Numerical solution for nonlinear Klein-Gordon equation by collocation method with respect to spectral method

J. Korean Math. Soc. 32 (1995), No.3, pp. 541-551

NUMERICAL SOLUTION FOR NONLINEAR KLEIN-GORDON EQUATION BY COLLOCATION METHOD WITH RESPECT TO SPECTRAL METHOD

IN JUNG LEE

1. Introd uction

The nonlinear Klein Gordon equation

(1)

where L1 is the Laplacian operator in R^d (d = 1,2,3), Vu (u) is the derivative of the "potential function" V, and f is a source term inde- pendent of the solution u, in various areas of mathematical physics.

Among the particular cases which are the practical relevance, we take V1I.(u) = lulQu with et > 0 (quantum mechanics), refer to [5].

The convergence of the Galerkin finite element method for second or- der hyperbolic equations has been studied by many authors: cf. among others Dupont[3], who obtained error estimates for time-discrete and time continuous approximations of linear problems, and Dendy [2], who examined nonlinear problems as well as various modified Galerkin methods. To compute the nonlinear term Vu (u), the product approxi- mation is used by Yves Tourigny [6J. This approximation is a technique which consists of replacing the nonlinear term by its interpolant in the finite-dimensional subspaces. This provides an interesting alternative to numerical quadrature and greatly eases the implementation of the Galerkin method.

In this paper, by the collocation method, when n =2, we get the numerical solutions of (1). We shall show the stability and convergence for using spectral method technic.

Received June 21, 1994. Revised November 18, 1994.

1991 AMS Subject Classification: 65M70.

Key words: collocation method, spectral method.

To define a collocation method with spectral method techics, we give as many distinct points

XI: kE J (a set of indices)

in the domain 0 or in its boundary ao as the dimension of the space PolN(O) in which the spectral solution is sought. At the number of these points, located onaothe boundary conditions are imposed. The remaining points are used to enforce the differential equation.

We assume that for any k E J, there exists a polynomial 4>1: E PolN(O), nessarily unique, such that

if k =m, if k #^m.

The 4>1:'S form a basis for the polynomials of degree N, since vex) = LI:EJv(xl:)4>I:(x) for all v E PolN(O). Let J be divided into two disjoint subsets Je and Jb, such that ifkE Jb, theXI:'s are on the part

ao of the boundary. Moreover, let LN be an approximation to the operatorL in which derivatives are taken via collocation at the points XI:'so The collocation soluton is a polynomial uN E PolN(O) which satisfies the equations

{

LNUN(Xk)= I(xk) BuN(Xk) = 0

for all k E J_{e ,} for allk E Jb.

The unknowns in a collocation method are the values of uN at the points XI:, i.e., the coefficients of uN with respect to the Lagrange basis. We introduce a bilinear form (U,V)N on the space Z = CO(O) of the functions continuous up to the boundaryof0 by fixing a family of weightsWk > 0 and setting .

(u, v) = L U(Xk)V(Xk)Wk

kEJ

The existence of the Lagrange basis ensures that (u, v)N is an inner product on PolN(O). Consequently, we define a discrete norm on POIN(O) as

for u EPOIN(O)

Numerical solution for nonlinear Klein-Gordon equation 543

The basis of the tPk'S is orthogonal under the discrete inner product.

We make the assumption that the nodes {xd and the weights {wd are such that

(U,V)N = (u,v) for all u,V such that uv EP01_2N-I(n).

In all the applicatios, this assumption is fulfilled since the ^Xk'S are the knots of quadrature formulas of Gaussian type.

Let XN be the space of the polynomials of degree less than or equal to N which satisfy the boundry conditions, i.e.,

Then the collocation method is equivalently written as

IfYN is the space spanned by the ^tPk'S with k E le, i.e.,

then can be wri tten as

2. Stability

Let n be an interval [-1,1J. We would like to approximate the solution of the following problem

8²u 8²u

&2 - 8x₂ +^lulQu =f ^m Q=n ^x [0,TJ (2) u(.,O) =^Uo and ( ; ) (.,0) =^UI in n

u(-1,t) = u(1,t) = ° ^{for t E [O,TJ}

where Uo E Ho(n), Ul E L(Q) and f E L(Q) are given functions, T>O.

The solution uN (x,t) of the Legendre Tau approximation of this problem is for all t >0 a polynomial of degreeN inx, which is zero at x = ±l and satisfies the equations,

[ : [uft"(x,t) - u~(x,t)⁺luN(x,tW~uN(x,t)] ^v(x)dx

=

[11

[11

[11 [uf (x, 0) - Ul(X)} v(x)dx= 0

Let we set XN = ^{{u E PNlu(-l)} = ^u(l) = ^{0, YN} = ^{PN-2 and}

(u,v) =J~lu(x)v(x)dx. For all uE XN we have

-1

-1

1

But, we know that the degree of luNlou N is greater than 2N - 1.

Here, we shall use the approximation of luNlou N in (3). We subsi- tute INluNlou N intead of luNlou N where IN : C(n) -+ XN is the interpolation operator.

We shall find the approximate solutionuN EXN such that

[11[uft"(x, t) - u~(x,^t)+INlu N(x, t)lou N(x,t)]v(x)dx

=

[11

[11

[11 [uf(x,O) - Ul(X)] v(x)dx = O.

Numerical solution for nonlinear Klein-Gordon equation 545 ,

THEOREM 1. For someT > 0,

IIPN-2uf(t)1I1..,(-1,1) +lIu~(t)1I2+(2/P,8)lIuN(t)II~'(_l,l) :::;{IIPN-2uf(0)1I1..,(-l,l)+_lIu~(0)1I2 +(2/P,8l1uN(O)II~'(_1,l)

+

l

Proof. Take v= PN-2Uf, from the left hand side first term in (4)

[11 uu(x, t)PN-2Uf (x, t)dx

=

1

-1

=(1/2)~IIPN-2uf(t)1I1..,(-1,1)' and the second term,

-1

-1

1

= un(x, t)PN-2-

d ^u (x, t)dx

-1 t

1

= U2: (x, t)-d^U2: (x, t)dx

-1 t

=(1/2)~lIu~^(t)1I2.

Now, forp = et+2, refer to [4],

(l/p)~^lIuN(x,t)II~"(_l,l)

= [11 luN(x, tWl'u N(x,t)u;V(x, t)dx

= [11 lu N(x,t)I^QuN(x,t)(1- X2)PN_2Uf(x,t)dx

We can choose thef3 which satisfies lIu N-INuNII;, = lIu N -PNuNII;, + IIRNuNII;,

..'(l/pldtJhtJ~.~DJlt~~~l,l~,._,."d .."-._..,,,,.~,,,,,--,~~.__.

~f3

[11

[11 [uft(x, t) - u~z(x,^t)+INlu N(x, t)IQu N(x, t)] PN-2uf' (x, t)dx

= [ :f(~,t)PN-2uf'(X,t)dx

weobtain

( / ) d ^{11 N 2} d N 2

1 2 dt PN-2^Ut (t)IILw(-l,l) +^{(1/2) dt}lIu^z (t)1I

+(l/Pf3)~^lIuN(t)lItp(-I,l)

~

[11

1 12 1 . N 2

~'2l1f(t)1L",(-l,l) +'2"PN-2ut (X)IIL",(-I,I)

IIPN-2uf(t)lIi",(-I,l)+lIu~(t)1I2+(2/P(3)lIuN

(t)1I1p(-1,1)

~IIPN-2uf(O)lIi",(-l,l) +lIu:(0)11²+(2/P(3)lIuN

(O)1I1p(-1,1)

+

Lt

Lt

This theorem shows the stability of the approximate solution ofuN for

0=

1

-1

-1 -1

Numerical solution for non linear Klein-Goroon equation 547

1

1

-1 -1

:::;c[11uo:c(x)uo",(x)dx <clluoll~Wl)

3. Convergence

Let RN be a projection operator from a dense subspace W of DB upon XN, where DB is a set which satisfies the boundary condition of (2). For each u E W, we further require RNu to satisfy the exact boundry conditions, i.e.,

We define the norm IIgIIE. = sUPueE.#O ~!',i2 ^{for all 9 E E* that is} dual ofE.

Let e(x, t) = uN (x, t) - RNu. We obtain the following theorem.

THEOREM 2. Assume that lulQ'u E H¹⁽-1,1) IIPN-2et(t)IIL<-1,1) +lI^ex(t)11²

:::; {IIPN-2et(O)IIL(-1,1) + Ilex(O)1I²+^{M2T} eT}

:::; {IIPN-2et(O)IIL(-1,1) +clleoll~A<'l) +^{M2T} eT}

Proof. From (3), we have

[11^{[uft - u~x +INlu N(x,t)lQ'uN(x,t) - I(x,t)] v(x)dx = 0}

t > 0, for all v EPN-2

We get (5)

~~IIPN-2et(t)1I1w(-I,I) +~~^Ilex(t)112

= J:l(utt - RNuft)et+(.1!-Nuft.-:- uxx)et +(lulQu - INluNIQuN)etdx

=IIPN-2(Utt - RNuft)III1 PN-2 et!lLw(-I,I)

. N

+IIPN-2(RNu x·x - uxx)IIIIPN-2 etIlLw(-I,I)

+IIlul^Qu- INluN IQ

uN

IIIIPN-2 et!lLw(_I,I) We refer to [I}: For each v EHJ(-1,1)

(PN-2(Utt - RNUtt),V)

=(Utt - RNUtt, v) - (Utt - RNUtt, v - PN-2V)

=«Utt-RNUit)x, (0-.RN0)x).-(utt-RNUtt, V-PN-2V) where 0is the only function in HJ(-1,1) satisfying -0xx = vthen we

*. . .

For eachv E HJ(-1,1)

(PN-2(U - Rnu)xx,v)

= - «u - Rnu)x,vx) - «u - Rnu)xx,v - PN-2V)

=«u - Rnu)x,vx) - (u xx - Pn- 2uxx ,v - P,i-2V)

Numerical solution for nonlinear Klein-Gordon equation 549

here we have used the fact that bothPN-2Uxz and(R,.u)xzare orthog- onal to v - PN-2V. Using the same approximation results as before, wededuce

In Legendre approximations, for all u EHm(-1,1) lIu - 1NUIlHt(-l,l)

~CN2l+!-mlluIlHm(_l,l) ^{for 0}~ ^{I$ m with m >} ~.

Asswne that lulQu EH¹(-1,1) let1=0. We get

IIlulou - 1NluNlouNIIL2(_l,l)

=lIlulou - 1nluI^QIIL2(_I,I) $ CN!-mllluIO'uIlHm(_I,I)'

We may asswne that m 2:: 2.

LetM = CN1-mlluttIlHm-2(_I,I)+CN1-mlluIlHm(_1,1)+CN-! IlIul1 UIlHm-2(_1,1) clearly M -+ 0 as N -+ 00. From (5)

IIPN-2et(t)1I1",(_1,1) +lIez(t)1I²

~IIPN-2et(O)IIL(-l,l) + lIe^z(O)1I²

+Lt M2

ds+

J

We know that lIez(O)1I² $ clleoll~~«(}) and applying Gronwall's inequal- ity we conclude the proof.

4. Numerical results

Set uN(x, t) = L:~o ai(t)li(x) is aN-degree Lagrange polynomial with N +¹nodes as -1 = Xo < Xl < X2··· < _Xn= 1. We substitute

uN(x,t) into (2), we get

Jla-(t) .

d;2 -l~(xi)al(t)+ ...+liV(xi)an(t) +la~(tWl!ai(t) = f(Xi' t) i = 0, 1,2, ... ,N

Applying the boundry condition and the difference equation with

Jlai(tj) _ alt;+!) - 2ai(tj)+ai(tj-1)

dt² - h²

ai(tO) =⁰

ai(tI) =hUI(Xi)

where h is a mesh size and tj =^jh, we obtain a system of N - 1 equations. For one example, let

f(x, t) = - ^2sin(1rx)+I(t - t 2)sin(1rX)I^Q(t - t2)sin(1rx) + 1r2(t - t2)sin(1rx)

'::: (x,O) = sin(1rx),

then we obtain the numerical solution as table 1.

Practically, this example has exact solution such that(t-t2)sin(1rx).

We can calculate errors. These errors are approximately less than C(k)(N-1) hN(N - 1)(~)(N-1)/ (~)! ^{in that} C (k)(N-1) is esti- mated from M which is in theorem 2., and N(N...:1)(~)(N-1)^/ (~)!

is calculated by a second order defferentiation of N-degree Laglange polynomial. Briefly, the errors are less than (i)^(N-1)hN(N-1)/ (~)!

and are independent of^Q.

Numerical solution for nonlinear Klein-Gordon equation 551

(

1)(N-l) (N)

K^="2 hN{N -^{1)/ "2 ! h =0.001}

numerical exact error k

N=8 9.91306E - 3 9.9E-3 1.3096E - 5 1.8229E - 5 N= 12 9.900053E - 3 9.9E-3 5.3E -8 8.9518E -8

table 1. The numericaJ. estimation of u{1/2, 0.01), error and error bound. For time valuet

we get results by 10th iteration.

References

1.Claudio Canuto M., Yousuff Hussaini Alfio quarteroni thomas A. Zang, Spec- tral Methods in Fluid Dynamics, Springer-Verlag, 1988.

2. Dendy, J. E.,An analysis of some Galerbn schemes for the solution of ninlin- ear time-dependent problems,SIAMJ. Numer. Anal..

3. Dupont, T., L estimates for Galerbn methods for second-order hyperbolic equa- tions,SIAM J.Numer. Anal..

4. Masanori Hosoya and Yoshio Yamada, On some nonlinear wave eguations I:

local e:tistence and regularity of solutions, J. Fac. Sci. univ. Tokyo Sect. lA, Math. 38 (1991), 225-238.

5. Perring,J. K.and Skyrme, T.R.H, A model unified field equation,Nud. Phys 31 (1962), 225-238.

6. Yves Truginy, Product a1'1'rozimation for nonlinear Klein-Gordon equations, IMA jourmal of Numerical Analysis 9 (1990), 449-462.

Department of Mathematics Hoseo University

ASM 337-850, Korea