On the convergence of the collocation method for non-linear boundary value problems

U.S.S.R. Comput. Maths. Maths Phys. Vol. 17, pp. 215-221 @ Pergamon Press Ltd. 1978. Printed in Great Britain.

0041-5553/77/1001-0215$07.50/O

ON THE CONVERGENCE OF THE COLLOCATION METHOD FOR NON-LINEAR BOUNDARY VALUE PROBLEMS* M. G. RASSADINA

and S. 0. STRYGINA

Kuibyshev (Received 16 February 1976) THE CONVERGENCE investigated operator.

of the collocation

method for non-linear

by means of a specially constructed

equation

boundary

value problems

with a completely

is

continuous

Of the available methods of projection type used to solve boundary value problems for ordinary differential equations, the collocation method is distinguished by the simplicity of its numerical realization. A large number of papers by various authors (see [l-6] , and also the review in [7] ) have been devoted to the validation boundary

value problems in conditions

of the convergence

where Green’s functions

problems exist. In this paper the collocation

of this method for two-point

of some linear boundary

value

method is used for an extensive class of non-linear

boundary value problems, including, in particular, the multi-point boundary value problem. The existence and convergence of approximations by the collocation method are proved with a more general assumption

about the difference

problem. The proofs are based on a transition method for an equation

with a completely

from zero of the topological to the consideration

continuous

characteristic

of the

of the abstract projection

operator in Banach space.

1. Statement of the problem and some auxiliary assertions 1. Statement system of ordinary

of the problem and properties differential

of the projection

operators.

Consider the

equations dx:,idl=j(t, x),

where s=R’li, f(t, x) is a vector function respect to both arguments interested

in solutions

(1.1)

which is defined and continuously

for tc [- 1,1 j, x from some bounded

differentiable

with

domain D~R”‘. We will be

of the system (1 .l) satisfying the boundary

conditions

I/x(t) =U.

(1.2)

where HX (t) = (Hlx (t), . , H”,x (t) ), the Hi are certain, in general, non-linear functionals in the space of continuous functions. (Below it is assumed that the H,, i= I, 2. . . , m. are bounded and continuously differentiable.) Let x*(t) be the solution of problem (1 .I), (1.2). To find it we can use the collocation method, consisting of the following. We fuc an integer n>O. On the segment [- 1, 1] we consider the system of points TO, . . . , ~--i, called collocation points. We will define as an approximation

*Zh. vjkhisl. Mat. mat. Fiz., 17, 5, 1315-1320,

1977.

215

216

M. G. Rassadina and S. 0. Strygina

of order n by the collocation

method to the solution x* (t), the linear combination

of Chebyshev

polynomials I

lEd’s

&l(t) =

(4

(1.3)

h=O

with coefficients

from Rm , satisfying the system dxn(t) dt

and the boundary

I

1,. . .) n-l

i=o,

=f[t,Xll(t)l/tPrjt

t=r,

(1.4)

condition Hx, (t) =O.

It is obvious that a system of non-linear As points of collocation

(1.5)

algebraic equations

is obtained

for finding

ao, .

. . , a,

we will consider

(n-1)

=j

n(2j+l) = cos -

j==O, 1, . . . . n-l,

’

2n

or

Tj

(n-1)

= cos-,

We will denote by L2=-L2[-I, are summable-square,

ni

j=O, 1,...,n-I,

n-l

C-1.

1] the space of vector functions

if (1-P) -‘h is taken as weight function.

z(t), whose components

The norm Ilz(t)l12 in this space

is defined as usual : 1

lIz(

=

max

Ilzj(t)ll~ =

j-1,2,....m

max j-1,3,...,?7&

[

n

’

J

jZj(t)J2 dt

-

(1-p)‘”

-1

1* “’

We will denote by CI=CI[-1, 11 the space of continuous vector functions z(t), possessing a derivative i(t) 6. We will define the norm Ilz(t)ll in this space by the equation

where lb (0 IIc =

max j=t,2,....m

max

IZj(t)

I.

--1ct
In the space L, we now consider the linear projection

operators P,, by means of the

equation n

where a,=R*

and are so selected that

P,x

(t)

=x

(t)

at the points

xj

(n)

(n) or Zj , j==O, 1,. . . , n.

217

Short communications

It is obvious that the system (1.4), (IS), obtained to find x,(t), an approximate solution of (1.3) by the collocation method, by means of the projectors P, _ 1 can be written in the form

-

We

dtn (1) dt

= Pn-If[C &I(t) I,

Hzfl(t)=O.

(l-6)

note a property of the operators P,. For any function x(t) 41 Ib-~n4lz~o(n)

MIS,

(1.7)

where a(n)=O(l/(n+l)).To prove the estimate (1.7) we have to consider the expansion of the function x(t) in a Fourier series m x(t)

-Co

+

E

C,T” Ct)

(1.8)

s=,

(co,. . . , c,, . . . ERR) which converges in L,. Changing now from the variable t to the variable 13by the formula t=c& 0 and following the scheme explained in [8], it is easy to obtain the connection

between the coefficients ck of the expansion (1.8) and the coefficients of the polynomial P,s (t) . This enables us to estimate in the space L, the norm of the difference P--P,,x in terms of the norm of the remainder of the series (1.8), which in turn is easily estimated by the right side of (1.7). The operators P, are uniformly bounded from C, into L,. 2. The equivalent equation. In this section we consider the operator equation equivalent to the boundary value problem (1. l), (1.2). We

denote by B the operator d

Bz (t) = --p,

z(t) ECI.

It is easy to see that for any y=Lz, OD

y=bo

+

b,T, (t), c

a=,

the equation Bz =y has a unique solution P)

z=B-‘y

=

c

(2s)-‘(b,-i_b,+i)T,(t),

%=I

belonging to the subspace of continuously differentiable vector functions, whose coefficient co in the expansion (1.8) equals zero. This means that on L, we can consider the operator -0, existing such that B- I, a constant IIB-‘YII =Wl yllz.

It is easy to verify that the operator B- 1 is completely continuous like the operator acting in Cl. USSR 17:5--o

(1.9

M. G. Rassadinaand S 0. Strygha

218

In the space C, we consider the system of equations Bx=vx,

Hx=O,

(1.10)

where cpis some continuous operator defined on the set W of functions of C, assuming values inD. The set of solutions of the system (1 .lO) is identical with the set of solutions of the equation x=Yox+B-‘qx+lix,

(1.11)

where the operator Y0 establishes a correspondence between the continuous function x(t) and the coefficient co in its expansion (1.8). Indeed, let x = x(t) be the solution of system (1. IO). Since BBox=O, then the first equation of system (1.10) is equivalent to the equation B(x-Box)

=cps.

From this ~--9~z=B-*cpx. Considering that Hx = 0, we obtain ~=90x+B-hpx+Hz, the solution of (1.11).

that is, x is

Conversely, let x =x(t) be the solution of Eq. (1.11). We will show that x satisfies the system (1.10). For this we apply the operator Y0 to both sides of (1.11). Then, since 9oB-*cp~=0, we obtain Hx = 0. If we now apply to both sides of Eq. (1 .l 1) the operator B, we have BX=CQX (since

BBaz=O).

This implies that the solution of problem (1.1 l), (1.12) is equivalent to the solution of the equation x=8x with an operator Six =POs(t) +B-*f[t, x(t)] +Hx(t) completely continuous in the space C,. The problem of finding an approximate solution x, (t) of system (1.6) is equivalent to the solution of the equation x,,=&‘~z,, where ZPI1x,(t)=9~,(t) +B-lPn-lf[t, x,,(t) I+ Hx,,(t) is a completely continuous operator acting in C, .

2. The existence and convergence of approximations by the collocation method to the exact solution of the boundary value problem 1. Convergence theorem. Let every initial condition z (-I) =x0 define a unique solution (I”O’P (t, -1; x0) of the system (1. l), defmed in [- 1, 1 ] . Then in the space Rm the shift operator [91 Ci=Hqg+&

ls defined. This operator is continuous, and its fmed points are the initial values of the solution of problem (1 .l), (1.2). Theorem 1

Let x*(t) be an isolated solution of problem (1. l), (1.2), the corresponding fmed point x*(-l) of the shift operator having a non-zero index. Then for sufficiently large n the approximations (1.3), determined from relations (1.4), (1 S), exist and as n-f m converge to x*(t) in the norm of the space C, .

219

Short communications

Roof: It is obvious that x*=x*(t) is a fured point of the operator 8. In the domain 9kC, a sphere S, with centre at the point x* can be chosen, within and on the boundary S, of which there are no solutions of problem (1 .I), (1.2) different from x*, the rotation 7 (0, S,) of the vector field @=I--8 on S, being identical with the index of the fured point x*(-l) of the shaft operator [lo] . But then x* will have a non-zero index (as the fmed point of the operator d ). As well as the field @=I-8

we consider on S, the vector fields (D,=Z-fYpn

and show that for sufficiently large IZthere is equality of the rotations y (0, S,) =y (@n, S,) . Since the field @ is completely continuous, there exists a constant

a>~,

such that

P.1)

Using the estimates (1.9) and (1.7) we obtain

where o(n-l)-tO, n+m. From this and (2.1) by RouchC’s theorem (see, for example, [ 1 l] ) it follows that for sufficiently large n the fields a,, are not degenerate on the sphere S, and Y(Q ,,, Consequently, [ 1 l] , for every n3no within the sphere S, there is at least S,) =y(@, 8,) *o. which is equivalent to the existence of approximate one fmed point of the operator d,,, solutions x, by the collocation method. The estimates (1.9), (1.7) and the complete continuity of the operator d enables us by following the reasoning usual in such cases (see, for example, [ 12]), to prove the convergence of the sequence xn(f) to x*(t) in the space C,. 2. Estimation of the rate of convergence. Let x*(t) be the exact solution of problem (1 .l), (1.2). We denote by F*(t) the Jacobi matrix 1 x=xa,l,l

(y)

i, k=l, 2 r...,m,

corresponding tof(t, x). We denote by H’x(t) the column vector;(Hl’[x*(t) lx(t),

, H,‘[s’(t) lr(t))‘.

Theorem 2 Let the linearized problem i(t)=F’(t)x(t),

H’x(t) =o

not have non-trivial solutions. Then for sufficiently large n the approximations x,(t) by the collocation method exist, converge to x*(t) and the estimate llx*-x,ll~cII

holds, where c is some positive constant.

(I-P,,-,)I[4

I*(t)

1112,

220

M. G. Rassadina and S. 0. Strygina

Proof: In the conditions of the theorem the Frechet derivative

of the operator 8 exists and the equation t=8’(s*)z does not have non-trivial solutions. In particular, this means that x* is an isolated fmed point of the operator 8’ of non-zero index (see, for example, [ 1l] ). Therefore, by Theorem 1, the approximate solutions x,(t) exist and as n+ UD they converge to x*(t). We prove the estimate. We note that the operators point x*. Let r,,=b,,x,,. Then x’-I,,=B-‘(I-P,,.-l)fx*+~f,,‘(s’)

8 a/ are also Frechet differentiable at the

(Y-x,,)

+o(cY,

Ilx,,-_z*II),

P-2)

where llo(z*, llr’-xl~ll) lI~o~Ill~*-~~lll, otL-+O, as IIx*--5,,ll+O; fs*=f[t, x’(t)]. In the conditions of the theorem the operator Z-Z’(.r*) is reversible. Therefore, for sufficiently large n the operators Z-8 ,,’(z’), are also reversible, and the norms of the inverse operators are bounded as a whole: II[I--8,,‘(I’)

I-‘Ilaw.

We rewrite (2.2) in the form Ir-~,,,‘(I’)l(~c’-r,,)=B-‘(I-P,,_*)fx’+o(s*,

11x’-s,,ll).

From this Ilt’-x,,ll~‘M~II

(z-P,,-,)f~.II?+Mo,,llz~-s,~ll.

For sufficiently large n this gives

The theorem is proved. Translated by J. Berry. REFERENCES 1.

VAINIKKO, G. M. On the convergence and stability of the collocation method. Differents. ur-niya, 1,2,2&J-254, 1965.

2.

VAINIKKO, G. M. The convergence of the collocation method for non-linear differential equations. Zh. vjGhis1. Mat. mat. Fiz., 6, 1,35-42,1966.

3.

KARPILOVSKAYA, E. B. On the convergence of the collocation method. Dokl. Akad. Nauk SSSR, 151, 4,766-769,1963.

4.

SHINDLER, A. A. Some theorems of the general theory of approximate methods of analysis and their application to the methods of collocation, of moments and of Galerkin Sibirskii matem. zh., 8, 2,415-432, 1967.

5.

RONTO, N. I. Application of the collocation method for the solution of boundary value problems. Vlcr. matem zh., 23,3,415-421, 1971.

6.

RONTO, N. I. Application of the collocation method to the solution of non-linear boundary value problems. In: Theory of electrotechnology and the computer design of electronic circuits (Tear. elektrotekhn. i mashinnoe proektirovanie elektronnykh tsepei), 103-113, IK Akad. Nauk Ukr SSR, Kiev, 1971.

7.

RONTO, N. I. Establishment of the existence of a periodic solution by the collocation method. In: A method of integrating manifolds in non-linear differenti equarions (Meted integr. mnogoobrazii v nelineinykh differents. ur-niyakh), 243-253, In-t matem. Akad. Nauk Ukr SSR, Kiev, 1973.

221 8.

ZYGMUND, A. Trigonomefric series (Trigonometricheskie

9.

KRASNOCEL’SKII, M. A. and LIFSHITS, E. A. The principles of duality for boundary value problems. Dokl. Akad. Nauk SSSR, 176,5,999-1001, 1964.

ryady), Vol. 2, “Mir”, Moscow, 1965.

10. STRYGINA, S. 0. The convergence of Gal&kin’s method for the multipoint boundary value problem. In: Proceedings of a seminar on differential equations (Tr. seminara po differents. ur-niyam), No. 1, 79-86, Izd-vo Kuibyshevskogo un-ta, Kuibyshev, 1975. 11. KRASNOSEL’SKII, M. A. Topological methods in the theory of non-linear integral equations (Topologicheskie metody v teorii nelineinykh integral’nukh uravnenii), Gostekhizdat, Moscow, 1956. 12. KRASNOSEL’SKII, M. A. et al. The approximate solution of operator equations (Priblizhennoe reshenie operatornykh uravnenii), “Nauka”, Moscow, 1969. U.S.S.R, Comput Maths. Maths Phys. Vol. 17, pp. 221-225 0 Pergamon Press Ltd. 1978. Printed in Great Britain.

0041-5553/77/1001-0221$07.50/O

A COMPLETELY CONSERVATIVE

DIFF;ERENCE SCHEME FOR GAS-DYNAMIC EQUATIONS* V. V. ZHAROVTSEV

Tomsk (Received 25 March 1976; Revised 26 May 1976)

FROM a family of three-layer completely conservative difference schemes approximating the one-dimensional equations of gas dynamics in Lagrangian mass variables, one scheme was successfully split into two systems of difference equations each of which, if the equations are considered in a definite sequence, can be solved explicitly. The results of trials of the scheme in the solrJtion of two test problems are given. The completely conservative schemes for integrating the equations of gas dynamics constructed in [l-3] are implicit, except for one which can be solved explicitly, but whose stability condition permits its use only with extremely rigorous constraints on the time step. Non-linear implicit difference equations are solved by iterative methods, but in a number of cases in order to ensure convergence of the iterative processes and also to obtain reliable results in the calculation of flows with shock waves, it is necessary to impose constraints on the time step, which leads to a considerable increase in the computing time for the problem. Moreover, as mentioned in [2], in practice the calculation is not continued to complete convergence of the iterations, and accordingly what is realized is not the initial difference scheme, but another close to it, and as a result an additional investigation must be carried out in order to estimate the unbalance of the terms of the difference equation realized. It is obvious that the construction of a completely conservative difference scheme permitting the integration of the gas-dynamic equations with a time step determined by Courant’s criterion is of definite interest. We will construct a difference scheme based on the system of equations alj au du dp z+s-=o,

ile

dX -_=JJ at

x--=0,

3

’

dV

x+e-g=@

*Zh. vFhis1. Mat. mat. Fiz., 17,5, 1320-1324.

1977

,=“,

at

s (1)