On correctness and numerical treatment of boundary value problems in differential-algebraic equations

U.S.S.R. Comput.~4aaths.~faPlath.Ph~s.,Vol.26,lio.l,pp.30-39,19860041~5553/86 $1o.00+0.oo Printed in Great Britain cl987 Pergamon Journals Ltd.

CiJCORRECTNESS AND t~U~ERICALTREATMENT OF BOUNDARY VALUE PROBLEMS IN DIFFERENTIAL-ALGEBRAICEQUATIONS*

Systems of differential-algebraicequations are considered. It is shown that when boundary value problems are correctly formulated a certain property of reducibility of the system is important. The use of the shooting, linearization and difference methods for solving wellposed boundary value problems is discussed.

1. Introduction. This paper is concerned with equations f(z'(f)*z(f), Q--O, @[fo,T1, (1.l) where f is SmoothandthepartialJacobimatrix ft'(y, t, t) has aconstantnon-trivial nullspace. Firstly, we trytostateboundaryvalue problems ascorrect operator equations in naturally chosen Banach spaces. Then we show that for such well-posed boundary value problems the linearization,shooting and difference methods, used in ordinary boundary value problems may be adapted to boundary value problems in differential-algebraicequations. We thus dispense here with the assumption that imf[(y,z, t) must be only time-dependent,which is used in /l--4/. For the correctness of a boundary value problem it is essential that the differentialalgebraic equations may be transferred into an ordinary differential equation and a finitedimensional attachment, These differential-algebraicequations are called transferable into a state variable system in /5/. This property generalizes the single nilpotency of a matrix pencil often used in case of linear differential-algebraicequation. For a special index-two problem the solvability is investigated. The related boundary value problems are shown to be ill-posed in the chosen basic spaces. This ill-posedness is caused by certain differentiations. Finally, it is shown that the ordinary integration methods become unstable even in the case of index-two linear constant coefficient equations and even if constant stepsize is used. Therefore, shooting and difference methods are not expected to work well in the case of ill-posed boundary value problems.

2. Choosing a natural operator-formulation. The basic assumptions used in this paper for ~q.il.1) are the following two. Condition 1. SsR”XR” be a region. The map f: %[f,, T]+R'" be continuous with continuous partials f,', fr' :IBX[t,, T]*L(R) satisfying the Lipschitz-condition lk'(Y,2,~)-f~(~.~,t)I~L(IY-Bl+l~-z"lft (Y,z,t),(y", %t)'+X[& T], +% 2. Condition 2. kerf,'(y, 2,f)=N, (Y,2,WQX[fl, Tl. Denote by Q a projection onto N, P=I-Q, so that we have QxEN, XER" for ZEN. Lemma 1.

and Qx=x, Px-0

Conditions 1, 2 yield f(Y, 8, l)==fPY,

(Y, 5, $1Kwto,

$3 t),

Tl *

Proof. For (v, 5, t) 49X [to, TJ

holds. Q.e.d. Consequently, for s~f?" we have

0

W(f), zf% f)~f((Pt)'(t),t(t),t). From this viewpoint the equation

f((W’V),sW,

t)-=a

Nto,

TJ

(2.1)

is a more precise formulation of (1.1). Put X=(zssC~Pz&'"}. Equipped with Ilzll-llzll~+ll(~z) 'If-* XETX, X becomes a Banach space. Lemma 2. ThesetXisinvariantunderthechoice projections are equivalent.

of Q.

Norms corresponding to different

Proof. Let Q,,Qt be projections onto N, Pi-I-Qt.t--l, 2. Denote by X$,X, the corresponding subsets of C. For 5=X,, i.e. t=C, P,xMY, (I-P,)z(f)-Q,~(~)EN,we have *Zh.vychisl.I~at.mat.Fiz.,26,1,50-64,1986 3.

31 P,z=P~P,zfP*(z-P*)t-P,P,x~C”‘, and therefore X,SX,. Analogously, we show X+X,. F&her, if ().I/o ]I$ denote the norms corresponding to Qlresp. 9~ then it holds for x=x=x,=x2 llrllz4141~+ll (Plz)‘114++11 (P,P,z)‘1l,~/l~ll~+II~~ll~ll (P9)‘Ilm~max(1, II~&) ll4lt and, on the other hand, Q.e.d. Define the map F:4f)*C on

ll4~m=f~,

ll~~~-)ll~l~~, IEX.

4a-(s~X)((Pz)'(t),z(t))E~, Mb Tll W T], s&D. (F;t) (t)=f((Px)'(t),z(t), t)* t=[t., Obviously,the operator equation Fz-0 is equivalent to (2.1). Note that 99 is open in X. NOW, we complete Eq.(2.1) by a suitable boundary condition r(x&), z(T))=O.

Here we assume

(2.2)

Condition 3. $,rR"XIP be a region, Q,a{(z(t,), r(T))/zfa). L be a subspace of R". The map r:8,+L be differentiable with Lischitzian partial derivatives. Define the boundary operator B:B+L by Br==r(s(to),z(T)),XGO. Additionally, we use the map s: 6-CXL, S-(F, B), so that we may formulate the boundary value problems (2.11, (2.2)in which we are interested as an operator equation

92-O.

(2.3)

Letma 3. Let condition l-3be true . Then F, 8 and flFr&het-differentiable on 9 with Lipschitzian derivatives_ Proof. Firstly, for a=X,a~b

we compute

(F'(s)a)(t)rA.(t)(Pz)'(t)fB.(t)z(t). where A,@)-f,'(@)'(t),s(t),t)l

W)-f,'(W)'(t), r(t),t).

Further, with

we derive Consequently,

IIP’(z)-F’(~)Il~max(l;,, Lz)ll~-211, 5, f&a. Analogously, we derive x=6, P=z ~'(~)~~r,'(~(~~),zfT))2(tp)+r~(t(t~),z(T))z(T), and

la'(~)-~(f)l
(2.4) holds. This is a consequence of the Implicit Function Theorem. Now, the bijectivity of Y(f.), yields the correctness of Eq.(2.3) in its X, (CXL)setting in the sense of Hadamard. Taking into consideration this fact we are mainly concerned with the linear map r(t.). Insect. 3we try to formulate conditions ensuring the bijectivity of ,W(z.) and so the correctness of (2.3). Insect. 4wedealwith problems (2.3)thatare ill-posed in their X, (CXL) setting. TO simplify scme formulations we assume Q to be the orthoprojection onto PI in the following. Then P is the orthoprojection onto NL. Because of Lemma 2 this does not restrict generality in any way. Besides P, Q we use the orthoprojections P,(y,r,.t), P&g, t, t) onto R(Y,+,t)-imf~(Y,.%~) and their orthogonal complement resepctively, (Y,z,t)&X[t.,T]. Sometimes we are interested in initial value problems for (2.1) with the initial condition

32 (2.5)

D,(x(t,)--a)==O,

where D,=L(R"), imD,==L, a=fP. This initial condition is a special case of (2.2) arising w+h r(u,u)=D,(u-a), U,u=R='. Obviously, condition 3 is true in this special case.

3.

Well-posed

boundary

Firstly, let us formulate correctness of (2.3).

Condition 5.

value

problem.

two assumptions

which will turn out to be essential

G(y,x,t)-Pnl(y,x,t)f?‘(y.r,t)ls

t)-'l&M*, (Y,G WWt0, Tl. (Here the index r means

be a bijection

the correspondingly

restricted

Theorem 1. Let conditions 1-5 be true, 2~6 are equivalent: 1) F(x) acts bijectively from X onto CXL; 2) S'(x) is injective and dimL=dimNl; 3) the matrix

from N onto

for the

R(Y,G~)', IG(y,r,

norm.)

be fixed.

Then the following

assertions

s=r,'(r(to),x(T))Z(tO)+r,'(x(to),x(T))Z(T) acts bijectively from N-L onto L, where the "fundamental-matrix" initial value problem. 4)

A,(t)(PZ)'(t)+B.(t)Z(t)=O, dimL=dim NA. SlNl is injective and

Proof. Consider

the linear boundary

Z(e) is the solution of the

P(Z&)--I)==O;

=[to,Tl,

value problem

S'(x)z=g,

i.e.

t=[to,Tl,

(F'(r)z)(t)=A&)(Pz)'(t)+&(t)z(t)=d(t),

B'(x)z=D,>.z(t,)+D,,,z(T)=b, where g-(d,b)==CXL,D,,.-ri'(x(to),x(T)), i=l,2. Now /l/ Theorem 1, yields the existence and uniqueness .??(.)EX (that means the columns of Z(.)belong to X). Moreover

the initial

of the fundamental

matrix

value problem

F'(z)z-0, P[z(t,)-al=0 are uniquely solvable on X and z(t,a)=Z(t)a=Z(t)Pa are their solutions. Consequently, we have ker (F'(x))--(Z(.)Pa(a4?'), dimker(F'(x))=dimNL. Further, the initial value problem F'(x)z=d,

Pz(&)==O

has a unique solution, too. We denote it by ZIEX. is the unique solution Now, z(.,a)+zs(.)==Z(.)a+zc,(.) F'(x)z=d, Using this representation

we find that

z=X

F'(x)z-a, if

z satisfies

the equation

of the initial

value problem

P[z(t0)-a]-0. is a solution

of our boundary

value problem

B'(z)z-b

F'(x)z==-dand its initial value

satisfies

the condition

[D,,b(t,)+D,,~(T)lPz(t,)-b-B'(z)za. Therefore,

9'(x) acts bijectively

L, that means 1) and 3) are equivalent.

from X onto CXL if.S acts bijectively from NL onto On the other hand, the equivalence of 3) and 4) is

obvious. Further, 1) yields 2) immediately. If there exists an element a&P with Pa#O,Sa-0, we obtain Finally, let 2) be true. a non-trivial z(.)-Z(.)a=kerb'(x), which contradicts the injectivity of 8'(m). Therefore, 2) yields 4). Q.e.d.

Theorem 2. Let conditionsl-5 be valid z.Ea be a solution of (2.3). S'(x.)be injective, dimL=dimNi. Then the boundary value problem (2.3) is correct in istx, (CXL)setting and the Lipscnitz-condition (2.4) holds. Troof. This assertion Theorem.

is a simple consequence

of Theorem

1 and the Implicit

Function

Theorem 3. Let conditions 1, 2, 4, 5 be true. In (2.5) let kerD,-N. Let X.EB satisfy Fx.=O. Denote a.=x.(t,). Then the initial value problem (2.1), (2.5) has a locally is sufficienctly unique solution ~(.,a)~49 for each a=U,- (c~R'"~~D,(c-~.)IGJ}, where 00 Moreover, ~(.,a) depends continuously on a, more precisely, we have 11z(.,a)-r(.,C?)/G small. Klo,(a-r)l, a, &mu,.

Proof. Firstly, here the matrix

S-D,Z(t,)==D,PZ(t,)=D,P acts bijectively

from

NA onto

L. Therefore, the Theorems 1, 2 yield the correctness of the initial value problem (2.1), (2.5) with u-a. as well as the Lipschitz condition (2.4). In particular, the perturbations which means that the initial value problem g=(O, D,(a-a)) belong to B(0, a) for a=U.

Fx-0, has a unique solution is identical

with

D,[t(to) -a.] =D,(a-a)

x(., a)-z(g)EB(x., p) for each (2.5).

~EU..

Note that this initial

condition

33

t&e Lipschitz

Finally,

perturbations.

condition

arises

as a special

case of (2.4) for the above special

Q.e.d.

Refiark 1. In /2/thedifferentiability satisfies the corresponding first variation

of z(r,a) with respect equation.

Ill-posed problems. Let us deal here with certain linear differential-algebraic

to a is derived.

(alaa)z(t.a)

4.

condition

5.

The simplest

example

we find with

equations

which

do not satisfy

m=2,

z,'(t)+zr(t)=dl(t), s,(t)-&(t), *[to, Tl. in this case. Further, conditions are allowed Obviously, no initial resp. boundary has to be differentiable for solvability. This example represents more a differentiation problem which is known to be ill-posed in C. If we put our example in the form

A(Pz)‘(t)+s(t)=d(t), we find that AEL(R*)

&

E=rto, Tl 9

has index two.

Theorem 4. Let A(.): [to, T]-+L(R) be continuously differentiable, ker A(t)=N, *[to, T]. The nilpotency-index* of A(t) be 2 for each %E[tO,Z'].Denote M=A(I-AAD), AD be the Drazin** inverse of A. I-M’(t)P be regular, EE[h,T]. Then the initial value probiem .

t=rto Tl,

A(t) V’z)‘(~)+M)-d(t), A(to)A(to)D[s(to)-a]

(4.1) (4.2)

-0

has for each a=R” and each dEXu -(s~CJMz~C”‘} exactly one solution on X. This solution depends continuously on a. For the linear map F:X+C induced by (4.1), i.e. we have

(Fx) W-4 (0 U’z)‘W+x(~),

imF=Xr.

Denote R==AAD,S==I-R. Here Proof. The properties AM-O,M’-O,PM=A+AM==O

AD,

=1t,,

Tl, x=x,

A+,R,S,M

follow

are continuously differentiable. from the index being equal to 2. Further,

x-xd7 (z~CIRt~C”‘). Let us show that (4.1), (4.2) is solvable for a=R"', d=XM. a&P', d=Xr. Take Firstly we determine the solution y(o)E C (I) of the ordinary initial value problem

y’+ADRy-R’PRy-R’PS(I-M’P)-‘[M(PR)’Ry-Sd+(Md)’]

(4.3a)

-ADd,

y(to)-A(to)A(W’a, then we define

z(.)“C

(4.3b)

by

z-(I-M’P)-‘[Sd-(Md)‘-M(PR)‘Ry]. Multiplying

Pz+Qz-hi’Pz==Sdwhich

(4.4)

the equation

is equivalent

to (4.4), by P and using

(Md)‘-MPR’Ry, PM-O

(4.5)

we obtain

Pz-PM’Pz+PSd-P(Md)‘. Taking

into consideration

that

PM’--P’M-0, P(Md)‘-A+A(Md) and

PSd=A+Md=C”’

‘--A+A’Md-

(A+) ‘AMd-0,

we find that Pz-PS&C”‘.

Obviously, x=y+z belongs (4.2) which we are looking (4.3) yields

to X. for.

We try to show that this element

is the solution

of

(4.11,

(Ry)‘+(Sy)‘+ADRy-R’PRy-R’PSz-A=‘d=O immediately.

RR’PSz-ADd=O.

Besides this, when multiplying (4.3) by R we derive The difference of these two equations is

(Ry)‘-

R’y+ADRy-RR’PRy-

(Sy)‘+R’y-SR’PRy-SR’PSz-0. NOW, we may use the properties SR’PS--S’RPS--S’RS-0, SR’PR--S’RPR--SIR, R’+S’R=R’-R’R-R’S to conclude

that

(Sy)‘+R’Sy-0. Because

of S(t,)y(t,)-0,the component

S(t)y(t) h as to vanish

identically.

*Here the nilpotency-index of quadratic matrix A denotes the size of its largest corresponding to zero eigenvalue. ** The definition of the Drazin inverse matrix see, for example, in /6, P.44/.

USSR 26:1-C

Jordan

cell

34 (4.5) by R we obtain

On the other hand, multiplying

Rz-RM’Pz=-R(Md)‘,

5 i.e.

Rz=RM’Pz-R(Md)‘=RM’PSd-R(MPSd)‘--RM(PSd)’=O Especially, we have Rs(t,)=Ry(to)=Ra, IJow we consider F1=_4(Pr)'+z.

i.e. x satisfies We have

the initial

condition

(4.2).

Fx=A(Py)‘+A(Pz)‘+y+z=A(Ry)‘+A(PSd)’+y+z= A (R’y-ADRy+RR’PRy+RR’PSz+ADd)+A(PSd)’+y+ SdfM’Pz-(Md)‘--M(PR)‘Ry=d+AR’y+ARR’PRy+ ARR’PSz+A(PSd)‘+M’Pz-(Md)‘-M(PR)’Ry. Finally,

using the properties ARR’PR=AR(RP)‘R=ARR’R=-ARS’R=ARSR’=0, ARR’PS=AR(RP)‘S=ARR’S=-ARS’-AR’S, AR’y-M(PR)‘Ry-AR’Ry-MR’Ry-ARR’Ry=-ARS’Ry=ARSR’y--O, A(PSd)‘+M’Pz-(Md)‘+AR’Sz-(ASd)‘-A’PSdf A’SPSd+AS’PSd-(ASd)‘+AR’PSz--A’RPSd+A(R’+S’)PSd-O

we obtain

Fx-d.

the solvability of (4.1), (4.2) for a=R", dEXM is shown. To prove the uniqueness of the solution we consider x=X satisfying problem. A(Px)‘+z-0, (Rx) (to) -0. Thus,

Multiplying ADx-0. Since

the differential-algebraic

equation

the initial value

by Jf resp. by AD we obtain

Ms-0

resp. R(Px)‘+

ADx=ADRx, R(Px)‘-R(PRz)‘+R(PSz)‘~R(Rx)‘+R(A+Mz)’~R(Rx)’~(Rx)‘-R’Rx,

we have (Rx)‘-R’R.z+ADRx-0,

(Rx) (to) -0.

That means Rz is the solution of a linear homogeneous Rz vanishes identically. lrloreover x=-A(Px)

ordinary

2.

therefore

‘--A(PRz)‘-A(PSx)‘=-A(A+Mz)‘-0.

The continuous dependence on a of the solution of (4.1), solution of (4.3). Finally, Fx belongs to C for ZEX and MFz=MzEC”‘. Q.e.d. i=l,

intial value problem,

(4.2) follows

from that of the

Consequently,

im F=X,.

we have

Ti>eorem 5. Let the assumptions of Theorem 4 for (4.1) be true. Let D,&(P), Then the boundary value problem in (4.1) with the boundary condition

imDi--L,

D,z(t,)fD*z(T)=b b=L,d=Xu

has for each

acts bijectively

a unique

from

imA(t,)A(tJD

solution

on X if the matrix

s=D,z(t,)+D,Z(T) onto L. Here the matrix

A (PZ)‘+Z-0,

2(*&X

A(to)A(to)DIZ(to)-ll-O.

Proof. As in the proof of Theorem 1 we find an expression (4.2) : Note,

(4.6) is chosen to satisfy

of the solution

of (4.1),

s(t, a)-Z(t)R(Qa+z,,(t). that here we have kerF=(Z(~)R(to)aIa&?“},

The initial

value problem-solution

dim ker F-dimim

r(., a)

satisfies

R(t,)
the boundary-condition

(4.G) if

[D,z(t,)+D,Z(T)lR(t,)a~b--D,~~(f~)-D,~~(T) holds.

p.e.d.

Remarks 2.

Obviously,

in Theorem

5 it has to hold that

dimL-dimimA(to)A(to)D
A(t)(P~)'(t)+B(t)z(O-d(t),t=r:e,Tl. of (4.1) by replacing A(t [h(t)A(t)+B(t)]-*A(t) in all formulations

(4.7) (cf./T/).

35 5. XM equipped with the norm Ilallu-lldll,+ll(Md)'ll,,d~X~, becomes a Banach space, again. Since liFzll~~llAfPx)‘+zll*+H(dlz)‘ll..cm~

~i+nM’ll,,

ll~u~~n~ll~)u~ll

holds f'd;all t=X, the map F is continuous in the X, Xu-setting. If S in Theorem 5 acts bijectively from irn~(~~)~(f~)D onto L, then the linear map F corresponding to the boundary (4.6) is ahomeomorphism from X onto X&Z, i.e. the linear boundary value value problem (4-l), problem (4.1), (4.6) are correct in their X, (X&%)-setting. from Remark 5 for non-linear Eqs.(2.l) 6. Trying to generalize the X, (X,Xl;)-formulation seems to be very formal and practically useless. Note that in la/ the solvability of linear constant coefficient differential-algebraic equations, isinvestigated for right-hand sides belonging to C'"'. Expressions for the general solution are given. In /6/ and /9/ boundary value problems in time-dependent linear differential algebraic equation are considered. For the solution z. of an initial value problem is found. an inequality of the type liz.li _gK{la(+lldll,+lld'l},+...Slld"'[l,) 5. On the numerical treatment of well-posed boundary value problem. Under the assumptions of Theorem 2 - Le. for correct boundary value problem (2.1), (2.2) in their X, (CXL)-setting- the numerical methods known from ordinary boundary value problem can be applied resp. modified for (2-l), (2.3). The necessary theoretical analysis for the shooting and difference methods is given in /2, 4/. A collocation method is proved in /lo/. For shootinq methods we choose the formulation of the initial value problem so that Theorem 3 is applicable, e.q. D,=P or D,=f,'(O, 0, to) in (2.5). Then the simple-shooting map S: R"-+R" S(a)==r(xW3, a)*E(T,a)), a=U,

is defined and smooth on u,. Further, we have S(a)-S(PU)EL,

kerS'(a)-N,

rankS'(a) -rank P=dimN-', W&J., withaW=(O, 0). The (simple) shooting equation represents a compatible least-squares problem with constant rank Jacobi matrix. To realize the shooting methods we may use generalized Newton methods. Analogously, multishooting is admissible /2/. The resul.tingnon-linear equation has a cyclic structure similar to that in the case of ordinary boundary value problem. But, as the matching conditions are taken in h'Lonly, in the case of (2.11, 12.2) some singular constant-rank blocks arise in the Jacobi matrix. Integration methods for initial value problems are described in /ll,l, 3/ as well as in many other papers. They work well under the assumptions of Theorem 3. At the same time, Theorem 2 provides sufficient conditions for the feasibility and convergence of the Newton-Kantorovichmethod for the operator Eq.(2.3). If we apply this linearization method to (2.3), starting with an initial approximation x@b which is sufficiently close to z.,weobtain the sequence of linear boundary value problems.

s-‘(24 z. --F(Z”), Z”**==x”+%“,

t5.11

n-=0, 1, . . . .

65.2t

Theorem 2 yields the well-posedness of (5.1) in the 8, (CXL)-setting and that z.+, remains in B,.Further, we have ~~~~-z.~~~O,n+rn.Modified damped Newton-methods can be used, too. For this, purpose, we replace *'(G) by ~(z~),OCk,Sn, in (5.1) and put z,+,-z,,+cc~z~ instead of (5.21, respectively. For initial value problem in special quasilinear equations arising in (2.1) with f fY* 2, f)--il(t)Y-_gk t) the wxhmped modified Newton method (k,==O, nZ0) is considered in /12/. In this case the resulting linear initial value problems are A(t)(Pt.+r)'(t)+B,(t)s+,(t)-B,(t)s,(t)fg(z"(t), t), where

Bo(t)=g,‘(zo(t), t).

Mb, Tl,

a (3.*,

(to) -a) -0,

Note that ftt=Iis used in /12/ while a is assumed to be a consistent initial value. In /12/ the nullspace of A(t) be time-dependent, too. Further, there AEC"', g=C'*', 2&s)-fz, z&?‘) is assumed to achieve that rt,belongs to C"). Finally, for sufficiently small /&o-z& it is proved that l}z~-s./__,-cO, n+=. The basic conditions used in /12/ are the "rank-degree" conditions, i.e. rankA(&)=degreedet[hrl( )-t-g.'(a, t,)], and rankA >rankA(t) which are equivalent to Condition 5 in essence he'ee. Difference methods for correct boundary value problem (2.11, (2.2) are investigated in /2, 4/. Especially the one-step methods are of interest here. Using the grid g: t,<...Ct,-?', and the stepsizes h.=l.-t,-,,e)l,is-max{k,le-l,Z,...,n} we consider the methods e&,2,...,+ f(h,-'(2,-z,,), &&*+b&-*. b,t.+b,f,-r)==O,

(5.3)

and h.-'P(~*:,-s,-,)--b,y.-~,y.-,~O,

f (y., 5.7 f.) ==O, Qy.-0,

cd,

2,....n.

(5.4a) (5.4b)

In general, b,+b,=l is assumed. Then these methods are consistent in the usual sense. The order is the same as in the case of ordinary differenti&. equations. For the realization of (5.4) we have to know or to compute the projections Q, P=i-Q. (5.4) may be written in the form f(ar., Qt.+Pr.-,+h.(b.~.+b,br,,,), t.)=O,

36

QYe=O,

Pze=JkJ-k

(boy.+bty*-,),

e==f, 2, . . . , n,

tiich shows the relationship of these methods to those in /5/. Compared with the ordinary differential equation-case we have two peculiarities. Firstly, the difference Eqs.(5.3),(5.4) have to be completed by the boundary condition r(s*,t,)-0 well a.s by certain additional equations defining the "missing" initial values. For instance, we can take f(Ya, 50,t0)-0, Qyo=O.

(5.5)

as

(5.6)

required additional initial conditions have to be derived from (2.1). Note that the non-linear P system P(zo-a)-0, f(Y0, ru,to)=O, Qw=O under the assumptions of Theorem 3 is isolatedly solvable. This fact is important for initial value problems, too, because it is well-known that the preparation of initial values in the case of differential-algebraicequatibns is hard in practice. Secondly, in (5.3) recursions in the finite-dimensionalequation belonging implicitly to (2.1) may give rise to instabilities. This becomes obvious if f degeneratesto f(y,z, t)=x. In this case (5.3) means boz,+b,x.-,=O, e- 1, 2,. . . , n, while (5.6) yields z,=O,YO=O. The resulting difference scheme is stable in the usual sense if /b,/b,lO and S-0 so that for each grid gas, for arbitrary The

and for

~,[‘~d3(x.(t.), i--l,Z,

p),

e=O,

I,...,

n,

y”‘EBt(Pz.)‘(t,),

P),

the inequality ItI max (z.1i1-z!*1 I-l-max IP(z, -zIZ )h.-'.-,A...,* .-S&l....,"

(5.7)

-zJ!:)h.-'I+IYl"-y~rlJ9S{ Ina& Id*‘P(z!"'

.-o,,,...**

zu!t’(+l Ultl_UItl [$I yI~l_y~~l~) holds. This stability definition pays rerJardto all kinds of errors arising in the realization of our difference scheme, i.e. local descretization errors and errors in the non-linear equations which have to be solved numerically. Inequality (5.7) leads to the usual error-inequality max Is.@.)-2.1+ mar. IA-'[(PA(Gf-(Pd ft.-All-‘.**....* *-w*...,A h.-'(Pz.-Pz.-J I< mftx lr.l. Il.“,,*,...,” where &denotes the local discretization error. It may happen that &belongs to a certain subspace of pwhile the rouding errors cannot be constrained to remain in this subspace. The stability of scheme (5.4)--15.6)may be defined analogously. Denote by s,(h) the class of all grids g with the sole restriction h
Proof. In /2/ the stability of the scheme (5.31, (5.5) and the equation Prl~(tol(o,to,tO)=O instead of (5.6) was proved under the additional assumption that im f,'(y.x,t) is independent of (Y,x1. Using (5.6) we may dispense with this assumption. For the proof we choose a suitable new map9 fn/2,Sect.3/, and proceed analogously as in /2/. p.e.d. in (5.4). Then the Theorem 7. Let all assumptions of Theorem 2 be valid and b,+b,=l difference scheme (5,4)-(5.6)is stable on s.(n) for sufficiently small W-0. Proof. The scheme (5.4)-(5.6)represents a modified difference scheme /If for the enlarged differential algebraic equation

(5.8) (W'(t)-y(t)-09 f(Y(Q,z(t);t1=0, -rb, Tl, instead of (2.1). The enlarged boundary value problem (5.81, (2.2) is correct, again. IIitbthis /2, Theorem 6c/, yields the stability of our scheme as a special case. Q.e.d. Note that (5.4) do not contain any recursions in the finite-dimensionalequations. In the consequence of the above discussion we find that only the scheme (5.4)-(5.6)with

37

b,=b,=‘/, is of order 2 and stable. unbr additional smoothness conditions the above stable schemes are proved to represent bounded additive commutative discretizationmethods /4/, so that the mesh independent principle is valid here. 6. On the numerical treatment of ill-posed differential algebraic equation problems. Let us deal now with some aspects of the numerical treatment of problems (2.11, (2.2) being ill-posed in their X, (CXL)-formulation. Firstly, we investigatetheimplicit Euler method which we obtain in (5.3) putting b,=i, b,=O. We restrict ourselves to the simplest case of such ill-posed problems, which have just been described in Theorem 4. Moreover, we assume A to be a constant index-two matrix here. In this case the implicit Euler method gives e=l, 2,..., n. Ahr-'(2.--r.-,)f~*=d(t,), (6.1) We complete (6.1) by the initial condition 14.2) As

AAD&-a) -0. an additional initial condition we may choose

(6.2)

M(xo--d(Q)-0. (6.3) (6.2) and (6.3) determine the initial value AZ o necessary for the first integration step. The value @?, is not needed here. To realize the implicit Euler method we have to solve the linear equation (A+h~Z)x,-AZ.-1+h,d(l.) (6.4 per step. Thereby, A+h,l is regular (but ill-posed) for sufficiently small stepsizes. Solving (6.41, we have to take account of rounding errors in all components,i.e. instead of r# we obtain &satisfying (d+fi.Z)~,~df._,+~,,d(t,)

+&.

The error 6, results from rounding errors, and, in general, we are not able to constrain 6. to belong to certain given subspaces. Consequently, the scheme (6.1)--16.3) is said to be stable on the grid-class Bif there is a constant S>O so that for each grid gms and for arbitrary &Ep,e==O,&...,n, and e==l, 2,. .., R,

w,=Ah,-‘(z.--Ze_i)+z., u==dd%‘ap,

(6.5)

V==MZ~,

(6.6)

the inequality max }z,~+lPzol+ max ih,-‘(Pz.-Ph-,) -l,*,...,” l-L*,...,” S{ maf 1u4+l4+l4~ .“.‘,ll,...,ll

IG

l

(6.7)

holds. Otherwise the scheme is said to be unstable. on the other hand, the local discretization error T#==h.-i[dz.(t,)-dz. (t.-,)]-(Az.)‘(t.) belongs to

imA

here.

Theorem 8. Let A be constant index-two matrix d=X,, o&F+‘. Then the scheme (6.1)-(6.3) is unstable even in the classes g,(Z) of equidistant grids with h
s-ZZ*Pr,Sh where

2

P(d+hf)-*w,_,,

e-4,2,.

. . ,n,

H-(d+hI)-‘A. Further, from (6.6) we derive Pzc-A+AufA+v.

Using the similarity transformation TAT-1p-J we decouple the above equations in the following way. Let I-diag(r,, 0, J,), where 1%contains all Jordan-blocks corresponding to non-trivial eigenvalues,Jo contains all second-order Jordan-blocks corresponding to h=O, while 0 means all first-order Jordan-blocks to h-0. We compute (A+hZ)-‘ET-*(HZ@-‘T, X-T-’ (i+hZ) -‘JT. Using the transformations &p=Tzh j--l, above representation into

+

2,...,

I(, +TPz,,

o,- Tw, f-4, Z,...,n,we decompose the

[ (HhZ) -‘J]‘~.+h r( f (J+hZ) -‘l]‘(l+hZ) -‘%-I,

38 into three parts corresponding to the structure of Y which are not coupled. Then, for Gz*components corresponding to the first two blocks of J, a stability-inequalitylike (6.7) is true. Consider the third part of our equations. Thereby we take into consideration that Jo'-0, (I,+hl)-'-L-' (I-h-110), (Jo+hl)-‘lPh-‘Jo. If ?)i denotes the third part of the components of &,and pr those ofcehwe calculate %-A[ (r,+kZ)-‘~.+k-‘r,(r,+hl)-‘~~,],

e>2,

~,-k-‘J,~p+fi(3,~kZ)-‘~,. This leads to

e>2. ~.-(I-k-'Jo)~L.fk"Jol".-i-~,-h-'(I~~,-l~~.-~), Now, it is easy to see that this part of our scheme is unstable. Q.e.d. and

f-0,fr...?, we have to put ror-t~~irnA.i~1~2,...,n, Remarks. 7. According to y=szy~,(t,), in (6.5) IS-=O,v==Oin (6.6). In this case (6.6) yields PxI-O and (6.5) leads to

.-I

I.

-

h

c

E*+‘A+c._,.

j-0

After the decomposition we obtain ?I=+~l, ll*-1.,c>2, where 7,denotes the corresponding part of components in TA+?h

Hence, we obtain the estimation

maz I+.-x.(t,)l+ nlaz h-'IPt.-PI._,-P+.(f,)f Pt.(t._,)[c *-l.*....,I *-Lt....," S mar If& ,-*,%....a which yields the convergence. 8. The convergence of constant-stepboundary value problem being applied to constantcoefficient linear differential algebraic equation was proved by various authors (e.g. in /ll/). However, if we take into consideration, roundinq errors, the error resulting from the implicit Euler method in the case of a differential algebraic equation with index ~32 has the size h‘*+n(O(h)+&.h-x), where 6 represents the rounding errors. 9. The constant-step boundary value problems work well if we succeed in making the influence of rounding errors unimportant. This is possible in the case of some relatively simple problems ~'111. 10. Analogously, other integration methods are unstable for higher index problems.even if consistent initial values are used. Our stability discussion related to the initial value problem showed that is is useless to apply shooting methods for solving the boundary value problems described in Theorem 5. Further, we must not expect more optimistic results for general linear resp. non-linear differentail algebraic equations. Up to now, the best way to treat such differential algebraic equations has been completing the problem so that it becomes well-posed. For this purpose, in /13/ differential algebraic equation of the form t+t*,Tl cp(s'(t),r(t), 1)=0, *(s(t), G"O, areunderstoodasordinarydifferentialequationson smooth manifolds. Then, well-known results about vector fields and dynamic systems on manifolds may be applied to prove the solvability of differential algebraic equations to complete the problems. For linear differential algebraic equations e.g. in /9, ll/, so-called reduction methods are proposed. There the derivatives making the problem ill-posed are taken analytically. By (analytical)differentiationsthe index of the system becomes smaller. Another way to solve linear boundary value problems indifferential algebraic equations like (4.1) is determined by the proofs of Theorems 1, 2. We may utilize the relation between the solutions of (4.11 and those of (4.3), (4.4). This type of methods is proposed in /6, 14/. However, they seem to depend very strongly on the accuracy of the Drazin inverse. Moreover, in /6, lS/ for linear differential algebraic equations some regularization methods are investigated. Finally, note that the collocation method described in /lo/ worked well in the case of some higher index test problems, while the convergence is proved for correct problems only. In this collocation method based on trigonometric functions the derivatives are taken exactly, too. In general, our knowledge about differential algebraic equation not satisfying the assumptions 4, 5 is very poor up to now. The theoretical analysis is only in its beginning. Consequently,

the same is true for the numerical

analysis.

REFEREXES 1. &Z

R., Multistep methods equations. Beitr. Numer.

for initial value problems Math., 12, 107-123, 1984.

in implicit

differential

algebraic

39

2. MARE R., On difference and shooting methods for boundary value problems in differential &%gebraic equations 2. angew. Math. und Mech., 64, 463-473, 1984. 3. MAR2 R., On initial value problems in differential algebraic equations and their numerical treatment. Preprint 44. Humboldt-Univ. zu Berlin, Sektion Math., 1982. 4. M&R2 R., On the numerical treatment of boundary value problems in differential algebraic equations. In: Numer. Treatment of ODE's, Seminarbericht NL. 46. Humboldt-Univ. zu Berlin, Sektion Math, 127-141, 1982. 5. GRIEPENTROG E., Zur numerischen integration steifer impliziter differentialgleichungssysteme. In: Numer. L&sung Differentialgleichungen,Rept R-MATH-01/83, AdW der DDR, Inst. Hath. 12-23, 1982. 6. BOYARINTSEV YU.E., Regular and Singular Systems of Linear Ordinary Differential Equations, Novosibirsk, Nauka, 1980. 7. CAMPBELL S.L., Index two linear time-varying singular systems of differential equations. SIAM J. Algorithm and Discrete Methods, 4, 237-243, 1983. 8. CAMPBELL S.L., Linear systems of differential equations with singular coefficients. SIAM J. Math. Analys. 8, 6, 1057-1066, 1977. 9. CHISTYAKOV V.F., Methods of solving singular linear systems of ordinary differential equations, in: Degenerate Systems of Ordinary Differential Equations, Novosibirsk, Nauka, 1982. WERNSDORF B., Ein Kollokationsverfahrenzur numerischen Berechnung von periodischen 10. LCisungenimpliziter nichtlinearer gewshnlicher DGLn: Dis. (A), Humboldt-Univ. zu Berlin, 1984. 11. PETZOLD L.R. and GEAR C.W., ODE methods for the solution of differential-algebraic systems. Rept. SAND 82-8051, submitted to SINUII. 12. CHISTYAKOV V.F., Linearization of degenerate systems of quasilinear ordinary differential equations, in: Approximate Iilethods of Solving Operator Equations and their Applications, Irkutsk, Sib. Power Inst. 1982. W.C., Differential-algebraicsystems as differential-equationson manifolds. 13. ~EI~OL~ Rept. ICNA-83. Univ. Pittsburgh, Inst. Comput. Math. and ~ppl. 1983. 14. BOYARINTSEV YU.E., Singular systems of linear ordinary differential equations and numerical methods of solving them, in: Applied Mathematics, Novosibirsk, Nauka, 1978. 15. CAMPBELL S-L,, Regularizations of linear time varying singular systems. Preprint. North Carolina State Univ., Dept Math. 1982.