Theoretical and Numerical Analysis of Differential-Algebraic Equations – Volume VIII of Handbook of Numerical Analysis – Elsevier North-Holland

359 Pages·2016·15.79 MB·English

by  Rabier

Rabier, P.J.; Rheinboldt, W.C. (2002) Theoretical and
Numerical Analysis of Differential-Algebraic Equations –
Volume VIII of Handbook of Numerical Analysis [Ciarlet,
P.G.; Lions, J.L. (Eds.)] – Elsevier North-Holland
Theoretical and Numerical
Analysis of
Differential-Algebraic
Equations
Patrick J Rabier
Departmento f Mathematics
University of Pittsburgh
Pittsburgh, PA 15260
USA
Werner C Rheinboldt
Departmento f Mathematics
University of Pittsburgh
Pittsburgh, PA 15260
USA
HANDBOOK OF NUMERICAL ANALYSIS, VOL VIII
Solution of Equations in Rn (Part 4)
Techniques of Scientific Computing (Part 4)
Numerical Methods for Fluids (Part 2)
edited by P G Ciarlet and J L Lions
© 2002 Elsevier Science B V All rights reserved
183

Contents
1 Introduction 189
PART A Theoretical Basis
CHAPTER I Model Problems 197
2 Network problems 197
3 Constrained mass-point systems 199
4 Constrained rigid-body systems 203
5 Singular perturbations 208
6 DA Es in the solution of PD Es 212
7 Control problems 214
8 Differential equations with invariants 216
CHAPTER II Linear DA Es: Classical Solutions 219
9 Linear DA Es with constant coefficients 219
10 Time-dependent coefficients 221
11 Transformation functions 223
12 Reduction of operator pairs 225
13 Classical solutions of linear DA Es 232
14 The theory of Kunkel and Mehrmann 235
15 Rectangular systems 239
16 Linear boundary value problems 241
CHAPTER III Linear DA Es: Generalized Solutions 243
17 Generalized inputs 243
18 Impulsive-smooth distributions 247
19 Discontinuous inputs 251
20 Inconsistent initial values 256
CHAPTER IV Geometric Reduction of Quasilinear DA Es 261
21 Motivation and preliminary results 262
22 Reduction procedure 264
23 Complete reducibility and tangent vector fields 269
24 Application to DA Es 271
25 An example: The pendulum 274
26 Index one DA Es 279
185
186 P.J Rabier and W C Rheinboldt
27 Local parametrizations 284
28 Other approaches 287
CHAPTER V Special Classes of Nonlinear DA Es 293
29 Algebraically explicit DA Es of index one 293
30 Algebraically explicit DA Es of index two 295
31 Algebraically explicit DA Es of index three 301
32 Examples 303
CHAPTER VI Other Reductions for Multibody Problems 309
33 Reduction to second-order OD Es on Rn 310
34 The second fundamental tensor of a submanifold 311
35 Reduction of Euler-Lagrange equations 314
36 Characterizations of the fundamental tensor 320
CHAPTER VII Singularities of Quasilinear DA Es 323
37 Singular quasilinear OD Es in Euclidean space 325
38 Existence theory for the reduced problem 327
39 Existence theory for singular quasilinear OD Es 330
40 Singular quasilinear DA Es of index one 334
41 Impasse points 338
42 Boundary singularities 342
43 Examples 346
44 Impasse points and foldpoints 348
45 Concluding remarks 352
CHAPTER VIII Discontinuous Solutions of Semilinear DA Es 355
46 Distribution solutions 355
47 Ambiguities 358
48 Difficulties with higher index problems 360
49 P-consistency: General ideas 361
50 P-consistency: Technical aspects 363
51 Examples 367
52 P-consistency and uniform approximation 378
53 Concluding remarks 382
CHAPTER IX Hopf Bifurcation and Stability in DA Es 385
54 The problem 386
55 Reduction of parametrized DA Es 387
56 Hopf bifurcation in implicit ODE s 391
57 Hopf bifurcation in quasilinear DA Es 393
58 Hopf bifurcation in concrete problems 398
59 Stability of equilibria 399
Contents 187
PART B Numerical Methods
CHAPTER X ODE Methods for DA Es 403
60 Multistep methods for index one DAE s 403
61 Multistep methods for higher index DA Es 409
62 Multistep software for DA Es 413
63 Runge-Kutta methods for index one DA Es 415
64 Runge-Kutta methods for index two DA Es 422
65 Rosenbrock methods 424
66 Projection methods 426
67 Extrapolation methods for DA Es 429
CHAPTER XI Local Parametrization Methods 433
68 Computation of manifolds 434
69 Methods for algebraically explicit DA Es 439
70 General quasilinear DA Es 448
CHAPTER XII Methods for Multibody Systems 451
71 Index reduction and stabilization 452
72 Local parametrization methods 456
73 Some computational examples 461
CHAPTER XIII Methods for Linear DA Es 469
74 The linear DAE solver GELDA 469
75 An extrapolation method for linear DA Es 473
76 Methods for linear DA Es with discontinuities 478
CHAPTER XIV Computation of Singularities 483
77 Methods for singularities of ODE s 483
78 Methods for singularities of DA Es 491
79 Discontinuities in initial condition dependence 499
80 Computation of Hopf points 503
CHAPTER XV Outlook 507
81 Boundary value problems 507
82 DAE optimization problems 513
83 Elastic multibody systems 516
84 Summary and perspectives 520
REFERENCES 525
SUBJECTINDEX 537

SECTION Introduction 189
1 Introduction
This article presents a fairly up to date survey of the research literature relating to
differential-algebraic equations (DA Es) and their numerical solution In contrast to or-
dinary differential equations (OD Es) the theory and numerical analysis of DA Es is still
largely covered only in a range of specialized journals and there are as yet no compre-
hensive expositions to speak of.
In fact, we are aware of only a few books about general (linear and nonlin-
ear) differential-algebraic equations: The 1986 monograph by GRIEPENTROG and
MARZ l 1986 l should be credited with having motivated much of the initial interest
in DA Es, but has a limited scope and is now outdated The book by BRENAN, CAMP-
BELL and PETZOLD l 1989 l (now in its second edition BRENAN, CAMPBELL and PET-
ZOLD l 1996 l) is specifically devoted to the numerical solution of initial value problems
for DA Es and was not intended as a theoretical reference The two volume monograph
set by HAIRER, NORSETT and WANNER l1 993 l and HAIRER and WANNER l1 996 l
represents a comprehensive reference on methods for solving OD Es and includes in the
second volume a fine survey of currently available ODE-based numerical methods for
DAE initial value problems But, of course, this work was not designed to develop the
theory of DA Es as such Similarly the text by ASCHER and PETZOLD l1 998 l grew out
of courses on the solution of ODE s and covers only in its last two sections some in-
troductory material on DA Es Finally, the two volumes by CAMPBELL l1 980 l, CAMP-
BELL l1 982 l, from the early eighties, address almost exclusively linear problems except
for some comments about the nonlinear case in CAMPBELL l1 982, Chapter VIl.
Over the years several approaches have been introduced for the study of local exis-
tence and uniqueness questions for DA Es While they exhibit major technical differ-
ences and are based on different assumptions, all these approaches agree with the basic
principle that a DAE is eventually reducible to an ODE and that this reduction should be
done via a recursive process In spite of the marked discrepancies, they can all handle
the same physical problems and only academic examples show up as different limit-
ing cases Thus, as regards local existence and uniqueness at regular points, the various
approaches are more or less equivalent for practical purposes although this is nowhere
emphasized.
Faced with this situation and since some interesting numerical issues for DA Es cannot
even be formulated without a solid theoretical basis, we begin this article in Part A with
a complete and hence lengthy theoretical treatment of DA Es that is hoped to provide a
rounded foundation for the field in general At the same time this treatment forms a firm
basis for the numerical material in Part B and is there used throughout as a reference.
The term "differential-algebraic equation", by now fully accepted, may be somewhat
misleading or even inadequate The name captures the fact that many practical problems
lead to mixed systems of OD Es and nondifferential equations such as, e g ,
{
-l = f(Xl, X 2) E m,
~=:g 1g (:l,q ~E2) lE~ rn~ ,-m, (1 1)
with (xl, x 2) E W" x IR n-m However, this system need not involve polynomial maps
and hence need not have any "algebraic" component in the traditional sense: Here, "al-
190 PJ Rabiera nd WC Rheinboldt SECTION I
gebraic" is just a substitute for "nondifferential" Furthermore, in more general exam-
ples, there need not be any clearly defined algebraic or differential part of the unknown
variable x, nor any definite splitting of the equations into differential or algebraic type:
This may depend upon the point in the vicinity of which the equation is considered.
This is best seen from the perspective of an implicit ODE
F(x, ) = 0, (1 2)
defined by a sufficiently smooth function F := F(x, p) :IR x > Rn If the deriva-
tive Dp F(x, p) is invertible at a point (xo, Po) then the ODE (1 2) with initial condition
x( 0) = xo, ( 0) = PO is, locally near (xo, po), equivalent to the explicit initial value
problem = O (x), x( 0) = xo, where O is uniquely determined and O( xo) = Po Hence,
if the invertibility condition is satisfied irrespective of the point (xo, Po), then the prob-
lem (1 2) is always locally reducible to an explicit ODE.
But when DpF (xo, Po) is not invertible then we have to look more closely at further
properties characterizing the non-invertibility In particular, the impact of the singularity
of DpF (xo, po) on the total derivative DF(xo, Po) with respect to both variables x
and p is crucial When DpF (xo, Po) is invertible, it is obvious that DF(xo, Po) maps
onto Ri The first important item is whether the singularity of DpF (xo, po) affects the
surjectivity of DF(xo, Po) If it does not, the next item is to check whether the rank of
DpF (x, p) is constant (and hence not full) on some neighborhood of (xo, Po) If both
these conditions hold, Eq ( 1 2) is called a differential-algebraice quation in the vicinity
of (xo, Po) In all other cases and in particular when (xo, Po) can be approximated by
points (x, p) where Dp F(x, p) is invertible Eq ( 1 2) is an ODE with a singularity
at (xo, po) and there are a numerous possible behaviors, far from being all known or
classified, for the solutions of (1 2) near (xo, P0 ) In contrast, when the equation is a
DAE, a local existence and uniqueness theory can be worked out which extends that
for OD Es Actually, this statement is correct only for so-called "index one" problems.
The general case requires asking the same questions not only for F but also for a finite
sequence of iteratively defined mappings This is related to the reduction procedure
mentioned above.
In the simple case ( 1 1) we have F(x, p) := (pl f(xl, x 2), g(xj, x ))T, whence
2
DpF (x, p) has constant rank m at every point The requirement that DF(x, p) is onto
IR n is equivalent to Dg(x) being onto Rjn-m, which is not a severe limitation in many
applications The system (1 1) is a special case of the class of quasilinearD A Es of the
form
A(x)i = G(x), A(x) E £(lR n), (1 3)
for which F(x, p) := A(x)p G(x) Nearly all DA Es arising in scientific or engineer-
ing problems are quasilinear, which strongly suggested to us to limit our exposition to
this case since it introduces very significant simplifications Moreover, any nonlinear
problem (1 2) can be written in the quasilinear form x = p, O= F(x, p) Accordingly,
"DAE" will henceforth mean "quasilinear DAE" in most of this article and the few fully
nonlinear examples will always be treated as quasilinear problems by the simple trick
just described.
SECTION Introduction 191
The DAE nature of the problem (1 3) resides essentially (though not solely) in the
assumption that the rank of A(x) is constant but not full in the domain of interest.
While there is added value to discuss nonautonomous problems
A(t, x)x = G(t, x), (1 4)
it still suffices to consider the simpler problem (1 3) since (1 4) can be made quasilinear
and autonomous by simply adding the equation i = 1 and changing the variable x into
(t, x) Analogously, as with OD Es, second-order DA Es can be reduced to the form (1 3)
by changing the variable x into (x, y) with x = y The generalization of that remark to
higher order DA Es is obvious.
One important exception to the rule that only autonomous DA Es have to be discussed
arises with linear DA Es
A(t)i + B(t)x = b(t), (1 5)
since of course such a system cannot be transformed into a linear autonomous problem.
As shown in Chapters II and III a direct investigation of (1 5) is needed to take full
advantage of the linear structure In those chapters both the classical and generalized
(distribution) solutions are considered and boundary value problems are covered.
Our presentation will show that the theory of differential-algebraic equations sim-
ply cannot do without the input of differential geometry It is a fact, not a mere point
of view, that a DAE eventually reduces to an ODE on a manifold The attitude of ac-
knowledging this fact from the outset leads to a reduction procedure suitable for the
investigation of many problems Actually, this differential geometric reduction turns
out to be the only one so far that has proved to be adequate for handling further as-
pects of importance such as singularities, discontinuous solutions, Hopf bifurcations, or
boundary value problems ' Moreover this reduction procedure is closer to the one used
in the linear case and has the advantage of leading to smaller and smaller systems as
the recursive reduction proceeds These combined features provided a compelling rea-
son to base our theoretical exposition on that reduction, which therefore is the only one
described here in full detail (Chapter IV) Some of the other known approaches are only
sketched with appropriate references to the literature for the interested reader.
The mechanism of the geometric reduction procedure completely elucidates the "al-
gebraic" and "differential" aspects of a DAE The algebraic part consists in the char-
acterization of the manifold over which the DAE becomes an ODE and, of course, the
differential part provides the reduced ODE It is rather remarkable that this can actually
be done in general since it means that, locally, the system ( 1 3) can be uncoupled in the
form g(x) = O (giving the manifold g-' (0 )) and i = X(x) where X is a tangent vector
field on g (0 ) In principle, g and X can always be constructed, but in practice this
construction may be quite demanding and may depend upon the higher derivatives of A
and G in ( 1 3) The difficulty in calculating g and X is directly measured by a nonneg-
ative integer, the index of the DAE, which is the number of recursive steps needed in
Although much work remains to be done in this area for nonlinear problems.