form [1-3] (1)

ut + Ux + aUUx - buxxt = 0,

where the subscripts t and x denote differentiation and a, b are positive real parameters, has been proposed recently[l]. Inhomogeneous Dirichlet boundary conditions u = u0 at x = Xo and u = 0 at x----XN are used to model wave forcing conditions. Peregrine[2], Eilbeck and McGuire[3] and others have set up finite-difference schemes for the solution of the R L W equation, some of which have been critically examined by Bona et al. [4]. An algorithm based on cubic splines within a splitting scheme has recently been discussed by Jain et al. [5]. In an earlier paper [6] we have used the B-spline finite-element method with cubic

(*) The authors of this paper have agreed to not receive the proofs for correction. 1487

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L . R . T . GARDNER and i. DAG

B-spline finite elements and periodic boundary conditions to model the interaction of R L W solitary waves [6]. Here we follow a similar technique using cubic B-spline finite elements within Galerkin's method to construct a stable numerical algorithm of high accuracy for the solution of the R L W equation with boundary forcing conditions. The results of our simulations are shown to compare favourably with [1] and [5]. 2. - Validation experiment: solitary-wave migration. In this experiment the boundary conditions are u --~ 0 as x -~ Xo, XN at the ends of the range. Taking a = b = 1 in (1) leads to the single-solitary-wave solution of the R L W equation [2] (2)

u(x, t) = 3csech 2 (k[x - vt - Xo]),

where (3)

k2_

c 4(1 + c)

and

v=l+c.

These solutions satisfy three conservation laws [7],

C1 = I u d x ,

(4)

C2 = I [u2 + b ( u x ) 2 ] d x '

Ca = I [ua + 3 u 2 ] d x " The invariants C1, C2 and C3 are monitored to check the conservation of the numerical algorithm. The Le and L~ error norms

iluo ct

h

oxac -

u~ 12]1/2

and max

(5)

[Iu e x a c t U n II~ =

J l u~xact -

V

I,

which measure, respectively, the mean error and maximum error in the numerical solution, are used to show how well the numerical scheme predicts the position and amplitude of the solitary-wave during its migration across the mesh. In the following simulations eq. (2) is taken as initial condition with range - 4 0 ~< ~< x ~< 60, h -- 0.1, dt = 0.1 and x0 = 0, with c = 0.1 and c = 0.03 so that the solitary waves have amplitudes 0.3 and 0.09 and comparison can be made with earlier work [5]. The simulations are run to time t = 20 and the L2 and L~ error norms and the invariants C1, C2, C3 recorded in tables I and II. The L | error norm may be estimated from fig. 1 and 2 of[5] as L~ = 5.6.10 -3 for c = 0.03, and L~ >i 68.10 -3 for c = 0.1 at time t = 20. These errors are considerably larger than the values L ~ = 0.432.10 .3 for

THE BOUNDARY-FORCED

REGULARISED

LONG-WAVEEQUATION

1489

TABLE I. - Invariantsfor single solitary wave, amplitude = 0.3, h = 0.1, At = 0.1, - 4 0 ~
C1

C2

C3

Lz'IO 3

L~'103

0 2 4 6 8 10 12 14 16 18 20

3.97993 3.97993 3.97993 3.97993 3.97993 3.97993 3.97993 3.97993 3.97992 3.97991 3.97989

0.810462 0.810461 0.810461 0.810461 0.810462 0.810461 0.810461 0.810462 0.810460 0.810461 0.810462

2.57901 2.57900 2.57900 2.57900 2.57900 2.57900 2.57900 2.57901 2.57900 2.57900 2.57901

0.000 0.023 0.046 0.068 0.089 0.112 0.134 0.155 0.176 0.197 0.217

0.000 0.008 0.018 0.026 0.035 0.044 0.052 0.061 0.069 0.076 0.084

TABLE II. - Invariants for single solitary wave, amplitude = 0.09, h = 0.1, At = 0.1, -40 ~
C1

C2

C3

L2" 103

L ~. 10~

0 2 4 6 8 10 12 14 16 18 20

2.10707 2.10702 2.10712 2.10708 2.10698 2.10686 2.10672 2.10652 2.10621 2.10574 2.10503

0.127303 0.127302 0.127302 0.127302 0.127302 0.127303 0.127302 0.127303 0.127302 0.127302 0.127302

0.388805 0.388804 0.388804 0.388805 0.388805 0.388805 0.388805 0.388805 0.388804 0.388803 0.388802

0.000 0.334 0.402 0.456 0.487 0.498 0.501 0.502 0.507 0.513 0.527

0.000 0.274 0.223 0.217 0.212 0.207 0.202 0.197 0.214 0.304 0.432

c = 0.03, and L ~ I> 0.084.10 .3 for c = 0.1 obtained at t = 20 with the p r e s e n t numerical method. The positions and profiles of the two solitary waves at t = 20 are indistinguishable from the analytic solution; a fact emphasized b y the smallness of the L2 e r r o r norms. F o r the solitary wave, of amplitude 0.3, the quantities C1, C2 and C3 change b y less than 2 . 1 0 - 3 % , 5-10 -3 % and 8" 10-4%, respectively, b y time t = 20. F o r the smaller wave, of amplitude 0.09, the changes are somewhat similar being less than 0.2%, 8 - 1 0 - 4 % , and 5 . 1 0 - 4 % , respectively. Thus the proposed method shows good conservation. We also compare the p e r f o r m a n c e of the p r e s e n t scheme with t h a t of the scheme used b y Chang et al. [1] and a Crank-Nicolson (CN) scheme [1], for a larger-amplitude single-solitary wave simulation: see table I I I w h e r e all e r r o r s are relative errors at t = 100. The wave has amplitude 3, h = 0.5, At = 0.5 and 8 iterations are used per time step with both Chang's scheme and the p r e s e n t one while 10 iterations are used with the C N scheme. The p r e s e n t scheme shows a b e t t e r result than either of the others except in the conservation of C1. I f the size of the time step is reduced to At = 0.1 and the n u m b e r of iterations p e r time step to 2 so t h a t a simulation takes roughly the

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L.R.T.

GARDNER

and

I. DAG

TABLE I I I . - Single solitary wave, amplitude 3: results and relative errors at t = 100. Method exact solution

Wave amplitude

Error %

3.0000

Wave velocity

Error %

2,0000

%

C2 error %

Ca error %

16.9706

31.3352

183.2821

C1 error

h = 0.5, At = 0.5, 8 iterations per time step Chang [1] scheme CN[1] scheme present scheme

2.9088

- 3.04

1.9600

- 2.00

0.0017

0.0478

2.8736

- 4.21

1.9550

- 2.25

0.0024

0.3055

2.9354

- 2.15

1.9600

- 2.00

0.0377

0.0107

0.0868

h = 0.5, At = 0.1, 2 iterations per time step present scheme

2.9850

- 0.50

2.0000

0.00

- 0.0007 0.0027

0.0031

same C P U time as before, the performance of the B-spline method is outstanding; see table III.

3. - S i m u l a t i o n

of wave generation.

The wave-maker boundary conditions applied to the semi-infinite region 0 ~< x < [8] are modelled by

t Uo - ,

0~
T

(6)

u(0, t) =

Uo,

v
to - t Uo--,

to-v~
The impulse grows linearly from 0 to U0 over a period v, it remains constant for a time t - 2v, after which it falls linearly back to 0 in a period v. The effect of the impulse is to generate solitary waves at x = 0, which grow at a rate determined by the magnitude of the forced boundary value. Solitary waves are continually generated while the forcing conditions prevail, then all growth slows and eventually ceases. A f u r t h e r (homogeneous) boundary condition is imposed at x = Xmax which is taken sufficiently far from x = 0 to r e p r e s e n t conditions at infinity. To compare with earlier work [1] we take a = b = 1 and apply an impulse U0 = 2, for a total period of to = 20, with build-up and run-down times of v = 0.3. In the simulation the space step h = 0.4, time step At = 0.1, range 0 ~< x ~< 260 and the experiment is continued until the first generated solitary wave approaches the end of the mesh. By time t = 20, when the forcing is switched off, five solitary waves have been generated. Subsequently, no new waves are born. Figure 1 shows the profile of the solution at time t = 100, five solitary waves have been formed and grown to

THE

BOUNDARY-FORCED

REGULARISED

LONG-WAVE

1491

EQUATION

4.0 3.5 3.0 2.5

~2.o 1.5 1.0

\J

05

0.0

r 100

80

. 120

140

160 180 distance

200

220

240

Fig. 1. - Solitary waves produced by boundary forcing of duration to = 20 and amplitude Uo = 2 at time t=100, h=0.4, At=0.1, a = b = l . 4.01 3.5 ~3.0 ~2.5 ~2.0 9,..-t

g 1.5 1.0 0.5 0.0

10

I

I

I

20

30

40

I

I

50 60 time

I

t

I

70

80

90

r

I

100 110

Fig. 2. - The evolution of the wave amplitudes for boundary-forcing conditions of duration to = 20 and amplitude U0 = 2. various amplitudes which confirm the observations of Chang et al., fig. 2 [1]. The development of the wave amplitudes as a function of time is recorded in table IV and fig. 2. The time-amplitude curves of fig. 2 show that the rates of growth are not constant and, in fact, differ from wave to wave as do the maximum amplitudes attained; see also table V. The time between births of solitary waves is, however, constant at AT8 = 4.18. F r o m the space-time curves of fig. 3 the average wave velocities, shown in table V, are estimated. These agree well with the velocities expected for free solitary waves of like amplitude. To examine the effect of mesh size on the observations, the space step is reduced to h = 0.1 and the experiment rerun. The resulting wave amplitude development is recorded in table VI. The closeness of the two sets of results conf~ms that convergence of the solution has been obtained.

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L . R . T . GARDNER and I. DAG

TABLE IV. Solitary-wave amplitudes with Uo = 2, period of forcing 0 <~ t ~ 20, h = 0.4, At=0.1, 0~
Time

1

2

3

4

5

2.5 5.0 7.5 10 15 20 40 60 80 100

2.23 2.77 3.06 3.24 3.44 3.54 3.69 3.74 3.75 3.76

2.23 2.52 2.90 3.13 3.47 3.52 3.52 3.52

2.43 2.74 3.11 3.06 3.05 3.06

2.05 2.38 2.35 2.33 2.33 2.33

1.95 1.08 1.07 1.07 1.07

TABLE V. - Observations of solitary waves, Uo = 2, time t = 80. Wave

1 2 3 4 5

Birth time

Generated-waves measured values

1.85 6.03 10.21 14.39 18.57

amplitude

velocity

Free-wave velocity

3.75 3.52 3.05 2.33 1.07

2.25 2.17 2.02 1.79 1.36

2.25 2.17 2.02 1.78 1.36

240 220 200 180 160 140: 120" ,-~ 100" 80" 6040" 200

10

20

30

40

50

60 70 time

80

90

100 12q

Fig. 3. - The space-time graphs for the solitary waves produced by boundary-forcing conditions of duration to = 20 and amplitude Uo = 2.

THE BOUNDARY-FORCEDREGULARISED LONG-WAVEEQUATION

1493

TABLE V]. Solitary-wave amplitudes with Uo = 2, period of forcing 0 <~t ~ 20, h = 0.1, A t = 0 . 1 , 0~
Time

1

2

3

4

5

2.5 5.0 7.5 10 15 20 40 60 80 100

2.24 2.78 3.06 3.24 3.44 3.55 3.69 3.74 3.76 3.76

2.23 2.52 2.91 3.14 3.48 3.52 3.52 3.52

2.01 2.43 2.74 3.11 3.07 3.07 3.07

2.05 2.38 2.36 2.34 2.34 2.34

1.95 1.08 1.07 1.07 1.07

TABLE VII. - Solitary-wave amplitudes with Uo = 2, from Chang et al. [1], see table I, h = 0.4, A t = 0 . 1 , 0~
1

2

2.5 5.0 7.5 10 15 20 40 60 80 100

2.24 2.76

1.86

3.24

2.52

3.71 3.75 3.76 3.76

3.44 3.49 3.51 3.51 3.51

3

4

5

2.73 3.07 3.07 3.07 3.07

2.37 2.32 2.32 2.32 2.32

1.86 0.98 0.98 0.98 0.98

1.5 1.0

0.5" o.o'~ _0.5 2 -1.0-1.5 -2.0

I

0

I

I

I

l

I

I

I

I

10 20 30 40 50 60 70 80 9010< time

Fig. 4. - The first derivative at the origin u~(O, t) as a function of time for boundary-forcing conditions of duration to = 100 and amplitude Uo = 2.

1494

L.R.T. GARDNERand I. DAG 1.0 84 ~D

0.5

0.0 -o.5. r r

-1.o

-1.5 -2.0

I

0

I

I

,

I

I

I

I

I

I

10 20 30 40 50 60 70 80 90 100 time

Fig. 5. - The second derivative at the origin Ux(0, t) as a function of time for boundary forcing conditions of duration to = 100 and amplitude U0 = 2.

The agreement with the earlier work of Chang et al. [1], recorded in table VII, is also close. The small differences which do occur may be explained by the coarseness of the space meshes. An additional difficulty in reproducing the exact conditions of the Chang experiment is the doubt concerning the actual impulsive function used. The value of to is quoted by Chang et al. [1], but the value of v is not given explicitly, although it can be seen from their fig. 1 that v has a very small positive value; we chose to take T = 0.3. Figures 4 and 5 show how the first and second spatial derivatives at the origin behave as functions of time when forcing is continued until t = 100. T h e r e is a large initial transient followed by oscillations of constant period 4.18 but decaying amplitude. If the R L W eq. (1) is evaluated at the origin x = 0 we find, with

4.0 3.5 ~3.0 ~2.5 ~2.0 "~ 1.5 1.0 0.5 0.0

10 20 30 40 50 60 70 80 90 100110120130140150 time

Fig. 6. - The evolution of the amplitudes of the first 14 waves for boundary-forcing conditions of duration to = 100 and amplitude Uo = 2.

1495

THE BOUNDARY-FORCED REGULARISED LONG-WAVE EQUATION

TABLE VIII. 0 ~
Solitary-wave amplitudes with Uo = 0.1, a = 3/2, b = 1/6 h = 0.1, At = 0.1,

Time

1

2

3

4

5

6

7

8

9

2 5 8 12 17 21 26 30 35 39 50 75 100 150 200

0.1046 0.1205 0.1245 0.1289 0.1321 0.1343 0.1374 0.1387 0.1415 0.1426 0.1474 0.1550 0.1619 0.1726 0.1784

0.1050 0.1100 0.1132 0.1148 0.1160 0.1174 0.1189 0.1199 0.1222 0.1283 0.1347 0.1464 0.1553

0.1022 0.1060 0.1082 0.1093 0.1100 0.1113 0.1122 0.1138 0.1174 0.1208 0.1289 0.1384

0.1018 0.1040 0.1060 0.1066 0.1079 0.1085 0.1099 0.1123 0.1146 0.1199 0.1264

0.1011 0.1034 0.1046 0.1055 0.1061 0.1075 0.1095 0.1109 0.1144 0.1190

0.1010 0.1024 0.1034 0.1041 0.1057 0.1074 0.1087 0.1114 0.1143

0.1007 0.1022 0.1028 0.1045 0.1061 0.1074 0.1095 0.1115

0.1006 0.1017 0.1035 0.1051 0.1064 0.1080 0.1097

0.1005 0.1025 0.1045 0.1056 0.1071 0.1082

u ( 0 , t) = 2, t h a t d 3ux(0, t) = -:-:. U~x(O, t). (It Thus, if ux(0, t) and ux~(0, t) are observed as functions of time, the maxima and minima in Uxx(O, t) should coincide with the zeros of ux(0, t); this is confirmed to occur in our simulation by fig. 4 and 5. The corresponding evolution of the amplitudes of the first 14 g e n e r a t e d solitary waves is given in fig. 6. Although m a n y of the early g e n e r a t e d waves a p p e a r to have attained a constant amplitude, this is by no means certain. Peregrine[2] has shown that when the R L W equation is used to model w a t e r waves in a shallow channel a = 3/2, b = 1/6 and wave amplitudes should be less than 1; u(x, t) is scaled to the w a t e r depth. We have, therefore, used these p a r a m e t e r s t o g e t h e r with a forcing b o u n d a r y value of U0 = 0.1 in an experiment the results of which are now described. The evolution of the amplitudes of the first 9 g e n e r a t e d waves in a long run, up to t = 200, when forcing is applied throughout the run are shown in table V I I I . I t is clear that all waves grow continuously but by t -- 200 no wave has achieved an amplitude as large as 0.18. The interval between solitary-wave births is ATB N 4.2, which is also the period of the oscillations in the derivatives at the origin.

4. -

Discussion.

I t has been shown t h a t the finite-element method based on cubic B-spline finite elements within Galerkin's method leads to an unconditionally stable numerical scheme for the solution of the R L W equation which shows good conservation. The scheme faithfully r e p r e s e n t s the amplitude, position and velocity of a single

1496

L.R.T.

GARDNER

and

I. DAG

solitary wave. The algorithm may be used with confidence for simulations of the motion of RLW solitary waves which are of long duration. When the Korteweg-deVries equation is subject to a constant boundary value, solitary waves are generated at a constant rate and all grow to the same limiting amplitude within a short time [8, 9]. After an initial transient, the derivatives at the origin oscillate with constant period and constant amplitude [9]. Similar behaviour is also exhibited by the modified Korteweg-de Vries equation when subjected to boundary forcing [10]. The forced RLW equation exhibits rather different behaviour. Solitary waves are generated at a constant rate but continue to grow throughout the experiment though at rates which become smaller with time, and the height reached by each subsequent wave in similar time periods is smaller. The derivatives at the origin, after an initial transient, also oscillate with constant period but with amplitudes which die away rapidly. These observations seem to imply that the rate of generation of solitary waves and the period of oscillation of the derivatives at the origin are constant and related, and do not depend upon either the values of the parameters a and b or upon the magnitude of the forcing function. The maximum amplitudes achieved by the generated waves appear to depend not only on the duration of forcing experienced but also on the magnitude of the first derivative at the origin.

REFERENCES [1] [2] [3l [4] [5] [6] [7] [8] [9] [10]

CHANGQ., WANG G. and Guo B., J. Comput. Phys., 93 (1991) 360. PEREGRINE D. H., J. Fluid Mech., 25 (1966) 321. EILBECK J. C. and McGUIRE G. R., J. Comput. Phys., 19 (1975) 43. BONAJ. L., PRITCHARDW. G. and SCOTT L. R., J. Comput. Phys., 60 (1985) 167. JAIN P. C., SHANKERR. and SINGH T. V., Commun. Num. Meth. Engin., 9 (1993) 587. GARDNERL. R. T. and GARDNERG. A., J. Comput. Phys., 91 (1990) 441. OLVER P. J., Math. Proc. Cambridge Philos. Soc., 85 (1979) 143. CHU C. K., XlANG L. W. and BARANSKYY., Commun. Pure Appl. Math., 36 (1983) 495. CAMASSAR. and Wu T. Y., Wave Motion, 11 (1989) 495. GARDNERL. R. T., GARDNERG. A. and GEYIKLI T., J. Comput. Phys., 113 (1994) 5.