regularization of index-1 differential-algebraic … · 46 p.k. moore et ai. definition 3.1....

Pergamon Computers Math. Applic. Vol. 35, No. 5, pp. 43-61, 1998

Copyright©1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0898-1221/98 $19.00 -I- 0.00 PII: S0898-1221(98)00004-2

Regularization of Index-1 Differential-Algebraic Equations with Rank-Deficient Constraints

P. K. MOORE Department of Mathematics, Tulane University

New Orleans, LA 70118, U.S.A.

L. R. PETZOLD t Department of Computer Science and Army High Performance Computing Research Center

University of Minnesota, Minneapolis, MN 55455, U.S.A.

Y . R E N $ School of Mathematical Sciences, University of Bath

Bath, BA2 7AY, U.K.

(Received April 1997; accepted June 1997)

Abstract--ln this paper, we present a regularization for semiexplicit index-I diiferential-algebraic equations with rank-deficient or singular constraints. We consider those problems for which the solution is well defined through the singularity. We give convergence results for the regularization applied to linear DAEs, and present some numerical experiments which illustrate its effectiveness.

Keywords--Differential-algebraic equations, Regularization, Singularity, Rank-deficient.

i . I N T R O D U C T I O N

In this paper, we consider semiexplicit differential-algebraic equations (DAE)

x' = f (x ,y) , (1.1a)

0 = g(x , y) . (1.1b)

The DAE (1.1) is index-1 [1], if gu = ~ is nonsingular. These types of systems arise, for example, in circuit analysis, chemical process simulation, power systems, and many other applications.

Problems with rank-deficient or singular constraints can exhibit a number of different solution behaviors. For example, if the constraints are rank-deficient (i.e., if g~ is rank-deficient) but constant-rank, the problems can be higher-index [1] or the solution may fail to exist at all, or there may be a well-defined solution. If the constraints become singular at a single point, the solution may bifurcate at that point, there may be an impasse where the solution does not exist

tThe work of this author was partially supported by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative Agreement Number DAAH04-95-2-0003/Contract Number DAAH04-95-C-0008, and by ARO Contract Number DAAL03-92-G-0247 and DOE Contract Number DE-FG02-92ER25130. SThe work of this author was partially supported by the Army High Performance Computing Research Center, and the Minnesota Supercomputer Institute at the University of Minnesota

Typeset by A~S-TEX

43

44 P . K . MOORE et al.

beyond the singular point, or the solution may be well defined through the singularity. In this paper, we are interested in problems for which the solution is well defined through the singularity and the problem is essentially index-1.

In Section 2, we introduce a regularization for rank-deficient or singular index-1 problems. The regularization is motivated by trust-region methods from numerical optimization and is related to some regularizations proposed for higher-index systems in [2]. In Section 3, we define more precisely for linear DAEs the class of problems which the regularization is designed to handle, and give convergence results. In Section 4, we present some numerical experiments which illustrate the effectiveness of the regularization for linear and nonlinear DAEs.

Regularizations for DAEs of index-1 have been studied in [3-6]. However, to our knowledge, these regularizations are not applicable if the system is rank deficient or singular.

Much work has appeared on regularizations for higher-index DAEs. See, for example, [6-8] which deal with problems in which there are no singularities. A regularization for Euler-Lagrange equations which is based on the augmented Lagrangian method from numerical optimization is proposed in [9-12]. An extension which is applicable for singular systems is given in [13].

A regularization for Euler-Lagrange systems is proposed in [14,15] which deals with singularities by first identifying them via Ganssian elimination and then adding to the vanishing and linearly independent constraints of their third derivatives. An alternative scheme for singular higher-index DAEs is a global coordinate mapping strategy, which reduces a differential-algebraic system to a singular ordinary differential equation system [16].

2. I N D E X - 1 R E G U L A R I Z A T I O N

In this section, we will derive a regularization of (1.1) which is appropriate for singular problems. Beginning with the regularization

x' = l ( x , y), (2.1a)

0 = g (x, y + hV) , (2.1b)

for index-1 problems without singularities, expanding g in Taylor's series yields

x' = f (x , y), (2.2a)

hguy' = -g(x ,y ) . (2.2b)

We note that when g~ is nonsingular y' is in a Newton direction, and when g~ is singular it is not possible to solve for y' in (2.2).

Now observe that the constraint in (2.1) is equivalent to the optimization problem

. I , 2 mm - Jig(x, y) + hgvy 112.

y' 2

Adding a trust-region constraint to deal with the singularity, we obtain the model optimization problem

1 1 Ilhe'll• < 6. min ~ llg(x, U) + hgz, Y'll~, subject to ~ _ y'

The Lagrangian function for this optimization problem is given by

L = 5 IIg(x, y) + hgyy'll] + IleY'll] - 1 , (2.3)

where e = h/6. Letting Vv, L = 0, we obtain an ODE for y',

(hgXv gv + eI) y' ----- -gXu g , (2.4)

Regularization of Index-1 45

which together with the ODE in (2.1a), yields the regularized system

x' = / ( x , y), (2.5a)

(hg g (2.5b)

When ~ = 0 and gu are nonsingular, (2.5b) is equivalent to (2.2b). At points (x*,y*) where g~ is r of (2.5b) becomes rank-deficient. We handle this problem by perturbing gu at singular, then gv

the singular point to obtain

• ' = y ( z , u),

(hg~ gu + = -

(2.6a)

(2.65)

Later we will see how to choose el and ~2 (in the linear case we can choose e2 = 0), although it will be understood throughout that h, et, and e2 are all nonnegative.

3 . C O N V E R G E N C E

In this section, we will consider convergence of the regularized system (2.6) for linear problems of the form (1.1),

x' = A(t)x + B( t )y + q(t), 0 = C( t )x + D(t )y + r(t).

x(0) = x0, (3.1a)

(3.1b)

We will consider two cases; first, when the null space of g~ is constant and second, when it is not constant.

We will require the following lemma.

LEMMA 1. Let D be an m x m matrix of rank r. Let Vl , . . . , Vm-r be an orthonormal basis

forAf (D) , the ntdlspace o lD . Then 3 U, an orthogonal matrix and vrn-r+l , . • •, vrn s.t. v l , . . •, vrn is an orthonormal basis for R m and s.t. UT D V is the singular value decomposition (SVD) of D.

PROOF. Let Vs,. . . , i 'm-r , Wm-r+l , . . . ,win be an orthonormal basis for R rn. Then Dwm-r+l , • . . , Dwm spans at most a r-dimensional subspace of R m. Let U l , . . . , Um-r be orthonormal such that u ~ D w k = 0, i = 1 , 2 , . . . m - r, k = m - r + 1 , . . . m , and let u s , . . . ,urn-r, x rn - r+ l , . . . , xm be an orthonormal basis for R m. Then

Z) I = U T1D V* =- u~-r T

Xrn-r+ l

0] D [ v l . . . , v m - r , w r n - r + l , . . . , w r n ] = D1 "

Since D = UIblV~, rank D = rank D, and the nullity D = nullity DI = m - r + nullity DI, which implies that the nullity DI = 0. Additionally, DTD = VID~b~V~ which implies that D and D1 have the same singular values. Now apply the algorithm of [17] to D1 to complete the proof, m

A more general result which implies Lemma 1 has recently appeared [18]. We have included our proof because it is simple and uses a different approach.

We are interested in problems for which the solution is well defined and the problem is essentially an index-1 DAE (i.e., if we were to remove the redundant constraints, the resulting problem would be index-l). Hence, we make the following definition.

46 P . K . MOORE et ai.

DEFINITION 3.1. Consider the linear semiexpllcit DAE (3.1). The problem will be called a rank-defic ient index-1 D A E i f the following holds.

1. Af ( D ) is constant, where Af ( D ) is the null space of D. 2. A f (V) C Af(B).

oo.,.oo , , , =, o) ,:, ,:

are obtained by Lemma 1) then UT C = ( 0 ) and uTr = ( 0 ) 0

4. The reduced problem is index-1. The reduced problem is the problem which remains after the redundant constraints have been eliminated and is given by (3.5).

We will also assume that V(t) is smooth.

THEOREM 3.1. Consider the rank-deficient index-1 DAE (3.1). I f x, q, r, and their derivatives are bounded then solutions ~ to the regularization

~' = A( t )~ + B ( t ) ~ + q(t),

(hDT (t)D(t) + e l i ) ~' = -DT( t ) (C(t)~ + D(t)~ + r( t ) ) ,

5(0) = x0, (3.2a)

~(0) -- Y0, (3.2b)

where el = o(h), converge to the solutions x of (3.1) for any t > O, as h --, O. Specifically, [Ix - ~[[2 = O(h), as h ~ O. We remark that under our assumptions, there is no guarantee that fl converges to y. However, the "nonredundant" portion of y can be recovered from ~ using the SVD.

PROOF. For convenience we drop the notation of the dependence on t. Let D = U ~ V T =

U \ ( ~ o ~ V T be the SVD of D where V and U are obtained via Lemma 1. Then V can be written 0 /

(by simple permutation) as V = (Vm,rVm,m-r) where the columns of Vm,m--r are an orthonormal basis for Af(D). By the constant nullspace assumption and Lemma 1,

V. I ,n,m-r -- O. (3.3)

Multiplying (3.1a) by U T yields

(3.4)

where z - ( z: ) = VVy" Thus, since Af(D) C A/'(B), (3.1) is equivalent to

x' = Ax + Bz l + q,

z, = - ~ - ' (Ox + ~) ,

(3.5a) (3.5b)

where/} = BV,,,r. On the other hand, multiplying (3.2b) by V T gives

where ~ = VT~. Now

From (3.3) it follows that

(hr.' + ~ : ) V T¢ = -r. (U Tc* + r.~ + ut,-) ,

v ~0' = ~ ' - ( v D ' v~.

0

Differentiating the identity V T V = I yields

(3.6)

(3.7)

(3.8)

(VT) ' V + V T V ' = 0. (3.9)

Regularization of Index-I 47

- 1 . 5

-2.8

-:L5

-3.8 -3.5 ~ ~

-4.g

-4.5

-5.8

-b.~l

-7.11 - 7 . 5 I I I I i I I I i , , i , , i , , i ,

i

t

1 8 e ~

8O

6 e

4 8

0

-2e f ~

-48

-68

-88

t

. - . . . . i , f • , , v . ,

- 1 . 5

- 2 . g

- 2 . 5

- 3 . 8

- 3 . 5

- 4 . ¢ q

x -4.5

-5.8

-5.5

-6.8

-6.5

-7.g _ 7 . ¢ j i i i i I I I I t i i i i i i i i i i

I i i , o , t i i i

t

1 0 ~ , i , i , i , i . , i , i , i , i ,

60

40

8

-28 ~

-411

- 68

-80

- l l { I g i i i i i i i i i 1 , , i i L i I

t

Figure 1. Numerical approximation of z (left) and y (right) computed via our regularizstion with DASSL for Example 4.1. Top: h = 10 -4, el = e2 = 10 -6. Bottom: h = 1 0 - 4 , el = 8 2 = 10 -7 •

Thus, using (3.3), (3.8), and (3.9),

~ ' 1 2 = - - T t T t ( v . v;2 + v2~ v~) = o. (3.10)

From (3.7), (3.8), and (3.10), it follows that

~] (3.11)

Then (3.6) is equivalent to

(h~, 2 + ,,I) (~i - ~11~,) = -r~ (c~ + g~, + ~),

e l ~ = O.

(3.12a) (3.12b)

Recall that £2 contributes only to the redundant portion of y and is therefore unnecessary.

4 8 P . K . M O O R E et al.

-1 .g

- 1 . 5

- 2 . ~

- 2 . 5

- 3 , g

- 3 . 5

- 4 . g

- 4 . 5

-5,1~

- 5 . 5

- b . g

- b . 5

-7.~1

i

t

1 N

8 ~

be

4 e

2g

g

-2e

- 4 1 D

- 8 1 m

, , i , i , i , , J , i , ,

- 1 , 5

- 2 . g

- 2 . 5

- 3 . $ ~'

- 3 . 5

- 4 , E l

-4 .5

-5.B

-5 .5

-5.11

-6 .5

- 7 . 0

-7 .5 i i i i i i i i i i i i i t i i i i i

0 t i i i i i ~ i i

t

8 ~

6i_ 4 0

2g

g

- 2 ¢

- 4 5

- 6 g

- e g

- l O e ¢D o, eo r~ ,o tn qr ~1 ¢~ ~ Q ~ ~ ¢~ qr t~ ,o r~ 0p c, m ~- i i i J : i i i i

t

Figure 2. Numerical approximation ofx computed via our regularization with DASSL for Example 4.1. Top: h = 10 - 4 , el = e 2 = 10 - 8 . Bottom: h = 10 - 2 , e l = e 2 - - lO-e.

Let

P(t) = (h~ 2 Jr E1/)-i ~2 _ ~rll '

Q(t) = (h~ 2 Jr E l / ) - l ~ ,

n(t) = (h~ 2 + ~1I)-1 ~,~,

(3.13a)

(3.13b)

(3.13c)

and let W(t) be the fundamental matrix for the matrix differential equation

W'(t ) + P ( t ) W ( t ) = O, W(O) = I. (3.14)

The solution of (3.12a) can be written as

Z l = W ( t ) z l ( O ) - f o t

Thus, (3.2a) is equivalent to

W(t)W-I($)(~,($)T,($) d$ - j~o t W ( t ) W - I ( 8 ) R ( 8 ) d s . (3.15)

X ' = A x J r . B Z l J r q , (3.16)

R e g u l a r i z a t i o n of I n d ex -1 49

- I . 5

-2.0 ~

-3.0

-3.5

-4.0

-4.5

-5.0

-6.11

-6.5

-7.6

- 7 . ~ i i I i i I I I i i i z i i i i i i i

i

t

1 0 0

6 0

4 0

0 - -

- 2 0

J f

- 4 0

-bO

- 8 6

- 1 0 0 ~

t

- 2 . 5 ~ ¢ ~

-3.0

-3.5

-4.11

-5.11

-5.5

-~, .0

-5.5

-7.0

- 7 . 5 i i I i i i [ I i , , 1 i , i i i i ,

8 0

bO

4 0

21}

>- 6

-4o I

-60

- 8 0

- I M 0

J

t

Figure 3. Numerical approximat ion of x (left) and y (right) computed via our regularization wi th DASSL for Example 4.1. Top: h = 10 -2 , el = e2 = 10 - s . Bot tom: h = 10 - 2 , e l = e 2 - - 1 0 - l ° .

together with (3.15) and (3.12b). Rewrite (3.5a) as

x' = A x - B W ( t ) W - l ( s ) Q ( s ) x ( s ) ds

[/o ] + ~ W ( t ) N - ~ ( s ) Q ( s ) x ( s ) as - ~ , -~Ox - ~ - 1 ~ + q.

(3.17)

Subtracting (3.16) from (3.17) gives

e' = Ae - [~ W(t)w-S(s)Q(s)e(s) ds

+ [~ [fotW(t)W-l(s)Q(s)x(s)ds- ~,-1Cx]

+ [~ [fotW(t)W-'(s)R(s)ds- E-l~] - BW(t)~I(O),

(3.18)

- 1 . 0

- 1 . 5

- 2 . 0

- 2 . 5

-3 . i ~

- 3 . 5

-4 .1 ]

- 4 . 5

- 5 . 0

- 5 . 5

- 6 . 5

-7.1~

- 7 . 5

-1 .e

- 1 . 5

- 2 . e

- 2 . 5

I ' l ' l , l ' , , I , l ' U ' l ,

I I l , , t l l l t t l l , l l l l l

I l l l l l l l l

t

-1 .iJ

-1 .~

- 2 . g

- 2 . 5

- 3 . ¢

- 3 . 5

-4 .11

- 4 . 5

- 5 . 1 1

- 5 . 5

i • u , i • , , i • i • i , i , i •

- 6 . 0

- 6 . 5

- 7 . 1

- 7 , 5 i i i i i t i t i t i i i i i I I I I

,7 i i i i i i i ~ t

t

I . I I t i . ] i • I .

- 3 . 0

- 3 . 5

- 4 . a x

- 4 . 5

-5.1~

- 5 . 5

- 6 . 1

- 6 . 5

- 7 . g

- 7 . 5

50 P .K. MOORE et al.

I I I I I l l l l l l l l l t l l l l

~ i O 0 0 1 1 } O I

t

- 1 . I t , , u , i , n , , , i , u ,

- 1 . 5

- 2 . 0

- 2 . 5

- 3 . $

- 3 . 5

- 4 . $

- 4 . 5

- 5 . 0

- 5 . 5

- 6 . 0

- 6 . 5

- 7 . 0

- 7 , 5 n , n , n , I o n I , I I I I I I [ I

I

t

Figure 4. Numerical approximation of z (left) and y (right) computed via our regulaxization with DASSL for Example 4.2. Top: h -- 10 -4, el ---- e2 ---- 10 -e . Bottom: h = 1 0 -4 , e l = e 2 = 1 0 - s .

where e = x - ~. For h sufficiently small , P(u) can be approximated by (1/h)I on 0 < u < t and thus W ( t ) W - l ( s ) ~ e-(Uh)(t-s)I. An application of Watson's l e m m a for the a s y m p t o t i c

expans ion of integrals [19] yields

o' w(t)w-~(s)q(8)e(s) d8 - £ - l e~(t) + O(h)¢,

0 t W(t)W-I(s)Q(s)x(s) ds -- ~ - l c z = O ( h ) z ' ,

f ' w ( t ) w - ~ ( s ) R ( s ) ds - ~ - ~ -- O(h).

(3.19a)

(3.195)

(3.19c)

Thus, (3.18) can be rewrit ten as

e' = [A - 9 ~ . - 1 0 + O(h)] e + O(h), (3.20)

with e(O) = O. Thus, e(t ) --* 0 as h --* O. |


2 . 0

1 . 8

1 . 7

1 . 6

1 .5

1 .4

1 .3

1 . 2

1 .1

1 .8

.9

8

i , l * , l l l l l J l l l l l l l l

~ t l * l l l l l l

t

2 . 8

1 .8

1 .6

1 .4

1 .2

. 8

.6

. 4

.2

, , . , , , . i . , , i . 1 , 1 , 1 ,

8 * ' ' ' ' *

~ l l l l O l l l l

t

- 2 . 0

- 2 . 5

- 3 . 8

- 3 . 5

- 4 . 8

- 4 . 5

- 5 . 8

1M8

~ 8

888

788

6 H

580

488

388

288

188

8

-188

- 288

- 380

- 5 . 5

-6.8 i i i i i l i I i i ] i i i , i i * , -488 i i i J i i i L i i i i i i i i i , i

t i i i * t i i i ~ ~ l i * * i i i i i

t t

Figure 5. NumericM approximation of =1 (upper leR), =2 (upper right), Yl (lower left), end Y2 (lower right) computed via our regularization with DASSL for Exam- ple 4.3 w i t h p = l , q = 2 , h = 1 0 -4 ,~1 = e 2 = 1 0 -e.

A different proof could be obtained by using the transformation method in [20] for singularly perturbed linear homogeneous initial value problems.

REMARK. The same analysis can now be used to show that

Zl - ~ l = O ( h ) . (3.21)

We now wish to consider the case that Af(D) is constant, except at one point to where its rank

decreases. The class of problems is defined by the following.

DEFINITION 3.2. Consider the//near semiexp//cit DAE (3.1). Assume the fo//owing.

1. ~ is nonsingular except at t*. 2. ~ - I ~ is continuous/or all t. 3. ~ - l f is continuous for all t.

52 P .K. MOORE et al.

2 . e

.9

.8

.7

.6

.5

,4

1.3

1.2

1.1

1.~1

.9

2 . 0

tl

i i i i i i i i i

t

-2.1~

- 2 . 5

-3.El

- 3 . 5

• ~ - 4 . e

-4+.5

- 5 . i l

- 5 . 5

- 6 . [ i

A . ~ t t

- O H

-1000 '

i i i i i i i i i i i i i i + i i i i

~ i i I I I I I I P

t

, ~ i J i i i i t i i ~ , ~ i I i i i i i i i

t t

Figure 6. Numerical approximation of xl (upper left), x2 (upper right), Yl (lower left), and 1/2 (lower right) computed via our regularization with DASSL for Exam- ple 4 . 3 w i t h p = 1 , q = 2 , h = 1 0 - 4 , ~ 1 = ~ 2 = 1 0 - s .

Then (3.1) has a solution for x near the singularity, and we will call the problem a k i n e m a t i c a l l y - s i n g u l a r i n d e x - 1 D A E .

A / / n e a r D A E (3.1) will be cal/ed a r a n k - d e f i c i e n t k i n e m a t i c a l l y - s i n g u l a r i n d e x - 1 D A E if after e//minating any redundant constraints as in Definition 3.1, the reduced problem is a kinematically-sir~ular index-1 DAE.

D E F I N m O N 3.3. A / / n e a r D A E (3.1) will be ca//ed a r a n k - d e f i c i e n t k i n e m a t i c a l l y - s i n g u l a r i n d e x - 1 D A E with a s i n g u l a r i t y o f m u l t i p l i c i t y m if near t = t*.

1. a~(t) = k i J t - t * l m' + O ( I t - t*lm++l), i = 1 , 2 , . . . , S .

2. m = m l > m2 > m 3 > . . . > m s > O . 3. t r i ( t * ) > O , i = s + l , . . . , r . 4. The elements ~ j of C sat is fy ~ j (t) = O( ( t - t* )"+ ), ni >_ O, i = 1, 2 , . . . , s, j = 1, 2 , . . . , r. 5. The elements ~+(t) of ~ satisfy ~i(t) = O((t - t*)"'), n~ >_ O, i = 1, 2 , . . . , s. 6. The elements b~j(t) of B satisfy bij(t) = O((t - t*)~), l = m a x l < ~ < s ( O , m - (m/m~)ni),

i , j = 1 , 2 , . . . , r .

We remark that assumptions (4)-(6) guarantee that (2) and (3) of Definition 3.1 wiZl be satisfied.

Regulariz&tion of Index-1 53

2.8 , i , n , , , , ' , ' , ' , ' + ' l ,=--

1.9

1.8

1.7

1.6

15

1.4

1.3

1.2

1.1

1 . I ~ i - - o ~ O

. 9 I I I I I l I I I i i l i i i i i i i

• T i i ~ + i i i , i

i .

2.

1.

1.

1.

1.

~ .

~ i i i i i i i i i i

l l l l l l l

- 2 . 8 . ~ ~ , •

- 2 . 5

- 3 . 8

- 3 . 5

- 4 . 8

- 4 . ' 5

- 5 . 8

- 5 5

_6.01 I I I I l I I i t I t i I + J I + I

t

1BII8

EIBg

658

488

288

8

-288

- 4 8 8

-61~8

-8815

- 1 8 8 8

, i , u i , i ,

i i | i I i i i t i i i t i L i i i

i ~ , i t i , i i ~

t

Figure 7. Numerical approximation of zi (upper left), z2 (upper right), Yz (lower left), and Y2 (lower right) computed via our regularization with DASSL for Example 4.3 w i t h p = 1 , q = 2 , h = 1 0 -4 ,ez = ~ 2 = 1 0 - 1 ° .

Also, note that I = m a x i < i < , ( O , m / m i ( m i - hi)) and m / m i >_ 1. If D is a/ready in S V D form (U = V = I ) then condition (6) simplifies to the following.

6'. The elements bij (t) of [~ satisfy bij (t) = O ((t - t *)t~ ), lj = max(0 , m# - n j ) , i = 1, 2 , . . . , r, j = 1 , 2 , . . . , s .

THEOREM 3.2. Consider the rank-deficient kinematically-singular index-1 D A E (3.1). Then for a n y T > 0, the so]ution ]c to the regularized sys tem (3.2) converges to the solution x of (3.1) as h ~ 0, i.e., ][x - 50 = O(h) as h --* O. Herein we consider the transformed sy s t ems outlined in

T h eorem 3.1.

PROOF. Fix T >_ t* >> 0 ( the case T < t* is proved by T h e o r e m 3.1). Henceforth, let cri(t) <_ as(t), i = 2, 3 , . . . , r, and as+z( t ) <_ as(t), i = s + 2 . . . . , r in a ne ighborhood of t*. Let

[min(O,t'-l/4),t'+l/4] , e I = h 2m+1, (3.22)

where k = maxi_<i<s ks, TL = t* -- h, Tu = t* + h, T t = t* - h m/m', and T b = t* + h m/m',

54 P .K . MOORE et al.

2 . 0 , i , i , i , ~ , ~ • ~ , ~ • i • i ,

. 8

. 7

. b

. 5

. 4

1 . 3

1 . 2

1 . 1

1 .0

t

2 , e

1 . 8

0

t

- 2 . 0 , ~ t ~ a

- 2 . 5

- 3 . 0

- 3 . 5

" ~ - 4 . 0

- 4 , 5

- 5 , 0

- 5 . 5

i i ~ i i i i i i i

t

4 0 0 0 5

3 0 0 0 0

2 0 0 0 0

10000

0

-10000

- 2 0 0 0 0

- 3 0 0 0 0

- 4 0 0 0 0 , l l l l l l l , l l l ~ l l l l l

~ J t l l l l J , t

t Figure 8. Numerical approximation of z l (upper left), z2 (upper right), Yl (lower left), and I/2 (lower right) computed vis our regularizstion with DASSL for Exam- ple 4.3 with p -- 1, q -- 4, h -- 10 -4, el -- 02 ffi 10 -10.

i = 1 , 2 , . . . , s . On the interval 0 < t <_ TL, we can apply T h e o r e m 3.1 s ince at t = TL, P(Tt~) ~ (1/h)I .

From (3.18), it follows that for t e (TL, Tv],

e' = Ae - g ( O W ( O W - l ( s ) Q ( s ) e ( s ) ds

+ ~( t )W( t )W- ( s ) Q ( s ) z ( s ) d s - / ~ ( t ) ~ - (t)O(t)x(t)

i t B(t)W(t)W-I(")R(8) ds - B(t)E-l(t)~(t) - BW(t)~I(TL). + JTL

(3.23)

From (3.13a) , the diagonal e l ements p.( t ) of P ( t ) satisfy

p - ( 0 -- ~ - e . > - m a x l e - ( 0 1 ÷ O(h). (3.24)

Regularizstion of Index-1 55

2.0 , , , , , , , ,' , , , , ,'" ! , , , , ,

1.9

1.8

1.7

1.6

.5

.4

.3

.2

.I

.e

.9 I i i i i i i i l i i i i i i , i i i

.;- i i i i , i i , i

t

2 . 0

1

$

t

- 2 . 0 , , , , , . [ •

- 2 . 5

- 3 . 0

- 3 . 5

• ; - 4 . e

- 4 . 5

- 5 . 0

- 5 . 5

- b . 0 l i J i I I I i I i , * , i i , i t i

t

40050

30000

20050

10006

$

- t 0000

-20000

-30600

- 4 0 9 0 0

, i , , • i , , ' ! ' i • , , i , i

A 0 ; , = f ~ - - - -D 0

l , i , i 1 | | i , J , | i i i , ,

t

Figure 9. Numerical &pproxin~tion of z l (upper left), z2 (upper right), Yl (lower left), and Y2 (lower right) computed via our regularization with DASSL for Exam- ple 4.3 w i t h p = l , q = 4 , h = 1 0 - 4 , e 1 = £ 2 = 1 0 -12.

The off-diagonal elements Pit (t) of P(t) are bounded by

Ip~j(t)l < m .ax I%0(t)l + O(h). (3.25) t,2

The Gershgorin Circle Theorem [17] implies that the real part of all the eigenvalues of -P( t ) is bounded by a positive constant. Thus, elements of W(t)W-l(s) which represent the solution of (3.14) with W(s) = I are bounded on [TL, Tv].

The elements ~ij of Q - B(t)W(t)W-l(s)O(s) have the form

] ~ = ~ c ~ + O(h) ~kwk, , l= l k = l

(3.26)

where wkz are the elements of W(t)W-l(s). If e I is the matrix whose entries are

5 6 P . K . M o o m z e t a/ .

2 .B

1 .9

1 .8

1 .7

. 6

. 5

. 4

. 3

. 2

,1

. e

T * * o i * i , o *

t

2.21 , , , , . , , , . , , , , , . , , , ,

1 . 8

1 .b

1 .4

1 .2

1 .B

. 8

. 6

. 4

, 2

0 i I ~ , ; l , l

. . . . A , ~ , ~ , .

- 3 . 0

- 3 . 5

"; - 4 . 0

- 4 . 5

- 5 , 6

- 5 . 5

- b . g

- 2 . 0

- 2 . 5

~ , l ~ I l l I l *

t

40000

30090

20009

10000

{

- I e lEBB

- 20000

-30000

-40000 J i i * * * * * *

t

, i i ,

i i J a n l a l a

Figure 10. Numerical approximation of Zl (upper left), z2 (upper right), ~/1 (lower left), and Y2 (lower right) computed via our regularization with DASSL for Exam- ple 4.3 with p -- 1, q --- 4, h 10 -4, el e2 = = = 10 -14.

^l O'l _ qij = Clj "~- i kWkl , (3.27)

then ¢~ = ~-]~Lz Qz. Similarly, the elements ~, and ~ of /~ -= B(t)W(t)W-Z(s)R(s) and ~z, respectively, have the form

|-----1 k = l

(3.2s)

and _ O" l

r, = + O(h) b~kwkl, (3.29)

where R = ~-~Lz/~z.

R e g u l a r i z a t i o n o f I n d e x - 1 57

.6 4 ~

. 4

-,2

- . 4 4 ~

4 -.6 ~

-.8 "\ - I . i } I I I I I I i I ~ ~ ' ' i , i i , i ,

. . . . . . . . . U U .-" .-' ,-" .-" U E ,-: .-' K

t

1 .0

. 8

.6

. 4

.2

- . 2

- . 4

- . 6

- R

-1 .t~

\.\ \

t

1 . 0

.8

.6

.4

.2

0

-.2

-.4

-.6

-.8

-1 .e

' [ ' = ' l ' t ' l ' l ' = ' l ' = ' "\ "\

"\ "\

"\ "\

"\ I l l l l l l l l l l l l I I I I I I

t

1•$

,9

. 8

,7

•6

>- .5

. 4

.3

.2

.1

$

• i , , , , , = , i , , , , , , , , ,

\ \

i t i i t I i t I / ~ t i t i L J i 1

t

Figure 11. Numerical approximation of z (left), y (right) computed via our regularization wi th DASSL for Example 4.4. Top: h = 10 - 7 , el = e2 = 10 - 9 . Bottom: h = 10 - v , el = 10 - 9 , and e2 = 10 -13 .

Substituting (3.26) and (3.28) into (3.23) yields

e' = Ae - QZ(s)e(s) ds 1=1 TL

+ ~ ~* Q'(s)x(s)~s- ~(t)~-l(t)e(t)x(t) I=1 Tt.

+ --~ [ t Rt(s) ds - B ( t ) £ - l ( t ) f ( t ) - [~(t)W(t)W-I(TL)W(TL)~I(TL). l=l J7 T L

(3.30)

We first consider the matrices {~z, l = s + 1,... ,r. Then from (3.22a), (4), and (6) of Defini- tion 3.3 and boundedness of wij, it follows that

C I~t~jl < -~, l = s + l , . . . , r . (3.31)

58 P . K . MOORE et a/.

1 .e

.2

- . 2

- . 4

- . b

- . 8

- I .l~

"\ "\ "\ "\ "\

" \ I l l l ~ I I I I I I I I I A I I ~

t

1. ~1

• El

.6

. 4

.2

g

- , 2

- . 4

- . b

\ i i i i i i i i i i i , i I I

t

Figure 12. Numerical approximation of z (left) and y (right) computed via our regulaxization with the modification sgn (g~)e2 and with DASSL for Example 4.4 with h -- 10 -~, e~ = ~2 = 10 -~.

1 .g

. 8

.6

.4

.2

e

- . 2

- . 4

- . 6

\ \

l l l l l l l l l l l l l l l l ~ l l

t

. 0

. 8

.6

,4

.2

),

1 .~

.8

\ \

.4 4 ~ .

.2 I I * l l l l l l l l l l ~ l l ~ " ' ~ ' l l A r " * ' -

t

Figure 13. Numerical approximation of x (left) and y (right) computed via our regularization with the modification sgn (gy)e2 and with DASSL for Example 4.5 with h = 1 0 - 6 , e l = ~2 = 10 -1°.

Bounds on the (~l, l = 1, 2,..., s can be obtained by considering the three regions [TL, T~,], [T~L, Tb], and [T•, Tu]. For t, s E [TL, TL l ] or t, s e [T b, Tu],

ck~ I~ - t ' l " I~ - t ' l " ' I~ - t * l ' + O(h) < - t*l " ' - ~ ' h' hk~ Is - t*l 2~' cls h + O(h) (3.32) Iq Jl <~

< h(m/m~)(nz-m~) -1 < -'~,

using (3.27) and (1), (2), (4), and (6) of Definition 3.3. On [T•, Tb], we obtain

ck , Is - t * l m ' Is - t * I" ' I t - t * l ' <~ -t- O(h) (k2 /100) h2m+l (3.33)

C h ( m / ' n ' ) ( n ' - ' n l ) h(,nl/mt) C < h + O(h) < -~,


using (3.27) and (1), (2), (4), and (6) of Definition 3.3 and the fact that m / m l >_ 1. The same analysis with (5) replacing (4) from Definition 3.3 establishes

C (3.34)

Substituting estimates (3.31)-(3.34) into (3.30), and using bounds on l}~]-lCx and / ~ - 1 ~ implied by (4)-(6) of Definition 3.3 yields

/: lZ; lie(u) II duds + O(h). (3.35) He(t)H _< He(TL)H + Co TL He(s)][ ds + ~ L L

Letting E(t) = He(t)H and F(t) = fTL E(u)du, and using the Gronwall inequality we obtain from (3.35)

E(T) <_ (E(TL) + O(h)) e (c(t-TL))/h. (3.36)

Thus He(t)[[ _< C[[e(TL)[[ < Ch for all t • [TL,Tu], where C may depend on parameters in the problem such as s, r, k~, i = 1, 2 , . . . , s, mi, i = 1, 2 , . . . , s, Hxll, and []A H but not on h. On the interval T > Tu, the result follows from Theorem 3.1 and (3.36). |

REMARK. In the kinematically-singular case, Zl may not exist at t* while ~1 does. However, the error in Zl outside the interval around t* can be shown to be O(h) using Theorem 3.1.

4. N U M E R I C A L E X P E R I M E N T S - - S I N G U L A R L I N E A R A N D N O N L I N E A R D A E S

In this section, we present results of our regularization applied to several linear and nonlinear, singular DAEs. Several examples meet the conditions of Theorems 3.1 and 3.2, while the others show that the regularization performs well in more general settings. The DAE code DASSL [1] was used in all examples. Absolute and relative error tolerances were set to 10 -6. The values of h, el, and e2 are specified in each example. The finite-difference approximation to the Jacobian in DASSL was used in all cases. In none of the examples was DASSL able to compute the correct solution through the singularity without the regularization.

EXAMPLE 4.1. Consider the linear DAE

x I = ty, - 1 < t < 1, (4.1)

0 = x - ty, x ( - 1 ) = - 1 , y ( - 1 ) = 1.

In this example, the rank of D -- - t is one except at t = 0 where the rank decreases to zero. Thus the problem is a rank-deficient kinematically-singular index-1 DAE with singularity of multiplicity 1. The rank of Bi t ) = t also decreases by one at the same point so this problem is covered by Theorem 3.2. We chose h = 10 -4, el = e2 = 10 -k, k = 6,7,8, which are less stringent than our proof of Theorem 3.2 required. Figures 1 and 2 show that our regularization accurately models the solution through the singularity. For el = e2 = 10 -k, k = 6, 7, the numerical solution experiences a small jump at the singularity. In Figures 1 and 2, we restricted the maximum absolute value of y and its numerical approximation so that the solution, away from the singularity, could be seen. Near the singularity, y becomes much larger (in terms of absolute value) than the numerical solution. However, this does not seem critical since we are not interested in accurately resolving the blow-up in y. To show the dependence on h we also solved the problem with h = 10 -2 and el = e2 = 10 -k, k = 6, 8, and 10. In Figures 2 and 3 it is seen that reducing el and e2 beyond a certain point does not lead to a more accurate solution, but that solution accuracy is limited by h.

EXAMPLE 4.2. Consider the linear DAE

x' = y, - 1 < t < 1, (4.2)

o = t " ( z - y ) , z ( - 1 ) = - 1 , y ( - 1 ) = - 1 , n = 1 , 2 , 3 .

60 P.K. MOORE et al.

This example also has D(t ) = - t n of constant rank except at t = 0. However, B ( t ) = 1 does

not decrease rank at t = 0, but D - 1 C -- 1. Specifically we consider the case n = 3 with el = e2 -- 10 -k , k = 6, and 8. The results for bo th variables are shown in Figure 4.

EXAMPLE 4.3. Consider the linear DAE system

xl 0

with x l ( - 1 ) = 2, x 2 ( - 1 ) = 2. The matr ix D is full-rank except at t = 0 where the rank

decreases to 1. The exact solution is given by xl = 1 + e - s ( t + 0 + 6(t + 1)e -s( t+l) and x2 =

2e -s( t+l) + 3(t + 1)e -s( t+l) with Yl = - 2 x l - x2 and Y2 = - t v - q ( x l + 2x2).

We begin by considering the case q = 2, p = 1 with h = 10 -4, el -- e2 -- 10 -k , k = 6, 8, and 10.

The results are displayed in Figures 5-7. As in the previous examples, the error can be reduced

by decreasing the size of el and e2. In the second case, q = 4 and p = 1. Now the singulari ty is

much more severe, as seen in Figures 8-10. In this case, we used the same h bu t el = e2 = 10 -k ,

k = 10, 12, and 14. Note tha t the error is reduced as the numerical approximat ion to Y2 becomes

more singular. The s i tuat ion will become even more severe when q is increased.

EXAMPLE 4.4. Consider the DAE [21]

dx d--t- = - 1 , (4.4a)

0 = y3 _ x, (4.4b)

with initial condit ion ix(0), y(0)) = (1, 1). ( 0, 0) is a singular point of this problem. The solution

shown in the top of Figure 11 was computed using h = 10 -7 and el = e2 = 10 -9. In this case,

the code handles the singulari ty correctly. Reducing e2 to 10 -13 results in the solution seen

in the b o t t o m of Figure 11. Round-off in the te rm g~ + e2 prevents the code from advancing

past the singularity. Even more disturbing, if y3 _ x is replaced by x - y3 and wi th the values

el -- e2 = 10 -9, the solution for y again gets stuck at 0. The lat ter shor tcoming can be overcome T by replacing the equat ion gy + e2 with g~+ sgn (gu)e2 (noting tha t in the scalar case gy = g~).

Then, with the same values of h, el, and e2, we obta in the solution shown in Figure 12. This

example shows tha t care is needed when applying the regularization to nonlinear problems and

points to a possible modification in the regularization strategy. Since our regularization models

Newton ' s method, we would expect difficulty at x = 0 since y = 0 is a multiple root of g in this

case.

EXAMPLE 4.5. Consider [21]

dx d t - y' (4.5a)

0 = x - (y - 1) 3, (4.5b)

Figure 13 shows t h a t the with initial condit ion (x(0), y(0)) = (1, 2). ~ is singular at y = 1.

solution passes th rough the singular point with h = 10 -6, el = e2 = 10 -10. We note t h a t ( - 1 , 0 ) is a stable equilibrium point of the DAE.

R E F E R E N C E S

1. K.E. Brenan, S.L. Campbell and L.R.. Petzold, Numerical solution of initial-value problems in differential- algebraic equations, Second edition, SIAM, (1995).

2. L.R. Petzold, Y. Ren and T. Maly, Regularization of higher-index differential-algebraic equations with rank- deficient donstraints, SIAM J. Scientific Computing (to appear).


3. S.L. Campbell, Regularizations of linear time-varying singular systems, Automatica 20, 365-370, (1984). 4. M. Hanke, Regularizations of differential-algebraic equations revisited, Rep. 92-19, Berlin, (1992). 5. M. Knorrenschild, Regularization of differential-algebraic equations motivated by electrical network analysis,

Aachen, (preprint), (1988). 6. R.E. O'Malley, Jr. and L. Kalachev, Regularization of nonlinear differential-algebraic equations, SIAM £.

Math. Anal. 25, 615-629, (1994). 7. E. Eich and M. Hanke, Regularization methods for constrained multibody mechanical systems, (preprint). 8. L. Kalachev and tLE. O'Malley, Jr., The regularization of linear differential-algebraic equations, SIAM J.

Math. Anal. 27, 258-273, (1996). 9. E. Bayo and A. Avello, Singularity-free augmented Lagrangian algorithms for constrained multibody dynam-

ics, Nonlinear Dynamics 5, 209-231, (1994). 10. E. Bayo, J. Garcia de Jalon and M.A. Serna, modified Lagrangian formulation for the dynamic analysis of

constrained mechanical systems, Computer Methods in Applied Mechanics and Engineering 71, 183-195, (1988).

11. E. Bayo and R. Ledesma, Augmented Lagrangian and projection methods for constrained multibody dynamics with no constraint violation, University of California at Santa Barbara, (preprint), (1993).

12. K.C. Park and J.C. Chiou, Stabilization of computational procedures for constrained dynamical systems, J. Guidance, Control and Dynamics II, 365-370, (1988).

13. U. Ascher and P. Lin, Sequential regularization methods for higher index DAF~ with constraint singularities: I. Linear index-2 case, SIAM J. Numer. Anal. 33, 1921-1940, (1996).

14. F.M.L. Amirouche and Chin-Wei Tung, Regularization and stability of the constraints in the dynamics of multibody systems, Nonlinear Dynamics 1, 459-475, (1990).

15. S.K. Ider and F.M.L. Amirouche, Numerical stability of the constraints near singular positions in the dynamics of multibody systems, Computers and Structures 33, 129-137, (1989).

16. P.J. Rabier and W.C. Raheinboldt, On impasse points of quasilinear differential algebraic equations, J. Math. Anal. Appl. 181,429-454, (1994).

17. G.H. Golub and C.F. Van Loan, Matrix Computations, Second edition, Johns Hopkins University Press, (1989).

18. C.-T. Pan and K. Sigmon, A bottom-up inductive proof of the singular value decomposition, SIAM J. Matrix Anal. Appl. 15, 59-61, (1994).

19. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, (1978).

20. R.E. O'Malley, Jr., Singular Perturbation Methods .for Ordinary Differential Equations, Springer-Verlag, (1991).

21. L.O. Chua and A.C. Deng, Impasse points. Part I: Numerical aspects, Inter. J. Circuit Theory and Applica- tions 17, 213-235, (1989).

22. U. Ascher and L. Petzold, Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM J. Numer. Anal. 28, 1097-1120, (1991).

23. U. Ascher and L. Petzold, Stability of computational methods for constrained dynamics systems, SIAM J. Sci. Comput. 14, 95-120, (1993).

24. U. Ascher and L. Petzold, Projected collocation for higher-order higher-index differential-algebraic equations, J. Comp. and Appl. Math. 43, 293-259, (1992).

25. U. Ascher, H. Qin and S. Reich, Stabilization of DAE and invariant manifolds, Numer. Math. 69, 131-149, (1994).

26. S.L. Campbell, Notes, (1992). 27. L.O. Chua and A.C. Deng, Impasse points. Part II: Analytical aspects, Inter. J. Circuit Theory and Appli-

cations 17, 271-282, (1989).

regularization of index-1 differential-algebraic … · 46 p.k. moore et ai. definition 3.1....

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