Two Energy Conserving Numerical Schemes for the Sine-Gordon Equation

Zhang Fei and Luis YQzquez

Departamento de Fisica Ted&a

Facultad de Ciencias Fisicas

Uniuersidad Complutense

28040 Madrid, Spain

Transmitted by John Casti

ABSTRACT

Two explicit conservative numerical schemes for the sine-Gordon equation are proposed. Their stability and convergence are proved. Numerical simulation of a sine-Gordon soliton shows that the new schemes are very accurate and fast.

1. INTRODUCTION

We consider the sine-Gordon equation (SGE):

Utt - U,X +sinu = 0, --oo<x<+w,

u(x,O) =tQ(~>> u,(x,O) = Ui(X).

It is well known that this equation allows soliton solutions, and it has application in physics; see [l].

One remarkable property of Equation (1) is that its solution conserved quantity, namely the energy:

(1)

a wide

has a

E= _/- +p[ +u; + +u”, + G(u)] dx = const, -co

where the function G is defined as

G(u) =l-cosu.

APPLIED MATHEMATICS AND COMPUTATZON 45:17-30 (1991)

(2)

17

0 Elsevier Science Publishing Co., Inc., 1991

655 Avenue of the Americas, New York, NY 10010 009%3003/91/$03.50

18 ZHANG FE1 AND LLJIS ViizQUEZ

Several numerical schemes have been proposed for the sine-Gordon equation [2-61. So far there are only two schemes known to be energy conserving [4]. The explicit one is the following:

SCEIEME 0 (SO).

.;+I -2u; + u;-’ u;+l-2u7 + uy-1 -

r2 h2 +G(

(3)

where u: = u(lh, nr), with r and h the time and snace steo-sizes. This

scheme was introduced in [5]. It has a constant discrete energy:

)( uY+;_u;)+ G(u;“)2+GWl 1 . (4)

In this paper, we generalize the techniques used in References [4], [5], and [7], and propose two new energy conserving numerical schemes. One of the new schemes is similar to Scheme 0; the other one is completely explicit in the sense that no iteration is needed. Numerical experiments show that the new schemes are as accurate as Scheme 0, while the completely explicit scheme is much faster.

The paper is organized as follows: In Section 2 we present the schemes. In Section 3, the stability and convergence are proved. Numerical simula- tions of a sine-Gordon kink are performed in Section 4. Finally, we draw conclusions in Section 5.

2. NUMERICAL SCHEMES

SCHEME 1 (Sl). Eight points and four levels:

n+3 Ul -(u;+"+u;+')+u; _ u;;:-2u)+"+u;-',"

2r2 2h”

u;=,’ -2u;+’ + u;tl’ -

2h” + G(

u,+,>- +;“+I) = o n+2 n+l

Ul -u1

(5)

Numerical Schemes fur Sine-Gordon Equation 19

SCXIEME 2 (S2). Five points and three levels:

u;+r -2U;’ + U;-’ U;+l -2u; + u;_r

r2 h”

+2 G(( .;:+‘+.;)/2)-G((u;-‘+u;)/2) =O, (6)

n + 1 II- I UI -u/

These two schemes have the following properties:

(PI) Sl is completely explicit, u;+3 can be directly calculated from the

difference equation (5), and the starting data are u;, u:, and u:. S2 is explicit in the sense that u;+ ’ can be found from (6) by iteration provided u;-’ and u; are given.

(P2) Both schemes are spatially and temporally symmetric. (P3) Their consistency can be proved by Taylor expansion. For Sl, we

have

n + 3 Ul

_(,y+, + u;+‘)+ u; pu

2r2 =-&lh,(n+;)~)+O(~‘), (7)

@& -&&;+a + U;‘; n;=; -2u;f’ -I- u;t:

h” +

h”

a2u =-&h,(n++)~)+O(r”hP-“), (8)

G(u;+“)-G(u;+‘) n+2 - .;+I

=sinu(Zh,(n+i)r)+O(~“), (9) f-J1

where a > 0, p > 0 are integers, (Y + /3 = 4. Therefore the truncation error for Sl is O(T’ + h2> provided T/h Q C, C a constant. A similar analysis can be done for S2.

(P4) For each scheme there is a constant discrete energy.

u;+l - u;)(u; - u;-‘)

+G .y+l + u;

( )I 2 .

(11)

20 ZHANG FE1 AND LUIS V;\ZQUEZ

Actually, if (5) is multiplied by (u;+” - u;+ ‘>h, then taking the sum over 1, after some manipulation, the identity E.&+” = E.t:’ will be obtained. E ‘!.+I 52

= Et2 can be derived similarly.

3. THE STABILITY AND THE CONVERGENCE

For convenience, first let us introduce some useful notation similar to that used in Reference [4].

Let h be the mesh size of the space variable, T be the mesh size of the time variable. We define

RII=(xlx=kh, k=0,*1,*2 ,...)

Let u”(r) be the value of the mesh function u(x) at the point x E A,, and time t = k7. We define the forward and backward difference quotients of u with respect to x and t,

We also introduce the inner product and norms,

Numerical Schemea for Sine-Gordon Equation 21

LEM\IA 1. Let w(k) and p(k) be nonrzegative mesh functions. lf C > 0 and p(k) is nondecreasing, and the inequality

k-l

w(k)<p(k)+C7. c w(l) 1 = 0

holds for all k, then for all k

This lemma is from [4j. It wiil play an important role in the study of the stability and convergence of the schemes.

Now we are going to analyze the stability and convergence of Scheme 1. With the notation defined above, Scheme 1 can be equivalentiy written as

$[&x)+u;~~(x)] -f[u~,+u”;,‘(x)] =Gl(u"(x)) (12)

where

Gl( u”( x)) = COSUk(x)-coSUk-‘(X)

U”(X) - P(r) .

Let Ck(x) and f(x) be the error of u~(x> and the right hand side of Equation (12) respeciively; then we have

where G”(x)= Gl(uk(x>+ iik(~))-G1(~k(~)) Multiplying Equation (13) by 2~:~ r(x) and taking the inner product, we

get

Summing the above equality for t = T, 27,. , kr, we obtain

22

By Schwarz’s inequality, we have

ZHANG FE1 AND LUIS VAZQUEZ

2 i (f’,z$‘j < ; (Ilf”ll” + lIt5-‘II”). I=1 1=1

On the other hand,

(15)

From the definition of 6l, it is easy to prove that

Therefore

llc?l12 =G 2(llfi’l12 + llPl12).

According to [4] [ see Equation (13) in that paper], for any 1, Zr < T, we have

l-l

lIdlIZ < 211~“112 +2TT c Iltql”, i=O

so we can obtain the following inequalities:

1-l

llC!‘l12 < 811fi”112 +8T7 c Ilii;ll”, i=O

(16)

2 ; (C?,c:-‘j < i (IIW+ Il~l,-ll12j l=l l=O

k-l

< 8kllfi0112 +(8T2 + 1) IFoilli:l12. (17)

Numerical Schemes for Sine-Gordon

Now let us estimate the third From Equation (13) we have

Equation 23

term of the left hand side of Equation (14).

-k u,t=-ii:i’(~)+ti~,(X)+~~f1(X)+2[~k(x)+fk(x)]

= . . .

=(-l)k-lfi’tt(x)+iik,&)-(-l)k-lii~X(X)

+21iz( - lY[d’(x) +f’(x)] >

from which we get

Now suppose that

7 6 -= h

Y()G--> 2

a <l; (19)

then from (18) using (161, we can obtain, for any k, k7 < T,

+ 16T7 i llfl12 + 128T3rki1 11~l,112. (20) 1=1 l=O

Substituting (15), (17) and (20) into Equation (14), we obtain the follow- ing inequality:

k-l

13kq5k+C7 c 81, l=O

(21)

24

where

ZHANG FE1 AND LUIS VAZQUEZ

tik = Iliill” + (1- a)lcklf,

pk = c, Iliql” + llcijll” + Ilii”ll” + lii’lf + 7 ; Ilf’ll” . i l=O I

C and C, are two suitable constants. Using Lemma 1, we arrive at the following result:

THEOREM 1. lf the condition (19) is fuljilled, then for all kr < T we haoe

This theorem expresses the generalized stability of the numerical scheme in the sense studied by Kuo Ben-Yu. See Reference [4] and references therein.

Now we study the convergence of the scheme. Let Uk(r> = U(x, k7) be the exact solution of the sine-Gordon equation (l), u~(x> be the numerical solution of Equation (5), and v”(x) = Us- Vk(r). Then

where fk(r> is the truncation error; from the previous section we know that Ifk<r>] = O(? + h”).

Applying Theorem 1 to Equation (22), we obtain the following result:

THEOREM 2. If the condition (19) is fulfilled, then for all k, k r < T,

l12;,k112 + 12jk12 < pkeCTT,

where

Pk = PO + P’;T

po = c,(llv,0ll” + Ilo:ll” + ll~“l12 + WIH),

p; = cp 5 Ilf Y. I=1

Nurnericul Schemes for Sine-Gordon Equution 2s

Here p,, und pf accourat for the iraitial errors and the truncation errors respecticely.

In particular, if in the nunerical simulation we take the initial values to be

sufficiently close to the true solution, such that p. = O(h’), then we have

Ilo;ll’ + lo”lf = 0( h’),

which shows the convergence of the scheme.

Because Scheme 2 is only a modification of Scheme 0, its stability and

convergence can be proved in the same way as in [4]. The same kinds of

results can be obtained.

4. NUMERICAL EXPERIMENTS

The sine-Gordon equation possesses soliton solution of the form

U(X,t) = 4tan -1/‘.xp( ;s 11. (23)

which describes a kink moving with a constant velocity V. The energy of the

kink, defined by Equation (2), is S/d-, and its momentum is

p=- cz / up, dx = -cc

(24)

In our numerical simulation, we used the mesh sizes T = 0.025, h = 0.05.

The initial values were simply taken as the exact solution given by (23). In

Figure 1 and Figure 2, the simulation of a moving kink with velocity V = 0.2

is presented.

During the whole computation, the discrete energy conservation is veri-

fied. We also compute the discrete momentum:

P”=-hz 1

u;+l--w ) ( u”l;u’_‘)~ (25)

The fluctuation of DP(n7) = P” - P’ is plotted in Figure 3 and Figure 4,

from which we can see that the variation of the discrete momentum is less

than 10m7.

26 ZHANG FE1 AND LUIS ViZQUEZ

FIG. 1. Evolution of a kink with velocity V = 0.2, simulated with Scheme 1.

FIG. 2. Evolution of a kink with velocity V = 0.2, simulated with Scheme 2.

1 I

-30 -20 -10

6.0

5.0

4.0

3.0

2.0

1.0

-10 0 10 20 X

We define the energy center of the kink as follows:

for Scheme 1:

r:,=-&(p)[i “;+:-“;)( u~-;q+2-2cos,;]

u;+1- u; 2

+(Z+@ h ( )I;

Numerical Schemes for Sine-Gordon Equution

5.0

0.0 co + W % -5.0

-10.0

-15.0

I!, I I I ! I 1 I I I I 1 I 1 I -20.0 0 10 20 30 40 50 60 70 80

TIME

FIG. 3. Variation of discrete momentum for Scheme 1

10.0

5.0

0.0 Do + W L -5.0

P

-10.0

-15.0

/ -20.0 0 10 20 30 40 50 60 70 80

TIME

FIG. 4. Variation of discrete momentum for Scheme 2.

for scheme 2:

27

28 ZHANG FE1 AND LUIS VAZQUEZ

TIME

FK:. 5. Variation of discrete velocity for Scheme 1.

I, I I, I II,,,,, 1 I

20 40 60 I TIME

0.10

0.00

,-0.10

FK:. 6. Variation of discrete velocity for Scheme 2

The corresponding velocities are

X n+l _ X”

v” = Cl

xn+l

” and ” - x&2

7 7 .

In continuous limit, X& = X& = Vt and V” = V. However, in the discrete case, V” is no longer a constant. Figures 5 and 6 show that the fluctuations of the discrete velocities are of order 10P6 for our schemes.

Numerical Schemes for Sine-Gordon Equation 29

0.25

I I I1 1 I, I I / I I 0 10 20 30 40 50 60 70

0.00 80

TIME

FIG. 7. Total errors for Schemes 0 (dotted line), 1 (solid line), and 2 (dashed line) versus

time.

Now let us compare the numerical solution with the exact one. We define the total error as the following:

ER(nr)= z/u;-U(&,n~)l.

From Figure 7 we can see that the errors for Schemes 0, 1, and 2 are not only quantitatively close to each other but also fluctuate in a similar way. In the computation we have 1200 discrete points in space; therefore the maximum average errors during the simulation time 0 < T < 80 for the three schemes are bounded by 0.25/1200 = 2 X lop4 (0.25 is from Figure 7), which shows the high accuracy of the schemes.

Concerning the speed of the three schemes, we found that the computer times for Schemes 0 and 2 are almost equal, but Scheme 1 is five times faster than the other two. The reason is that Scheme 1 is completely explicit and no iteration is needed.

5. CONCLUSIONS

In this paper, we have proposed two new conservative finite difference schemes for the sine-Gordon equation, which has wide application in physics. One of the schemes is completely explicit, while the other one is explicit in

30 ZHANG FE1 AND LUIS VAZQUEZ

the sense that the iteration is needed in order to solve for the unknown variable at each discrete point. We have discussed some properties of the two schemes and proved their stability and convergence. Numerical simulation of a sine-Gordon soliton (kink) has been performed using the new schemes as well as the conservative scheme previously proposed in [4] and [5]. Through comparison we find that the three schemes are almost equally accurate, while the newly proposed complete explicit scheme (Sl) is five times faster than the other two.

One of the authors (2. F.) is very grateful to the Ministry of Education and Science of Spain for financial support (FPZ-PG89). This work is partially supported by the Direction General de Znvestigacion CientiLfica y Tecnica through project TIC 73 / 89.

REFERENCES

R. K. Dodd, J. C. Eibeck, J, D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic, 1982. M. J. Ablowitz, M. D. Kruskal, and J. F. Ladik, Solitary wave collisions, SlAM J.

Appl. Math. 36:428-437 (19791. F. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, S. Ruffo, and A.

Vulpiani, Approach to equilibrium in a chain of nonlinear oscillators, J. Physique 43:707-713 (19821. Guo Ben-Yu, P. J. Pascual, M. J. Rodriguez, and L. Vazquez, Numerical solution of the sine-Gordon equation, Appl. Math. Comput. 18:1-14 (1986). W. A. Strauss and L. VQzquez, Numerical solution of a nonlinear Klein-Gordon

equation, J. Comput. Phys. 28:271-278 (1978). S. JimCnez and L. Vazquez, Analysis of four numerical schemes for a nonlinear Klein-Gordon equation, Appl. Math. Comput. 35:61-95 (1990). Zhang Fei and Luis VQzquez, Some conservative numerical schemes for an

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