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Numerical simulations of Klein-Gordon solitary-wave interactions

(1_{) Mathematics Department, Suez Canal University - Ismailia, Egypt}

(2_{) School of Mathematics, University of Wales - Bangor, Gwynedd LL57 1UT, UK} (ricevuto il 9 Ottobre 1996; approvato il 15 Gennaio 1997)

Summary. — Solutions of a non-linear Klein-Gordon equation are studied

numerically using a cubic B-spline finite-element method. Test results indicate that, when solitary waves interact, the final state obtained depends on their relative velocity. The simulations confirm existing observations and produce new results. The numerical algorithm developed is efficient with an undemanding stability criterion. PACS 02.60.Cb – Numerical simulation, solution of equations.

PACS 11.10 – Field theory.

1. – Introduction

In this paper we consider a particular form of the Klein-Gordon equation referred to as the phi-four equation in particle physics. This equation has been solved numerically using a variety of techniques [1-6]. Kudryavtsev [1] obtained the first stable numerical scheme and studied the interaction of kink and anti-kink solitary waves with velocities of 0.1. He showed that at this energy the two kinks formed a long-lived oscillating bound state, which decayed slowly by radiating energy. He also reported that for higher collision velocities the kinks repelled each other, and part of their energy was lost by radiation. Aubry [2] found that the formation of the bounded state ( kink-antikink pair ) exhibited a more complicated dependence on their relative velocities. Ablowitz, Kruskal and Ladik [3], showed that for velocities greater than 0.25 the collisions of the kink pairs were inelastic. For velocities less than 0.25 the interaction of a kink-antikink pair resulted in an oscillating state, except very near the value 0.2, where inelastic collisions were again observed. Mitchell and Schoombie [4] and Dodd et al. [6] studied similar configurations. Campbell et al., as reported in [6], have undertaken a comprehensive study of kink collisions using a fourth-order finite-difference method; they found that for collision velocities less than 0.193 an oscillating bound state is formed, for velocities greater than 0.258 an inelastic collision occurs and for the region in between a variety of states are observed including a double oscillation. We have previously used B-spline finite-element methods to simulate the interaction of solitary waves of both the Regularised Long Wave equation [7] and the

Korteweg-de Vries equation [8]. Here we apply a collocation method using cubic B-spline finite-elements to obtain a numerical solution for a Klein-Gordon equation. Studies of the interaction of solitary waves of various velocities are undertaken and compared with published data.

2. – The B-spline finite-element scheme

A Klein-Gordon equation of the form

Utt2 Uxx4 U( 1 2 U2) ,

(1)

where the subscripts x and t denote differentiation, is discussed. To solve the initial-value problem (1) numerically we first replace it by a system which is first order in the time derivative,

Ut4 V ,

(2)

Vt4 Uxx1 U( 1 2 U2) .

(3)

Natural boundary conditions

Ux(a , t) 4Ux(b , t) 40

(4)

are used to model the physical condition that U KNU0N as NxN K Q.

Let us consider a 4x0E x1R E xN4 b as a partition of [a , b] into N equally sized

intervals h and fi(x) as those cubic B-splines with knots at the points xj. Then the set

of splines ]f21, f0, R , fN, fN 11( forms a basis for functions defined over [a , b] [9].

Finite elements for the problem are identified with the intervals [xm, xm 11] and the

element nodes with the knots xm, xm 11. The variation of U over the element [xm, xm 11]

is given by Ue 4

!

where u_{m 21}, um, um 11, um 12 act as nodeless parameters and the splines fm 21,

fm, fm 11, fm 12act as element shape functions. In terms of a local coordinate system

z given by hz 4x2xm, where 0 GzG1, expressions for the cubic B-splines covering

the element [xm, xm 11] are [7, 10]

f 4 (123z13z2

2 z3, 4 26z2

1 3 z3, 1 13z13z2

2 3 z3, z3₎T_.

(6)

The nodal variables for the function Um, U 8m and U 9m are given in terms of the

parameters um by Um4 U(xm) 4um 211 4 um1 um 11, (7) U 8m4 U 8 (xm) 4 3 h[um 112 um 21] , (8) U 9m4 U 9 (xm) 4 6 h2[um 212 2 um1 um 11] . (9)

A cubic B-spline finite element has the same nodal variables Um, U 8m and U 9m and

hence the same continuity properties as a quintic polynomial Hermite element. A collocation approach may be interpreted as a Galerkin finite-element method with delta-function weights. Convergence of the method as the element size is reduced is guaranteed since the B-spline trial functions have the property that

!

j 4m21fj4

constant, and so ensure that when the nodal values and, consequently, the element parameters u_{m 21}, um, um 11, um 12are constant for a given element then U is constant

over that element [9].

In implementing the collocation approach for the system (2)-(3), identify the collocation points with the nodes. Expand U and V in terms of cubic B-splines through (5) and use eqs. (7) to (9) to produce the required nodal values,

Um4 um 211 4 um1 um 11, (10) Vm4 vm 211 4 vm1 vm 11, (11) h2 U 9m4 6[um 212 2 um1 um 11] , (12) h2_{V 9}m4 6[vm 212 2 vm1 vm 11] , (13)

where U(x, t) and V(x, t) have, for the given finite-element array, the global trial functions UN(x , t) 4

!

!

Use eqs. (10)-(13) to derive from (2)-(3) a system of 2 N 12 coupled ordinary differential equations: ¯ ¯t[um 211 4 um1 um 11] 4vm 211 4 vm1 vm 11, (16) ¯ ¯t[vm 211 4 vm1 vm 11] 4lm[um 211 4 um1 um 11] 1 6 h2[um 212 2 um1 um 11] , (17) where lm4 1 2 [um 211 4 um1 um 11]2, (18) m 40, 1, R, N.

3. – Leap-frog approximation in time

In the approach adopted here, a recurrence relationship based on a leap-frog approximation in time is derived. Suppose that the nodal values Um, and so um, are

known at even time steps and Vm, and vm, at odd time steps so that eqs. (16) and (17)

and Dt is the time step, un 11 m 211 4 umn 111 um 11n 114 um 21n 211 4 umn 211 um 11n 211 2 Dt[vm 21n 1 4 vmn1 vm 11n ] , (19) v_{m 21}n 12_{1 4 v}mn 121 vm 11n 124 vm 21n 1 4 vmn1 vm 11n 1 2 Dtln 11m [um 21n 111 4 umn 111 um 11n 11] 1 (20) 112 Dt h2 [u n 11 m 212 2 umn 111 um 11n 11] , where ln 11 m 4 1 2 [um 21n 111 4 umn 111 um 11n 11]2, (21) m 40, 1, R, N, and n41, 3, 5, R .

The natural boundary conditions are used to eliminate u₂₁, u_{N 11}, v₂₁, v_{N 11}from the recurrence relationships (19)-(20) which then become matrix equations for un 11₄

(u0, R , uN)T and vn 124 (v0, R , vN)T: Aun 11_{4 Au}n 21_{1 2 Dt Av}n_, (22) Avn 12_{4 Av}n 1 2 Dt A(lm) un 111 12 Dt h2 Bu n 11_, (23)

where A and B are the (N 11)3 (N11) tridiagonal matrices,

A 4

C

`

`

`

`

`

D

E

`

`

`

`

`

F

C

`

`

`

`

`

D

E

`

`

`

`

`

F

These matrix equations are efficiently solved using the Thomas algorithm. A linear stability analysis of the difference scheme leads to the stability condition

Dt E h 2k3 . (24)

4. – The initial state

From the initial condition we must determine for the leap-frog scheme the two vectors u0_{and v}1_{so that the time evolution of u}n 11_{and v}n 12_{using (22) and (23), can be}

started. Rewrite the global trial functions equations (14)-(15) for the leap-frog start-up conditions, as UN(x , 0 ) 4

!

!

where ui0and vi1 are unknown parameters to be determined. To determine the uj0

require UNto satisfy the following constraints.

a) It must agree with the initial condition at the knots; eq. (7) leads to N 11 conditions, and b) the first derivative of the approximate initial condition shall be zero at both ends of the range; eq. (8) produces two further equations. The start-up vectors are then determined as the solution of two matrix equations,

Au0

4 b , (27)

where

b 4 (U00, U10, U20, R , UN0)T,

and a similar equation relating v1_{to V}1_{, both of which are tridiagonal and can be readily}

solved by the Thomas algorithm. The boundary conditions lead to u14 u21, v14 v21, uN 114 uN 21, vN 114 vN 21.

5. – Numerical results

Simulations of the motion and interaction of solitary waves are used to validate the numerical algorithm. Solutions of the KG equation conserve both energy E and momentum P which are given by the functionals

E 4 1 2



k

l



TABLE_{I. – Travelling-wave simulations c40.5, 230 GxG30, t410. Various h, Dt combinations.} expt h Dt P E L23 103 LQ3 10 3 1 0.2 0.05 20.2721171 13.96137 15 11 2 0.1 0.02 20.2721563 13.93634 3.8 3.0 3 0.05 0.01 20.2721601 13.92384 0.93 0.73 4 0.02 0.005 20.2721578 13.91634 0.10 0.08

Values of these quantities will be monitored throughout the simulations to test the conservation properties of the numerical method.

a) A travelling-wave solution has the form U(x , t) 4tanh

y

k

z

where c is the velocity of the wave, NcNE1. The associated function V(x, t) is obtained

Fig. 1 – Wave interaction c 40.1, region 260 GxG60, h40.05, Dt40.01, x04 6. Temporal evolution of the wave maximum.

by differentiating with respect to t: V(x , t) 4Ut(x , t) 4 2c k2 1

k

y

k

z

For the travelling wave it is easy to integrate (29) over the infinite range 2QGxG1Q to obtain a closed expression for the momentum P in terms of the wave velocity c :

P 4 c 3

k

l

which for c 40.5 leads to P40.272166.

Simulations of this solitary wave for c 40.5 over a range 230 GxG30 are obtained using as initial conditions eqs. (30)-(31) evaluated at t 40. In table I the error norms and energy and momentum values for experiments using various space-time step combinations are recorded. As the step size is reduced the error norms decrease in size but the values of the energy and momentum are not necessarily better conserved nor more accurate. The wave geometry is accurately described by both experiments 3 and 4 in table I and good conservation is shown. However, the combination h 40.05/Dt40.01 leads to a momentum value of 20.2721601 which is closest to the theoretical value, and has error norms less than 1023_{, therefore this step size combination will be used in the}

interaction experiments that follow.

b) The interaction of travelling waves with equal and opposite velocities. This process is studied using as initial condition

U(x , t) 4tanh

y

k

z

y

k

z

TABLEII. – Incident c and final cfwave velocities, energy E and momentum P collisions between waves of equal and opposite velocities.

c cf cf[3] E0 E120 NPN 0.1 osc osc 28.11744 28.11748 G 1026 0.15 osc 28.10531 28.10538 G 4 3 1027 0.175 osc 28.09732 28.09741 G 1026 0.195 0.108(2 osc) 28.09000 28.09026 G 6 3 1027 0.2 0.160(2 osc) 0.155 28.08799 28.08821 G 1027 0.22 osc 28.07956 28.07975 G 7 3 1027 0.25 osc 28.06502 28.06529 G 1026 0.26 0.020 28.05976 28.05988 G 1026 0.3 0.1375 0.137 28.03588 28.03602 G 1026 0.5 0.388 0.384 27.83520 27.83542 G 1026 0.7 0.600 0.594 27.37217 27.37241 G 3 3 1024 0.9 0.8275 0.826 25.68690 25.68725 G 3 3 1025

Fig. 2 – Wave interaction c 40.3, region 260 GxG60, h40.05, Dt40.01, x04 6. Temporal evolution of the wave maximum.

and V(x , t) 4 2c

k

{

y

k

z

y

k

z

}

In the following interaction experiments we use a range 260 GxG60 and h40.05, Dt 40.01, x04 6 and run for a time t 4 120. The result of the interaction depends upon

the value of the wave velocity c. When the wave velocities are small, such as c 40.1, the colliding waves stick together and an oscillating bound state is formed which decays slowly by radiation [1-3]. When wave velocities are large, such as c 40.5, an inelastic collision occurs and the waves repel one another [1, 3]. For velocities in the range 0.193 GcG0.258 a variety of states has been observed including double oscillations [6], particularly near c 40.2 [3]. The results of our experiments given in table II confirm these observations.

For a simulation with c 40. 1, the energy has a mean value of 28.117 and a variation of about 60.0004 while the momentum varies from zero by up to 1026_{. The function}

maximum Umax shows clearly the oscillations taking place, see fig. 1. Similar bound

oscillatory states are also formed for c 40.15 and 0.175, see table II.

Fig. 3 – Wave interaction c 40.2, region 260 GxG60, h40.05, Dt40.01, x04 6. Temporal evolution of the wave maximum.

see fig. 2. During the experiment the energy shows little variation from a mean of 28.0359 with oscillations of approximately 60.001 about it. The momentum varies about zero by less than 61026_{. It is estimated that the waves rebound from the collision with}

their velocities reduced to cf4 0.138 6 0.001 in agreement with Ablowitz et al. [3].

Similar conclusions can be drawn for c 40.26 and larger values of c; results for c4 0.026, 0.5, 0.7 and 0.9 are given in table II. It is calculated that when c 40.7 the waves rebound after the collision with speed cf4 0 .600 6 0 .003 which compares well with the

value 0.594 given in [3]. We also observe that when the incident velocity is c 40.9 the rebound velocity is cf4 0 .828 6 0 .005 ( 0 .826[ 3 ] ), when c 4 0 .5 the rebound velocity is

cf4 0 .388 6 0 .001 ( 0 .384[ 3 ] ) , when c 4 0 .26 the rebound velocity is cf4 0 .020 6 0 .001.

For wave velocities in a critical region between 0.193 GcG0.268 intermediate processes can occur. The solitary waves seem to bind together for a short period while making 2 oscillations (see fig. 3 for c 40.2) after which the solitary waves emerge from the interaction with their final velocities somewhat reduced. For a simulation with c 4 0 .2 the energy shows a small variation of approximately 60.003 from a mean of 28.0882. The momentum varies from zero by up to 1026_{. It is estimated that the waves rebound}

from the collision with their velocities reduced to cf4 0 .160 6 0 .001 which is close to the

value of 0.155 given in [3] where the double oscillation was not reported. Similar conclusions can be drawn for c 40.195 which also shows the double oscillation. The waves rebound with velocity cf4 0 .108 6 0 .001. Campbell et al. [6] have also observed

the double oscillation before rebound and also other intermediate states for velocities c in this critical region. The incident and rebound velocities for the series of simulations are collected together and compared with earlier observations [3] in table II, together with details of the energy at times t 40 and t420 and the variation in momentum observed in the experiments.

6. – Conclusions

A numerical solution algorithm for a Klein-Gordon equation has been constructed using a leap-frog approximation in time and spatial collocation over cubic B-spline finite elements. The stability criterion places no severe restriction on the time steps that may be used. A travelling wave is well represented by the algorithm; the error norms increase monotonically throughout the simulation but are still less than 1023_by

time t 410. The energy and momentum are both conserved satisfactorily.

The interaction of wave pairs is studied through numerical experiments. Inter-actions between waves with equal and opposite velocities have results which are mutually consistent and broadly in line with earlier results. The simulations satisfactorily conserve both momentum and energy and the three regimes of behaviour found in similar studies by Campbell et al. [6] and Zaki [11] have been confirmed. It is concluded that the proposed numerical algorithm is very suitable for the solution of the Klein-Gordon and similar equations.

R E F E R E N C E S

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[3] ABLOWITZM. J., KRUSKALM. D. and LADIKJ. F., SIAM J. Appl. Math., 36 (1979) 428. [4] MITCHELLA. R. and SCHOOMBIE S. W., Finite element studies of solitons, in Numerical

Methods for Coupled Systems, edited by R. W. LEWIS, P. BETTISSand E. HINTON(J. Wiley) 1984, pp. 465-488.

[5] MAKHANKOVV., Comp. Phys. Comm., 21 (1980) 1.

[6] DODDR., EILBECKC., GIBBONJ. and MORRISH., Solitons and Non-linear Wave Equations (Academic Press) 1982.

[7] GARDNERL. R. T. and GARDNERG. A., J Comp. Phys., 91 (1990) 441.

[8] GARDNERG. A., GARDNERL. R. T. and ALIA. H. A., Modelling non-linear waves with B-spline finite elements, in Mathematical & Numerical Aspects of Wave Propagation Phenomena, edited by G. COHEN, L. HALPERNand P. JOLY(SIAM, Philadelphia) 1991, p. 533.

[9] SEGERLINDL. J., Applied Finite Element Methods (J. Wiley, New York) 1976. [10] PRENTERP. M., Splines and Variational Methods (J. Wiley, New York) 1975.

[11] ZAKI S. I., Proceedings of the International AMSE Conference “Signals and Systems”, Centinje, Yugoslavia, 3-5 September, 1990, Vol. 3, pp. 17-26.