On Asymptotics in Case of Linear Index-2 Differential-Algebraic Equations.
Asymptotic properties of solutions of general linear differential-algebraic equations (DAEs) and those of their numerical counterparts are discussed. New results on the asymptotic stability in the sense of Lyapunov as well as on contractive index-2 DAEs are given. The behavior of the backward differentiation formula (BDF), implicit Runge--Kutta (IRK), and projected implicit Runge--Kutta (PIRK) methods applied to such systems is investigated. In particular, we clarify the significance of certain subspaces closely related to the geometry of the DAE. Asymptotic properties like A-stability and L-stability are shown to be preserved if these subspaces are constant. Moreover, algebraically stable IRK(DAE) are B-stable under this condition. The general results are specialized to the case of index-2 Hessenberg systems.
Introduction
. The present paper is devoted to the study of asymptotic properties
of solutions of differential-algebraic equations (DAE's) on infinite intervals and
those of their numerical counterparts in integration methods. It is rather surprising
that, in spite of numerous papers on numerical integration, there are very few results
in this respect.
For index-1 DAE's, asymptotic properties on infinite intervals have been investigated
by Griepentrog and M-arz [4]. Among other things, the notion of contractivity
and that of B-stability were generalized to the case of DAE's and criteria for total
stability were formulated. Algebraically stable IRK(DAE) were shown to be B-stable
for index-1 DAE's, too, provided that the nullspace N of the leading Jacobian was
constant. If this nullspace rotates, stability properties may change.
In this paper, we study general linear index-2 DAE's
exclusively, where the nullspace N := ker A(t) is assumed to be independent of t. A(t)
and B(t) are assumed to be continuous in t. Equation (1.1) is not assumed to be in
Hessenberg form and the coefficients A(t) and B(t) need not commute. Recall that
Hessenberg index-2 DAE's have the special form
This corresponds to the special coefficient matrices in (1.1)
Moreover, it corresponds to a trivially constant nullspace N , since A(t) itself does not
vary with t.
This paper is a heavily revised and enlarged version of an earlier manuscript with the same title
(Preprint 94-5).
y Humboldt-Universit?t zu Berlin, Institut f?r Mathematik, D-10099 Berlin, Germany
M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
Presenting statements on the linear case we hope, as in the case of regular ordinary
differential equations (ODE's), that it will be possible to carry over some properties
to nonlinear DAE's via linearization.
As far as we know, in case of index-2 DAE's stability analyses of integration
methods on infinite intervals have only been presented for linear systems (see M-arz and
Tischendorf [13], Wensch, Weiner, and Strehmel [16]). The latter paper is restricted
to special Hessenberg form systems and relies on the so-called essentially underlying
ODE introduced in Ascher and Petzold [2] for these special systems. We consider this
case in Section 3 and describe the close relation between the inherent regular ODE
which we will take up from [8] in Section 2, and the essentially underlying ODE in
detail.
Although the above mentioned paper [2] is not concerned with asymptotic stability
on infinite intervals, it contains an observation that is highly interesting for us:
Among other things, Ascher and Petzold point out that the backward Euler method
applied to (1.2) may yield rather an explicit Euler formula for the essentially underlying
ODE, and they discuss the influence of the blocks B on this phenomenon.
We will show that not the derivatives of B 12 (t); B 21 (t), but the derivatives of the
projector matrix H(t) := B 12 (t)(B 21 (t)B constitute the essential term,
i.e., the rotation velocity of the subspace described by H(t) is the decisive feature.
Our paper is aimed at explaining the importance of additional subspaces for
answering questions concerning the asymptotic behaviour of integration methods.
Hence, besides introducing the necessary fundamentals, Section 2 provides new results
on the asymptotic stability of DAE solutions in the sense of Lyapunov as well as on
contractive DAE's.
In Section 4, BDF, IRK, and PIRK are investigated in detail. Asymptotic properties
like A-stability and L-stability are shown to be preserved if a certain subspace
it does not rotate. Moreover, we show that an algebraically
stable IRK(DAE) is B-stable under these conditions.
Section 5 illustrates our results by means of examples.
For convenience of the reader, the short appendix provides the basic linear algebra
facts once more.
2. Linear continuous coefficient index-2 equations. Consider the linear
equation
with continuous coefficients. Assume the nullspace of A(t) 2 L(R m ) to be independent
of t and let
Furthermore, set
Obviously, S(t) is a subspace of R m which contains the solutions of the homogeneous
form of the DAE (2.1). Note that the condition
ASYMPTOTICS IN INDEX-2 DAE'S 3
characterizes the class of index-1 DAE's (see Appendix for related facts from linear
algebra). Equation (2.2) implies that the matrix
is nonsingular for all t 2 J , where Q projector onto N . Let
Higher index DAE's are characterized by nontrivial intersections S(t) " N or
equivalently by singular matrices G 1 (t).
Definition. The DAE (2.1) is said to be index-2 tractable if the following two
conditions
hold, where
In the following, let Q 1 (t) denote the projector onto N 1 (t) along S 1 (t), and
(t). Due to the decomposition (2.5), Q 1 (t) is uniquely defined.
Remarks.
1. It holds that dimN 1
2. Due to Lemma A.1, (2.4) and (2.5) imply that the matrix
is nonsingular. But G 1 (t) is singular, independently of how Q is chosen [8].
3. Applying Lemma A.1 once more we obtain the identities
4. Each DAE (2.1) having global index 2 is index-2 tractable with a continuously
differentiable assuming Q 1 to belong to the class C 1 in the sequel is
not restrictive.
The conditions (2.4), (2.5) imply the decompositions
which are relevant for the index-2 case. Taking this into account we decompose the
DAE solution x into
Multiplying (2.1) by PP
respectively, and carrying out a
few technical computations, we decouple the index-2 DAE into the system
4 M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
Equation (2.10) represents the inherent regular ODE of the DAE system.
On the other hand, if we consider (2.10) separately from its origin via the decomposition
(2.9), we know that imPP 1 (t) is an invariant subspace of this explicit ODE
in u. To be more precise: If we have
at some t 0 2 J , then (2.10) implies Furthermore, (2.12) and (2.11) lead
to respectively. Thus, solving (2.10) - (2.13) and setting
we obtain the solutions of the DAE (2.1).
Inspired by the above decoupling procedure, we state initial conditions for (2.1)
as
This yields
but we do not expect x(t 0
Next, we shortly turn to the case of a homogeneous equation (2.1): For the
system
The matrix \Pi(t) :=
is also a projector, and
is said to be the canonical projector for the index-2 case.
Now, the following assertion is easily proved by means of the decoupling explained
above.
Theorem 2.1. Let (2.1) be index-2 tractable with continuously differentiable Q 1 .
Then it holds:
(i) The initial value problems (2.1), (2.14) are uniquely solvable in
N
provided that q 2
(ii) If x(:) solves the homogeneous equation, then it holds that
(iii) Through each x  exactly one solution of the homogeneous
equation at time t   2 J . The solution space M (t) is a proper subspace of S(t)
and
Remarks.
1. The inherent regular ODE (2.10) is determined by the complete coefficient
not only by its first term PP
2 B. If PP 1 (t) varies
rapidly with t, the second term (PP 1 may be the dominant one. This should also be
taken into account when considering the asymptotic behaviour of solutions of (2.1).
ASYMPTOTICS IN INDEX-2 DAE'S 5
2. In general, the linear DAE (2.1) appears to be much simpler if the relevant
subspaces N , N 1 , S 1 and the two projectors Q, Q 1 are constant. In that case (2.10)
simplifies to
3. The value x 0 involved in the initial condition (2.14) is not expected
to be a consistent initial value. What we have is
As shown above, a consistent initial value for the homogeneous
equation always belongs to M (t 0 ), which is precisely the set of consistent initial values
then.
4. If the product PP 1 is time invariant, we have
hence
Note that (QP 1 G \Gamma1
2 B)(t) is also a projector onto ker A(t). It should be mentioned
that the solution space M (t) remains time-invariant provided that both projectors
are constant.
Now we turn to the asymptotic behaviour of the solutions of the homogeneous
equation. Considering the decoupled system (2.10) - (2.12) once more, we see that the
component represents the dynamic one. Supposed the canonical projector
remains bounded on the whole interval 1), the asymptotic behaviour of
the solution
is completely determined by that of its component u(t). Clearly, if u solves a constant
coefficient regular ODE, we may characterize asymptotics by means of the corresponding
eigenvalues. This is what we try to realize for the DAE in the following
theorem.
Theorem 2.2. Let (2.1) be index-2 tractable, Q 1 be of class C 1 , (PP 1
(i) Then the pencil -A(t) + B(t) has the eigenvalues - uniformly for
implies each homogeneous equation solution to
tend to zero as t !1, provided that the projector \Pi(t) remains uniformly bounded.
Proof. Due to our assumptions, the inherent regular ODE has the constant coefficient
2 B. On the other hand, the nontrivial eigenvalues of \GammaP P
(that
is, eigenvalues that do not correspond to ker PP 1 ) are exactly the pencil eigenvalues
of -A +B (cf. [10]).
Let U (\Delta) denote the fundamental solution matrix of u
Taking the solution representation
into account, the assertion follows right away.
Roughly speaking, the assumptions that PP 1 and PP
have to be constant
mean that there is a constant coefficient inherent regular ODE and a possible time
dependence of the system may be caused by (time dependent) couplings only.
6 M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
Next, what about contractivity in case of index-2 DAE's? In the regular ODE
theory, contractivity is well-known to permit very attractive asymptotic properties of
numerical integration methods. Corresponding results are obtained for index-1 DAE's
in [4] by means of an appropriate contractivity notion. In particular, this notion says
that a linear index-1 DAE (2.1) is contractive if there are a constant c ? 0 and a
positive-definite matrix S such that the inequality
holds true for all
Here, we have used the scalar product hz; vi S := hSz; vi and the norm jzj S := hz; zi 1=2
S .
Clearly, this reminds us of the one-sided Lipschitz condition used for contractivity
in the regular ODE case (i.e. I in (2.1)). In the latter case we have,
with
However, things are more difficult for index-2 DAE's. First, considering the decoupled
system again, we observe that each solution of the homogeneous
DAE (2.1) satisfies the identities
inspired by the notion of contractivity given for the index-1 case in [4], we state
the following definition.
Definition. The index-2 tractable DAE (2.1) is called contractive if the following
holds: There is a constant c ? 0 and a symmetric positive-definite matrix S such that
imply
As usually, with this notion of contractivity, too, we aim at an inequality
for all solutions of the homogeneous DAE, that shows the component
decrease in that norm. The following theorem will show: If the canonical projector
\Pi(t) is uniformly bounded, then the complete solution x(t) decreases.
Theorem 2.3. Let (2.1) be index-2 tractable, Q 1 belong to C 1 , \Pi(t) be uniformly
bounded on J and (2.1) be contractive. Then, it holds for each solution of the
homogeneous equation that
where fl is a bound of j\Pi(t)j S .
ASYMPTOTICS IN INDEX-2 DAE'S 7
Proof. We have
Not surprisingly, we obtain
Corollary 2.4. Let (2.1) be index-2 tractable with continuously differentiable
uniformly bounded \Pi(t). If the condition
is satisfied for all u the estimate (2.21) is valid.
Proof. It may be checked immediately that (2.19) and (2.22) lead to (2.20), i.e.,
implies contractivity.
Note that there is no need for assuming (2.22) for all u . For the assertion of
Corollary 2.4 to become true, it is sufficient that (2.22) holds for all u 2 im PP 1 (t),
only.
Inequality (2.22) looks like the usual contractivity condition for the regular ODE
(2.10), i.e., the inherent regular ODE of (2.1). The only difference is that the values
are taken from the subspace imPP 1 (t) instead of all of R m . Roughly speaking, one
has: The DAE (2.1) is contractive if the inherent regular ODE (2.10) is contrctive on
the subspace imPP 1 (t).
As a direct consequence of other results on stability ([15], e.g.) one can deduce
counterparts for linear index-2 DAE's, e.g. the well-known Poincar'e-Lyapunov Theorem

3. Specification of the projector framework for index-2 Hessenberg-
form DAE's. Most authors restrict their interest to so-called Hessenberg-form equa-
tions, i.e., to systems
In our context this
corresponds to
I
Obviously, z
lar, which is the well-known Hessenberg-form index-2 condition. Under this condition
the block
is also a projector. It projects onto im B 12 (t) along ker B 21 (t).
. It holds
Furthermore, one has
8 M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
The canonical projector \Pi is
Recall that M precisely the solution space of the homogeneous
form of (3.1). It is time dependent if the projector H(t) is. However, M (t) may also
rotate with t even if H(t) is independent of time. Note that PP 1 is easier to compute
than \Pi.
Furthermore, the nontrivial part (i.e., dropping the zero rows) of the inherent
regular ODE (2.10) reads now as
where us emphasize once more that quickly varying subspaces
may cause the term H 0 to dominate within this regular ODE. Clearly, H 0 u 1 corresponds
to the term (PP 1
Theorems 2.1 and 2.2 apply immediately. In particular, we obtain: Suppose
H(t) and are time-invariant. Then the eigenvalues of
determine the asymptotic behaviour of the solution.
us turn to the discussion of aspects of contractivity. For index-2 Hes-
senberg-form DAE's (3.1), relation (2.20) applies to the first components only, i.e.
should be satisfied if (cf. (2.19))
Moreover, (2.22) simplifies to
for all
Again we see that the constant-subspace case H 0 (t) j 0 becomes much easier.
It should be stressed that the above decoupling as well as the inherent ODE are
stated in the original coordinates. In particular, the subspace M (t) ae R m is precisely
the one that contains the solutions of the original DAE. No coordinate transformation
is applied and only a decomposition into characteristic components is employed.
Ascher and Petzold [2] use a different approach to decouple characteristic parts of
linear index-2 Hessenberg systems: They use a coordinate change z such that
and
ASYMPTOTICS IN INDEX-2 DAE'S 9
(cf. also [16]). In [2] the matrices R and S are constructed in the following way. Let
First, a matrix R with linearly independent rows is chosen so
that
is satisfied. As a consequence, the m 1 \Theta m 1 block
is nonsingular. Choosing S in such a way that
hold true, we have
The relation
z 3
the main idea of that transformation. R(t) and S(t) are assumed to be smooth.
Carrying out a few straightforward computations one obtains a regular ODE for the
component z namely
z 0
Equation (3.4) is said to be the essentially underlying ODE (EUODE) of the DAE
(3.1).
What does the EUODE have in common with the inherent regular ODE? What
is the difference?
Multiplying the EUODE (3.4) by S and taking into account that
is given, we obtain (3.3). On the other hand, scaling the inherent regular ODE (3.3)
by R leads back to the EUODE (3.4) because
Thus, the EUODE turns out to be nothing else but a scaled version of the inherent
regular ODE and vice versa. Due to
the uniformly
traced back to R m1 \Gammam 2 . Thus, the EUODE has the advantage to be written in the
minimal coordinate space R m1 \Gammam 2 . Unfortunately, the matrices R and S are not
uniquely determined. Consequently, the EUODE is strongly affected by the choice of
R, S. Note that once an R is chosen, we may multiply by any regular K 2 L(R m1 \Gammam 2 )
to obtain another one by ~
R := KR.
From this point of view, the inherent regular ODE (3.3) seems to be more natural,
since all its terms are uniquely determined by the original data.
is a direct component of the original variable x 1 , but the ODE (3.3) lives in the
higher-dimensional space R m1 , and im(I \Gamma represents an invariant
subspace.
Ascher and Petzold [2] observed that the Euler backward method applied to the
may behave like an explicit Euler method. Choose
which simplifies the EUODE to
z 0
(3.
M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
Via the transform z, the Euler backward formula applied to this special DAE
If additionally B 12 (t) and R 0 (t) do not vary with t, (3.6) simplifies toh
This is the explicit Euler formula for (3.5). Clearly this phenomenon is closely related
to time-varying blocks R(t) and S(t) of the coordinate transformation. Let us mention
again that this behaviour depends on the choice of R and S.
In the following section we show that the behaviour of the characteristic subspace
in the general case is decisive for understanding what
really happens.
4. Asymptotic stability of integration methods. A number of widely used
notions for the characterization of asymptotic properties of integration methods for
explicit ODE's relies on the complex scalar test equation
The asymptotic behaviour of a numerical method applied to (4.1) characterizes the
asymptotics in the case of linear constant coefficient systems
Here, the role of - is replaced by the eigenvalues of \GammaB. The justification for restricting
the consideration to (4.2) is given by Lyapunov's theory: The linearization of a
nonlinear autonomous explicit system at a stationary point provides criteria for the
asymptotic behaviour of solutions. In essence, the same is true for index-1 and -2
DAE's [12]. Therefore, we are led to the constant coefficient DAE
with regular matrix pencil -A +B. This equation can be transformed into the Kronecker
canonical form
z 0
I
z
0:
J 0 is a nilpotent matrix (J k
discretization and transformation
to (4.4) commute for many methods, the numerical solution for z vanishes
identically, whereas y is discretized like an explicit system. Hence, numerical methods
applied to constant coefficient linear DAE's trivially preserve their asymptotic stability
properties that are based on the test equation (4.1) (e.g. A-, A(ff)-, L-stability).
Thus, at first glance, one could expect the well-known concepts of asymptotics in
the numerical integration of explicit ODE's to be sufficient for DAE's, too. How-
ever, as described in Sections 2 and 3, DAE's have a more difficult structure than
explicit ODE's, even in view of numerical integration. Roughly speaking, we should
ASYMPTOTICS IN INDEX-2 DAE'S 11
not expect the numerical methods to match the subspace structure exactly if those
subspaces rotate. The scalar test equation (4.1) turns out to be an inappropriate
model in case of DAE's.
Similar results about B-stability are more difficult to obtain. It is well-known
that so-called algebraically stable Runge-Kutta methods are B-stable [6, p. 193] for
explicit systems. In [4, p.129] a similar result is shown to be true for index-1 DAE's
provided that (i) the nullspace N (A(t)) of A(t) does not depend on t, and (ii) the
Runge-Kutta method is a so-called IRK(DAE) (a stiffly accurate method [6, p. 45]).
There are simple linear examples showing that the backward Euler method loses its
B-stability if (i) is not valid.
We recall the notion of B-stability for DAE's having a constant leading nullspace:
Definition [4]. The one-step method x called B-stable if
for each contractive DAE the inequalities
and
jQx (1)
are satisfied. Here, K ? 0 is a constant and x (1), x (2)are arbitrary consistent initial
values.
4.1. BDF applied to linear index-2 DAE's. The k-step BDF applied to
reads as
At each step, equation (4.5) provides an approximation x ' of the exact solution value
Recall that the nullspace of A(t) is assumed to be constant.
Supposed (2.1) is index-2 tractable, we may decouple (4.5) and (2.1) simultaneously
(cf.
ARZ
where we have used the above decomposition again, i.e.,
In particular, if the inhomogeneity q vanishes identically, then the Q 1 -components
are both zero, and one hash
for approximation of
and
for approximation of
The following proposition is an immediate consequence.
Proposition 4.1. Let (2.1) be index-2 tractable with continuously differentiable
. Then the BDF method applied to (2.1) generates exactly the same BDF method
applied to the inherent regular ODE (4.11) if and only if the projector PP 1 (t) does not
vary with t. For a constant projector PP 1 , the BDF methods retain their asymptotic
stability properties for index-2 DAE's provided the canonical projector \Pi(t) remains
uniformly bounded.
On the other hand, varying subspaces may cause the term PP 0
1 to dominate the
inherent regular ODE itself. For instance, the backward Euler method provides thenh
which shows that u(t ' may or may not happen. As it was mentioned
in Section 3, Ascher and Petzold [2] have observed this phenomenon in case of linear
index-2 Hessenberg systems (3.1) (cf. also Section 3). However, this is not surprising
since we cannot expect any discretization method to follow the subspaces precisely
without profound information on the inner structure of the DAE.
Naturally, similar arguments apply to Runge-Kutta methods, too.
4.2. Implicit Runge-Kutta methods and their projected counterparts
applied to linear index-2 DAE's. According to the originally conceived method
for the numerical solution of ordinary differential equations, an implicit Runge-Kutta
(IRK) method can be realized for the DAE (2.1) in the following way [14]: Given
an approximation x '\Gamma1 of the solution of (2.1) at t '\Gamma1 , a new approximation x ' at
obtained via
s
ASYMPTOTICS IN INDEX-2 DAE'S 13
'i is defined by
and the internal stages are given by
s
s:
The coefficients a ij , b i , c i determine the IRK method, and s represents the number
of stages. Assume the matrix A := (a ij
i;j=1 to be nonsingular and denote its inverse
s
s
Equations (4.12) - (4.14) are equivalent to
s
s
s
s:
Looking at (4.16) we observe that the internal stages do not depend on Qx '\Gamma1 .
The special class of IRK methods (IRK(DAE)) with coefficients
is shown to stand out from all IRK methods in view of their applicability to DAE's
in that case, the new value x belongs to the obvious
constraint manifold
Therefore we have
For Hessenberg equations (3.1), relation (4.18) simplifies to
In general, if (4.17) is not fulfilled, then we have % 6= 0, and (4.18) resp. (4.19) are
no longer true. Since this behaviour is a source of instability (for h ! 0), Ascher and
Petzold [1] propose another version for the application of IRK methods to index-2
Hessenberg DAE's (2.18), the so-called Projected IRK (PIRK). Actually, after realizing
the standard internal stage computation, the recursion (4.15) is now replaced
by
s
s
and - ' is determined by
14 M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
If we multiply (4.20) by I \Gamma H(t ' ), - ' can be eliminated:
s
s
On the other hand, (4.21) is equivalent to
It should be mentioned that for IRK(DAE) the projected version is exactly the same
as the original one, since (4.17) implies -
Considering (4.22) - (4.23) in association with the projector formulae (3.2), an
immediate generalization of PIRK methods to fully implicit linear index-2 systems
(2.1) is suggested by
s
s
Since the internal stages -
X 'j do not depend on Q-x '\Gamma1 , there is no need to compute
Q-x ' at this stage.
Now return to the standard IRK (4.15) - (4.16) and decouple (4.16) in the same
way as (2.12). For that, we decompose
A straightforward computation yieldsh
s
s
\GammaP
s
s
(4.26)h
s
s
s
The recursion (4.15) can be decomposed simply by multiplying by the projections:
s
s
ASYMPTOTICS IN INDEX-2 DAE'S 15
s
s
s
s
s
s
s
s
Now, consider the homogeneous case, that is we set If the inhomogeneity q
vanish identically, then v does so, too. Moreover, all values V 'i are equal to zero.
However, if % 6= 0, this is no longer true for This means that, in general,
the resulting x ' has a nontrivial component in contrast to the exact solution
that fulfills Q
In more detail, (4.26) reduces toh
s
s
\GammaP
s
which supposedly approximates
Moreover, (4.27) yields
s
\GammaQ
s
for approximating
In the consequence, the following result holds true for IRK methods analogously to
Proposition 4.1 for the case of BDF methods:
Proposition 4.2. Let (2.1) be index-2 tractable with continuously differentiable
. Then the IRK method applied to (2.1) generates exactly the same IRK method
applied to the inherent regular ODE (4.30) if and only if PP 1 (t) does not vary with
t.
For constant PP 1 , the solution
(4.
M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
of the homogeneous equation is approximated at t ' by
s
s
(4.39)h
s
s
s
Starting with a consistent initial value x 0 (with the components v ' vanish
step by step, too.
For IRK(DAE), (4.42) provides
that is, in the case of constant PP 1 , the approximation x ' belongs to the solution
manifold M (t ' ) given in Theorem 2.1.
Let us briefly turn to PIRK methods (4.24), (4.25). For homogeneous equations,
The decoupled system parts (4.32), (4.34) remain
valid also for the "-" values.
Proposition 4.3. Proposition 4.2 is true for PIRK methods, too.
It should be mentioned that, for constant PP 1 , in PIRK methods we have simply
instead of (4.41). Ascher and Petzold [1] have not considered a recursion for the
component Q-x ' for Hessenberg systems (2.19). Nevertheless, if one is interested in
approximations Q-x ' , a recursion like (4.42) will come up again. In that case, the only
difference between PIRK and IRK methods is the determination of the Q 1 -components
versus (4.41)). Note again that PIRK and IRK are identical for IRK(DAE).
Next, concerning B-stability, the following assertion shows the notion of contractivity
given in Section 2 to be useful.
Theorem 4.4. Let (2.1) be index-2 tractable with continuously differentiable Q 1 ,
each algebraically stable IRK(DAE)
applied to (2.1) is B-stable.
Proof. Denote . Due to the algebraical stability,
positively semi-definite matrix.
Since we deal with linear DAE's only, it remains to show the inequalities jP x '
for the case of the homogeneous equation (2.1).
ASYMPTOTICS IN INDEX-2 DAE'S 17
therefore Additionally, with an IRK (DAE) we also have
s
s
since
holds true for all ~ t. Hence, using the contractivity (cf.
Section 2) we obtain the inequalities
s:
Now, following the standard lines, we compute
s
s
'i
s
s
s
'i
s
s
'i
s
s
Finally, x
It should be noted that Theorem 4.4 does not apply to PIRK. While the first
part, i.e., jP x ' holds true analogously, the necessary relation for the
nullspace component is not given at all for % 6= 0.
5. A numerical counterexample. In the previous sections we have seen that
BDF and Runge-Kutta methods preserve their stability behaviour if PP 1 is constant.
The following example shows that these properties get lost if PP 1 varies with time.
Consider the DAE
with
M. HANKE AND E. IZQUIERDO MACANA AND R. M -
ARZ
where are constant. Note that (5.1) is an index-2 Hessenberg system. One
easily computes (using
such that
Compute the projections
Taking into account that in (5.1), the inherent regular ODE (2.10) reads
The solution subspace M (t) (cf. Theorem 2.2) is given by
subject to consistent initial values (2.13) may be
reduced to the scalar ODE,
together with
Consequently, the asymptotic stability of (5.2) is governed by the sign of - (indepen-
dently of j 2 R). The parameter j measures the change of N 1 (t). fi serves only for
mixing the P component with the nullspace component. Now the complete solution
of (5.1) can be easily computed using (2.10) - (2.12). If x 0 2 R 3 is a consistent initial
value at
the solution of (5.1) is
was solved using the 5-step BDF (Fig. 5.1) and an algebraically stable 2-stage
Runge-Kutta method introduced by Crouzeix (cf. [5, p. 207]) with ae - \Gamma0:73
5.2). The figures show the norm of the numerical solution at the end of the
different values of j and -. Note that, for
a constant coefficient system. The results indicate that the asymptotic behaviour of
the numerical solution depends not only on the asymptotic stability of the differential
equation (5.3) (controlled by -), but also on the geometry of the problem (controlled
by j).
ASYMPTOTICS IN INDEX-2 DAE'S 19


Appendix

Basic linear algebra lemma. A basic connection between the
spaces appearing in the tractability index and the choice of the corresponding projectors
is given by the following lemma, which may be directly obtained from Theorem
A.13. and Lemma A.14. in [4].
Lemma A.1. Let -
be a projector onto ker( -
A). Denote -
A)g. Then the following
conditions are equivalent:
(i) The matrix -
Q is nonsingular.
A).
If -
G is nonsingular, then the relation
holds for the canonical projector -
along -
S).
Proof. (i) ! (ii) The space R m can be described as -
A), because
holds for any z 2 R m . z 2 obviously lies in ker( -
Q is a projector onto
A). For z 1 we obtain
S.
It remains to show that -
f0g. To this end, let x 2 -
A).
Qx holds and there exists a z 2 R m such that -
Qx and
Qx. Consequently,
This holds trivially by definition.
chosen such that -
Ax and so
S. On the other hand, -
Qx lies in ker( -
A). Thus, x 2
holds due to the
assumption. That means, -
Q). Then has to be true, and
G is nonsingular.
Because of the uniqueness of the decomposition ( ), the latter assertion follows
immediately.



--R

Projected implicit Runge-Kutta methods for differential-algebraic equations
Stability of computational methods for constrained dynamic systems
Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations

Solving Ordinary Differential Equations I
Solving Ordinary Differential Equations II
Approximation von Algebro-Differentialgleichungen mit bereich Mathematik






Order results for implicit Runge-Kutta methods applied to differential algebraic systems
Nonlinear Differential Equations and Dynamical Systems
Stability investigations for index-2-systems
--TR

--CTR
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Hong Liu , Yongzhong Song, Stability of numerical methods for solving linear index-3 DAEs, Applied Mathematics and Computation, v.134 n.1, p.35-50, 10 January
I. Higueras , R. Mrz , C. Tischendorf, Stability preserving integration of index-1 DAEs, Applied Numerical Mathematics, v.45 n.2-3, p.175-200, May
I. Higueras , R. Mrz , C. Tischendorf, Stability preserving integration of index-2 DAEs, Applied Numerical Mathematics, v.45 n.2-3, p.201-229, May
Roswitha Mrz, Differential algebraic systems anew, Applied Numerical Mathematics, v.42 n.1, p.315-335, August 2002
asymptotic properties;backward differentiation formulas;differential-algebraic equation;runge-kutta method;stability