we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the ...
REGULAR SOLUTIONS OF NONLINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS AND THEIR NUMERICAL DETERMINATION
PETER KUNKEL3
AND
VOLKER MEHRMANNy
Abstract. For a general class of nonlinear (possibly higher index) di erential-algebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed. Several numerical examples are included. Key words. Nonlinear di erential-algebraic equations, regular solutions, strangeness index, existence and uniqueness, consistent initial values, numerical methods. AMS subject classi cations. 65L99, 34A09. 1. Introduction. If physical systems contain constraints such as links in mechanical systems or if they are governed by some conservation laws as Kirchho 's laws for electric circuits, their modelling usually leads to nonlinear equations of the form (1) F (t; x; x_ ) = 0; so-called di erential-algebraic equations (DAEs), with F 2 C (I 2 Dx 2 Dx; Rn), I R (closed) interval, Dx; Dx Rn open. Obviously, ordinary di erential equations form a special case of (1). But although the theory and numerical methods are well developed for the latter class, similar results for (1) are still rare. In the last years, great e ort was made in understanding the properties of (1) and nding existence and uniqueness results that are as general as possible and, from a numerical analysis point of view, suitable for building a base for numerical methods. Remarkably, this not only concerned the nonlinear case. Already the linear case (2) E (t)_x = A(t)x + f (t); where E; A 2 C (I; Rn;n), f 2 C (I; Rn), required some hard analysis. In the last decade, several more or less di erent approaches were taken to understand special or general nonlinear DAEs. Most of them require that some step-by-step index reduction process is possible, see, e. g., [7, 15, 17, 18, 19, 20]. The assumptions needed for this process mainly consist of constant rank conditions for each level of the reduction process in the neighborhood of the solution under investigation. Although these approaches give a good insight into a large class of nonlinear problems, in general they do not lead to practical numerical procedures. Only very recently, a rather general theory together with a method for the numerical treatment of nonlinear DAEs was developed in [3, 4, 5]. This approach is theoretically based on the concept of the di erentiation index which played a central role in the understanding of linear DAEs. _
_
3 Fachbereich Mathematik, Carl von Ossietzky Universit at, Postfach 2503, D{26111 Oldenburg, Fed. Rep. Germany y Fakult at fur Mathematik, Technische Universitat Chemnitz-Zwickau, D{09107 Chemnitz, Fed. Rep. Germany. This work has been supported by Deutsche Forschungsgemeinschaft, Research grant Me 790/5-2 Di erentiell-algebraische Gleichungen.
1
The aim of this paper is to present a local existence and uniqueness theory for a class of nonlinear DAEs that is more general than in the above approaches and that also leads to suitable numerical procedures for the computation of the solution. The paper is organized as follows. In Section 2 we brie y discuss the linear case as far as it is relevant for the nonlinear case which is treated in Section 3. On the basis of the obtained results we then develop appropriate numerical methods in Section 4 and discuss some numerical tests in Section 5. Finally we give some conclusions in Section 6. 2. The linear case. The standard method to solve nonlinear problems is by means of Newton's method. In principle, it consists of linearizing around some given initial guess, solving the so obtained linear problem and then correcting the inital guess. The main problem therefore is to be able to treat the arizing linear subproblems. Moreover, for a Newton-Kantorovich like convergence theorem these subproblems must be regular in some sense, see, e. g., [6]. Following an idea of Campbell [2], we can di erentiate (2) for suciently smooth coecient functions to obtain in ated DAEs (3) M` (t)_z` = N` (t)z` + g`(t) with (M`)i;j = 0ji1E i0j 0 0j i 1A i0j0 ; i; j = 0; : : :; `; A i for i = 0; : : :; `; j = 0; ( N ) = ` i;j (4) 0 else, (z`)i = x i ; i = 0; : : :; `; (g`)i = f i ; i = 0; : : :; `; where we use the convention that 0ji1 = 0 for i < 0, j < 0 or j > i. The most general class of linear problems that possess the desired properties seems to be characterized by the following hypothesis. There exist integers ^, a^, and d^ such that the in ated pair (M ; N) associated with (E; A) has the following properties. 1. For all t 2 I we have rank M (t) = (^ + 1)n 0 a^, such that there exists a smooth matrix function Z^ with orthonormal columns and size ((^ +1)n; a^) satisfying Z^T M = 0. 2. For all t 2 I we have rank A^ (t) = a^, where A^ = Z^ T N [In 0 1 1 1 0]T , such that there exists a smooth matrix function T^ with orthonormal columns and size (n; d^), d^ = n 0 ^a, satisfying A^ T^ = 0. 3. For all t 2 I we have rank E (t)T^ (t) = d^, such that there exists a smooth matrix function Z^ with orthonormal columns and size (n; d^) yielding that E^ = Z^T E has constant rank d^. When the coecient functions of a linear DAE (2) satisfy Hypothesis 2.1, we call the least possible ^ the (global) strangeness index (or abbreviated as s-index in the following) of (2). In [12] it has been shown that requiring Hypothesis 2.1 is equivalent to requiring that the di erentiation index (see, e. g., [1]) is well-de ned, but it needs less smoothness of the coecient functions. Compared with so-called index reduction methods (see, e. g., [2, 7, 11, 18]) which represent recursive procedures that iteratively lower the index by one until one arrives at a DAE with vanishing s-index, systems that (
)
+1
(
1)
( )
( )
( )
Hypothesis 2.1.
^
^
^
2
2
^
2
2
2
^
2
2 2
2
1
1
2
1
satisfy Hypothesis 2.1 allow for an index reduction to a DAE with vanishing s-index in a single step (such that no assumptions on ranks are needed for intermediate steps). De ning ^ ^ ^ (5) E^ = E 0(t) ; A^ = A^ (t) ; f^ = f^ (t) A (t) f (t) with A^ = Z^T A and f^ = Z^T f , f^ = Z^T g for suciently smooth f , Hypothesis 2.1 obviously yields that all solutions of (2) are also solutions of the DAE (6) E^ (t)_x = A^(t)x + f^(t): But one can also show that even the reverse is true, see [12]. 1
1
1
1
Theorem 2.2.
2
1
1
1
2
2
^
2
Consider the linear DAE
(7)
E (t)_x = A(t)x + f (t);
together with the initial condition
(8)
x(t0 ) = x0 :
(9)
A^2 (t0 )x0 + f^2 (t0 ) = 0:
Let (E; A) satisfy Hypothesis 2.1 with ^ and ^a. In particular, let E; A 2 C ^+1 (I; Rn;n) and f 2 C ^+1 (I; Rn). Then the following holds. 1. An initial condition (8) is consistent (i. e., the corresponding initial value problem has at least one solution) if and only if (8) implies the a^ conditions 2. Every initial value problem with consistent initial condition has a unique solution.
According to [8, 13], the so obtained strangeness-free DAE (i. e., with vanishing sindex) can be formulated as an operator equation with an underlying regular operator. Let D: X ! Y be de ned by (10) (Dx)(t) = E^(t)_x(t) 0 A^(t)x(t); t 2 I; Theorem 2.3.
where the Banach spaces
^ (t ) = 0g; Y = C (I; Rn) ^ 2 C (I; Rn); E^ Ex (11) X = fx 2 C (I; Rn) j E^ Ex +
1
+
0
are equipped with the norms
(12)
^ )k ; kf k = max kf (t)k1 : kxk = kxk + k dtd (E^ + Ex t2 X
Y
Y
Y
I
Then D is continuous and invertible with continuous inverse.
Note that homogeneous initial conditions can be obtained without loss of generality by replacing x(t) with x(t) 0 x . The operator E^ E^ is de ned pointwise by ^ )(t) = E^(t) E^(t)x(t) where E^(t) denotes the Moore-Penrose pseudoinverse (E^ Ex of E^(t). +
+
0
+
+
3
3. The nonlinear case. Turning to the general nonlinear problem (1), we assume that a solution x3 is given. To guaratee local uniqueness of x3 and the applicability of Newton's method for the determination of x3, we need some regularity 3 property of x which can be extended to a whole neighborhood of x3. The results of the previous section propose that the linearization of (1) along x3 should satisfy Hypothesis 2.1. But requiring Hypothesis 2.1 for all linearizations along x in a neighborhood of x3 may be too restrictive. The DAE 1 0 x_ 0 x = 0 x x_ 1 may be seen as a regularly transformed linear DAE satisfying Hypothesis 2.1 with ^ = 1, a^ = 2, and d^ = 0. Its linearization is given by x 0 x_ 0 1 1_x = 1 0 1x + 0x_ x_ + x x_ 0 x : 1x x_ 01 1_x x_ 0 x x_ For (x ; x ) = (0; 0) we nd that M` is always rank de cient whereas for (x ; x ) = ("; 0) with " 6= 0 it always has full rank. Thus Hypothesis 2.1 does not hold uniformly for all linearizations in a neighborhood of the solution (x3; x3) = (0; 0). In general, we must therefore expect that away from the solution the constant rank assumptions as required in Hypothesis 2.1 for the linear case do not hold for the linearization. For these reasons, we generalize Hypothesis 2.1 to nonlinear problems (1) in the following way denoting by (13) F`(t; x; x_ ; : : :; x ` ) = 0 the in ated nonlinear DAEs obtained by successive di erentiation and by M` (t; x; x_ ; : : :; x ` ) = F` x;:::;x ` (t; x; x_ ; : : :; x ` ); (14) N` (t; x; x_ ; : : :; x ` ) = 0(F` x (t; x; x_ ; : : :; x ` ); 0; : : :; 0) its Jacobians. There exist integers ^, a^, and d^ such that for all values (t; x; x_ ; : : :; x ) 2 L with (15) L = f(t; x; x_ ; : : :; x ) 2 I 2 Rn 2 Rn 2 : : : 2 Rn j F(t; x; x_ ; : : :; x ) = 0g Example 3.1.
2
1
2
1
1
1
1
1
1
1
2
1 2
2
1
2
2
2
1 1
2
1
1
2
2
( +1)
( +1)
( +1)
( +1)
( +1)
;_
;
( +1)
Hypothesis 3.2. ( ^+1)
( ^+1)
^
( ^+1)
associated with F the following properties hold. 1. We have rank M^ (t; x; x_ ; : : :; x(^+1) ) = (^ + 1)n 0 ^a, such that there exists a matrix function Z^2 being smooth on L with orthonormal columns and size ((^ +1)n; ^a) satisfying Z^ T2 M^ = 0. 2. We have rank A^2(t; x; x_ ; : : :; x(^+1) ) = a^, where A^2 = Z^ T2 N^ [In 0 1 1 1 0]T , such that there exists a matrix function T^2 being smooth on L with orthonormal columns and size (n; d^), d^ = n 0 ^a, satisfying A^2 T^2 = 0. 3. We have rank Fx_ (t; x; x_ )T^2 (t; x; x_ ; : : :; x(^+1) ) = d^, such that there exists a matrix function Z^1 being smooth on L with orthonormal columns and size (n; d^) yielding that E^1 = Z^ T1 E has constant rank d^.
It is clear that Hypothesis 3.2 reduces to Hypothesis 2.1 for linear problems. The main di erences between Hypothesis 3.2 and the assumptions in [4, 5] are that we 4
do not require constant rank in a neighborhood of the solution in the whole space but only on a submanifold, and that we need less smoothness of the function F . The latter observation is described in detail in [12]. Note that due to Taylor's theorem we obtain the same matrix functions when we rst di erentiate F and then linearize along some x or when we rst linearize along x and then di erentiate. The rst immediate consequences of Hypothesis 3.2 are the following. Remark 3.3.
Lemma 3.4.
We have
rank F x;x;:::;x (t; x; x_ ; : : :; x ) = (^ + 1)n for all (t; x; x_ ; : : :; x ) 2 L. Proof. The claim follows directly from F x;x;:::;x = (0N [In 0 1 1 1 0]T ; M ): (16)
^;
( ^+1)
(^ +1)
_
( ^+1)
^;
(^ +1)
_
^
^
The set L R1+n+(^+1)n forms a (smooth) manifold of dimen-
Corollary 3.5.
sion n + 1.
For the nonlinear problem from Example 3.1 written in the form x_ 0 x = 0; x_ x_ 0 x_ x + x = 0; di erentiation yields x 0 x_ = 0; x x_ + x_ x 0 x x 0 x_ + x_ = 0: Thus for ^ = 1 we nd 3 2 0 1 01 0 7 6 x_ 7 (0N [I 0]T ; M )(t; x; x_ ; x) = 64 00x_ 01 0x_ 1 0 0 1 5 0x 0 x 0 2_x x + 1 x_ 0 x x_ and L = f(t; x ; x ; x_ ; x_ ; x ; x ) j x = x_ ; x = x_ x 0 x_ x_ ; x_ = x ; x_ = x x + x_ 0 x x_ 0 x_ x g = f(t; x ; x ; x_ ; x_ ; x ; x ) j x = 0; x = 0; x_ = x ; x_ = 0g: Hence, we have rank M = 2 and rank(0N [I 0]T ; M ) = 4 on L and the above problem satis es Hypothesis 3.2 with ^ = 1, a^ = 2, and d^ = 0. Note that L forms a manifold of dimension three which can be parametrized by (t; x ; x ). Following Section 2, the idea now is to transform the original problem to a new one such that x3 remains a solution but such that the new problem allows for the application of Newton-Kantorovich like theorems (see, e. g., [6]). According to Corollary 3.5, the set L can be locally parametrized by n + 1 variables. These can be chosen from (t; x; x_ ; : : :; x ) by checking which columns of F t;x;x;:::;x can be discarded without rank drop. Following Hypothesis 3.2, the variable t can always be taken as parameter. Let now y = (t ; x ; x_ ; : : :; x ) 2 L be xed. Because of corank M(y ) = a^; rank Z^ (y )T N (y ) = a^; Example 3.6.
2
2
1
1
1 2
1
1 2
1 1
1 2
2
1
2
1
2
2
1
2
1
1
2
1
2
1
2
2
2
1
1
2 1
1 1
1
1
2
1
2
2
2
1
1
1 1
2
1
1
2
1 2
1 1
1
2
1
2 1
1
1 2
1
1 2
2
1
1
2
0
0
( ^+1)
^;
_
(^ +1)
0
^
0
2
5
0
0
^
0
( ^+1) 0
(N (y ); M(y )) is of full rank and there are a^ variables from x, say (without loss of generality) x from x = (x ; x ), such that discarding the d^ = n 0 a^ columns of F t;x;x;:::;x that belong to x indeed does not lead to a rank drop. The rest of a^ variables, say v, can therefore be chosen from (x;_ : : :; x ). Let (t ; x ; v ) be that part of y that corresponds to the variables (t; x ; v). The above considerations imply that there is a neighborhood U Rn of (t ; x ; v ), without loss of generality an open ball with radius " > 0 and center (t ; x ; v ), and a neighborhood V of y such that L \ V = f'(t; x ; v) j (t; x ; v) 2 Ug where ': U ! L \ V is a di eomorphism. In other words, the relation F (t; x; x_ ; : : :; x ) = 0 can locally be solved according to (t; x; x_ ; : : :; x ) = '(t; x ; v): In particular, there are locally de ned functions G and G such that (17) F (t; x ; G (t; x ; v); G (t; x ; v)) 0: Let us now consider F^ = Z^T F (18) with Z^ as de ned by Hypothesis 3.2. From F^ (t; x ; G (t; x ; v); G (t; x ; v)) 0 it follows by di erentiation with respect to v that (omitting arguments) d ^ F = (Z^ T x F + Z^ T F x )G v + (Z^ T x;:::;x F + Z^ T F x;:::;x )G v dv = Z^T F x G v 0 on U. By construction, x was selected such that Z^T F x is nonsingular, hence G v (t; x ; v) = 0 for all (t; x ; v) 2 U. Therefore, x = G^ (t; x ) = G (t; x ; v ): Di erentiating now F^ with respect to x , we get (without arguments) d ^ F = (Z^ T x F + Z^ T F x ) + (Z^ T x F + Z^ T F x )G x dx + (Z^T x;:::;x F + Z^T F x;:::;x )G x = Z^T F x + Z^T F x G x = 0Z^T N [In 0 1 1 1 0]T G I x 0 ^
0
^
0
2
^;
1
2
(^ +1)
_
1
( ^+1)
0
10
0
0
1
+1
0
0
10
10
0
0
1
1
( ^+1)
^
( ^+1)
1
2
^
1
2
1
3
2
2
3
1
^
2
2
2
2; 2
2
^
1
2
3
2;
^; 2
2
1
2; _
1
(^ +1)
^
2
^; 2
1
0
2
(^ +1)
^; _
3;
2;
^; 2
2
2;
1
1
2
2
1
2
1
2
2; 1
2
1
^
2
^; 1
2; 2
^
(^ +1)
2; _
2
^; 1
2
^; 2
2; 1
2
6
^
2; 1
^; 2
2
^
2
^; _
(^ +1)
2; 1
3; 1
0
on U. Following Hypothesis 3.2, we therefore can choose
I (19) T^ (t; x ) = G^ (t; x ) x and hence Z^ as a function in (t; x; x_ ). Finally setting F^ = Z^ T F; (20) we now know that (t ; x ; x ; x_ ; x_ ) as part of y solves the system (21) F^ (t; x ; x ; x_ ; x_ ) = 0; x = G^ (t; x ): Writing this in short form as (22) F^ (t; x; x_ ) = 0; we have for all (t; x; x_ ) satisfying F (t; x; x_ ) = 0 that ^T ^T F^x (t; x; x_ ) = Z Fx (t; x ; x ; x_ ; x_ ) Z Fx (t; x ; x ; x_ ; x_ ) 0 0 and 3 : ^Fx(t; x; x_ ) = ^ 3 G x (t; x ) 0I Because of T T ^T rank G^ Z^x (Ft;xx ) Z^ 0FIx = rank G^Z x F(t;xT^x ) 00I = n by Hypothesis 3.2, the DAE (22) has vanishing s-index (or di erentiation index at most one). Di erentiating the second equation in (21) and inserting into the rst equation, we can then locally solve for x_ . Finally eliminating x from this new equation, we obtain a system of the form (23) x_ = G^ (t; x ); x = G^ (t; x ): In summary, we have shown that the rst part of each y 2 L solves an equation of the form (23). Therefore, given a suciently smooth solution x3 of a DAE (1) satisfying Hypothesis 3.2, x3 locally also solves a DAE of the form (23). But the latter DAE now has the desired regularity property. De ning F: D ! Y, D X open, by ^ F (x)(t) = x_ (t) 0 G^ (t; x (t)) (24) x (t) 0 G (t; x (t)) with (25) X = fx 2 C (I; Rn) j x 2 C (I; Rd ); x (t ) = 0g; Y = C (I; Rn); 2
1
1
2; 1
1
1
0
10
20
1
10
1
1
_
2
20
1
0
2
1
_1
1
2
2
1
2
1
_1
1
1
1
_2
_
1
1
2
1
2; 1
1
1
1
_2
1
2; 1
2; 1
2
2
1
1
2
2
1
0
1
1
1
1
2
2
1
^
1
7
1
0
2
1
2
its Frechet derivative Fx(x): X ! Y at some x 2 D given by ^ Fx (x)(1x)(t) = 1_x (t) 0 G^ x (t; x (t))1x (t) 1x (t)0 G x (t;x (t))1x (t) (26) ^ ( t; x ( t )) 0 G 0 I x d = 0 0 1_x(t) 0 G^ x (t; x (t)) 0Ia 1x(t) is a homeomorphism according to Section 2. In this way, we have obtained the following result where, without loss of generality, we again use that we have a homogeneous initial condition. 1
1; 1
1
1
2
2; 1
1
1
^
1; 1
1
2; 1
1
^
Let a suciently smooth solution x3 2 D of
Theorem 3.7.
(27)
F (t; x; x_ ) = 0; x(t0) = 0; be given, where F satis es Hypothesis 3.2. Then x3 is a locally unique solution of the
(strangeness-free) problem
F (x) = 0:
(28)
Proof. The claim follows by means of Newton's method in Banach space, see, e. g., [6, Theorem 15.6]. Since the formulation (28) may allow for less smooth solutions, we may see (28) as a weak formulation of (27) and de ne all solutions of (28) to be weak solutions of (27). 4. Numerical methods. The numerical treatment of initial value problems for DAEs mainly consists of two parts. First, the consistency of the given initial value must be checked. If consistency does not hold, a consistent initial value must be computed. Second, starting from a consistent initial value we must stepwise integrate the DAE. In Section 3, it has been shown that each y 2 L can be locally extended to a solution of (1). Thus, consistency of an initial value x at t means that (t ; x ) is part of some y 2 L. To determine a consistent initial value or to check it for consistency, we must therefore solve (29) F (t ; x; x_ ; : : :; x ) = 0: for (x; x_ ; : : :; x ). By Hypothesis 3.2, the Jacobian of this underdetermined system of nonlinear equations has full row rank at every solution. Thus, the Gau-Newton method will converge locally and quadratically (see, e. g., [16]). This remains valid when we do not allow t to change during the iteration. Following Section 3, there are d^ additional values from x that can be kept xed. For ODEs x_ = f (t; x), this corresponds to the procedure to choose y = (t ; x ; x_ ) 2 L by taking (t ; x ) from the inital condition and computing x_ by x_ = f (t ; x ). Of course, this approach is preferable to that of simply solving the underdetermined problem x_ 0 f (t; x) = 0 which may also alter x . Denoting by T~ some approximation (with orthonormal columns) to T^ at the desired solution, the method of choice here is to solve (30) F (t ; T~ T~T x + (I 0 T~ T~T )x; x_ ; : : :; x ) = 0 0
0
0
0
0
0
^
( ^+1)
0
( ^+1)
0
0
0
0
0
0
0
0
0
0
0
0
0
2
^
2
0
2
2
0
2
8
2
( ^+1)
for (x; x_ ; : : :; x ). Since T~ T~T is an orthogonal projector of rank d^, (30) applied to ODEs becomes x_ 0 f (t ; x ) = 0 which does not depend on x. Hence, the GauNewton method will never change the corresponding starting value and we indeed obtain the preferred value y = (t ; x ; f (t ; x )) 2 L. Let F of (1) satisfy Hypothesis 3.2. Then the Jacobian J of (30) at a solution (x; x_ ; : : :; x ) has full row rank provided T~ is a suciently good approximation to T^ corresponding to this solution. Proof. We have (without arguments) J = [0N [In 0 1 1 1 0]T (I 0 T~ T~T ) j M ]: By Hypothesis 3.2, it follows rank J = (^ + 1)n 0 a^ + rank[A^ (I 0 T~ T~T )]: For T~ = T^ , this reduces to rank J = (^ + 1)n 0 a^ + rank A^ = (^ + 1)n; such that T~ must approximate T^ in such a way that A^ (I 0 T~ T~T ) keeps full rank. To integrate the DAE (1) numerically, we must discretize it. For simplicity, we concentrate here on BDF methods. It is well-known that these are suited for the numerical solution of general nonlinear DAEs with di erentiation index at most one, see, e. g., [1]. For DAEs with higher index, BDF directly applied to (1) may give wrong results or may not de ne a numerical solution at all. In the following, let Dh denote the backward di erence operator which is of the form k 1X (31) l xi0l Dh xi = ( ^+1)
2
2
0
0
0
0
0
0
0
Theorem 4.1.
( ^+1)
2
2
^
2
^
2
2
2
2
2
2
2
2
2
2
2
2
h l=0
where h is the stepsize, xi denote numerical approximations to x(ti) with ti = t + ih, and l are the coecients of the k-step BDF method. The only property of Dh we need is that it is linear with respect to the actual approximation xi with nonzero derivative =h. Our method of choice to proceed from t to t + h then is to solve (a) F(t + h; x; x_ ; : : :; x ) = 0; (32) (b) Z~ T F (t + h; x; Dhx) = 0 for (x; x_ ; : : :; x ) and to take the part x as approximation x to x(t ). Here Z~ denotes some approximation (with orthonormal columns) to Z^ at the desired solution. According to the considerations of Section 3, system (32) can be interpreted as follows. The rst part (32.a) yields a solution for which x = G^ (t + h; x ) holds. For a suciently good approximation Z~ , the same arguments as for (21) show that Z~T F (t; x ; x ; x_ ; x_ ) = 0; x = G^ (t; x ) (33) has di erentiation index at most one such that we can apply BDF. But discretizing (33) by BDF is nothing else than (32). 0
0
0
^
0
( ^+1)
0
0
1
( ^+1)
1
1
1
1
2
2
1
1
1
2
1
2
2
9
2
1
0
1
Theorem 4.2. Let F of (1) satisfy Hypothesis 3.2. Then the Jacobian J of (30) at a solution (x; x; _ : : :; x(^+1)) has full row rank for suciently small h provided Z~1 is a suciently good approximation to Z^1 corresponding to this solution. Proof. We have (without arguments)
J=
0N^ [In 0 1 1 1 0]T M^ : Z~T ( F + F ) 0 1
By Hypothesis 3.2, it follows
0 h x_
x
^ rank J = (^ + 1)n 0 a^ + rank Z~T ( AF + F ) : x h x For suciently small h, we then obtain rank J = (^ + 1)n + rank[Z~T FxT^ ]: For Z~ = Z^ , this reduces to rank J = (^ + 1)n + d^ such that Z~ must approximate Z^ in such a way that Z~T F xT^ keeps full rank. The approach developed here consists of rst dicretizing the nonlinear DAE and then solving the nonlinear discrete problem by linearization. For the BDF discretization, however, one can show that linearization and discretization commute, i. e., that we obtain the same method when we rst linearize (linearization in function spaces is sometimes called quasi-linearization) and then discretize by BDF. Solving underdetermined nonlinear systems of equations by the Gau-Newton method, the most important property one has to look for, is full row rank at the desired solutions. This property then not only extends to a neighborhood of the solution set and guarantees local and quadratic convergence to some solution but also allows for some techniques known from Newton's method to improve the eciency of the method such as simpli ed Gau-Newton method ( xing the Jacobian) or Quasi-Gau-Newton method (Broyden rank one updates of the Jacobian), see, e. g., [16]. If it is known that Z^ of Hypothesis 3.2 only depends on t, then one can choose Z~ = Z^ and system (32) can be reduced to (a) Z^T F(t + h; x; x_ ; : : :; x ) = 0; (34) (b) Z~T F (t + h; x; Dhx) = 0: Due to Hypothesis 3.2, the dependence of (34.a) on (_x; : : :; x ) can be neglected such that (34) represents a system which only depends on x. In particular, this simpli cation applies to ODEs where Z^ is an empty matrix and Z~ = In such that (34) becomes the standard BDF method applied to the ODE. Moreover, it applies to linear DAEs where (34) becomes nothing else than (6) such that we have rediscovered the techniques of [10, 14] for linear problems with unique solution. Note that already for the determination of consistent initial values, we must know at least the value of ^. In the linear case, it can be computed even when we have no consistent value because the linearization here does not depend on the trajectory along which we linearize. But in the nonlinear case we must take into consideration 2
0
1
1
1
_
2
_
1
1
1
1
_
2
Remark 4.3.
Remark 4.4.
Remark 4.5.
2
2
2
2
^
( ^+1)
0
0
1
( ^+1)
2
1
10
Table 5.1
Input parameters
t0 z~0 ^; d;^ ^a h N kmax m
initial time initial guess for (x ; x_ ; : : :; x ) characteristic values due to Hypothesis 3.2 stepsize number of steps to be performed maximal order of BDF to be used mode parameter m = 0 : linear problem (use system (34) and perform only one Gau-Newton step) m = 1 : special nonlinear problem (use system (34) according to Remark 4.5) m = 2 : general nonlinear problem (use system (32)) 0
0
( ^+1)
that we can compute ^ (as well as a^ and d^) only when we already have a consistent value due to possibly di erent ranks of the quantities in Hypothesis 3.2 away from the manifold L. In case the values of ^, a^, and d^ are not known in advance the only possibility is to try to x them during the determination of a consistent initial value. One may proceed as follows. Set ^ = 0, a^ = 0, and d^ = n 0 a^. In each iteration step for solving (29) or (30), perform the numerical rank checks (a) corank M a^; (b) rank Z~T N = ^a; (35) (c) rank FxT~ = d;^ say by singular value decompositions. Here Z~ denotes a matrix whose orthonormal columns span the corange of M perturbed to a matrix of corank a^ and T~ denotes a matrix whose orthonormal columns span the kernel of Z~T N perturbed to a matrix of rank a^. These perturbations can easily be obtained from the singular value decompositions by neglecting the super uous nonzero singular values. A violation of the numerical rank check in (35.a) then indicates that there may be additional constraints and a^ should be increased by one (and d^ decreased by one), a violation of the numerical rank check in (35.b) indicates that local uniqueness may not hold for the given problem and one should terminate, and a violation of the numerical rank check in (35.c) indicates that one is near a higher index problem and ^ should be increased by one. The determination of the characteristic values ^, a^, and d^ is successful if the iteration methods ends with accepting a consistent initial value under the rank checks (35). 5. Numerical examples. We have implemented the techniques introduced in Section 4 in a FORTRAN subroutine on a xed stepsize xed order basis. Additionally to the subroutine evaluating F and its derivatives with respect to t together with their Jacobians, the program uses the input parameters given in Table 5.1. Note that this code is on the level of a research implementation. We therefore give no comparisons even when the speci c problem is of a form that can be treated by a di erent code. We should emphasize that currently there are no production codes for ^
^
2
2
_
2
^
2
2
11
^
Table 5.2
Rank checks during the determination of the characteristic values for Example 5.1
trial 1 2 3 4
characteristic values violated rank check ^ = 0, d^ = 2, a^ = 0 (35.a) ^ = 0, d^ = 1, a^ = 1 (35.c) ^ ^ = 1, d = 1, a^ = 1 (35.a) ^ = 1, d^ = 0, a^ = 2 none
general nonlinear DAEs with arbitrary index available to compare with. A production code on the basis of the research code discussed here is under development. Because we get back the techniques of [10] for linear problems (i. e., for mode parameter m = 0) and those for the well-known code DASSL (see, e. g., [1]) for strangeness-free problems (i. e., for s-index ^ = 0), we concentrate here on results for nonlinear problems with ^ 1. All computations were performed on an HP715/33 workstation in FORTRAN double precision. We rst consider the problem of Example 3.1 with inhomogeneity in the form 1 0 x_ 0 x 0 et0 = 0: x_ 1 x 0t Example 5.1.
2
1
1
1
2
Working in mode m = 2, we took the initial guess z~ = (1; 0; 0; 1; 0; 0) for t = 0. The determination of the correct characteristic values ^, d^, and ^a by the procedure described at the end of Section 4 brought a sequence of rank checks given in Table 5.2. For N = 100, h = 0:01, and k = 1, the solution x(1) = (0; 1) was reproduced without diculty. The computing time was 0.53 seconds. A model of a ring modulator is described in [9]. It consists of n = 15 di erential-algebraic equations. The characteristic values are given by ^ = 1, d^ = 10, and a^ = 5. Starting with the initial guess z~ = 0, we obtained after N = 2000 steps with h = 5 1 100 and k = 1 in mode m = 1 a numerical solution whose rst component is shown in Figure 5.1 consuming 352 seconds computing time. The computed solution is similar to that given in [9] for a nearby system of di erential equations. One easily recognizes in Figure 5.1 the superposition of two oscillations with di erent amplitudes and frequencies. It should be mentioned that in this example the arising Jacobians are badly scaled. For the rank decisions, we had to scale rst the independent variables to about the same size and then to equilibrate the corresponding Jacobians with respect to the rows. A planar model of a truck is given in [21]. It is of the standard Euler-Lagrangian form of multibody systems with characteristic values ^ = 2, d^ = 20, and a^ = 3. As initial guess for t = 0, we took z~ = (x ; 0; 0; 0) where x is the consistent initial value given in [21]. The solution given there was reproduced using the parameters N = 500, h = 0:04, k = 1, and m = 2 within 776 seconds of computing time. Figure 5.2 shows the vertical motion of the driver seat with respect to its equilibrium position when the truck moves along a street which is not completely plain. The last example is a path prescribed control problem for a twolink planar robot arm taken from [4]. The characteristic values here are ^ = 4, d^ = 0, and a^ = 8. Table 5.3 gives a possible sequence of characteristic values tried for z~ = 0 0
0
max
Example 5.2.
7
0
max
Example 5.3.
0
0
0
0
max
Example 5.4.
0
12
. First solution component of Example 5.2
Fig. 5.1
while performing the procedure described at the end of Section 4 to nd the correct ones. Note that some of the rank checks indicated a rank de ciency of two which was used to reduce the number of necessary steps. The computations for N = 100, h = 0:01, k = 6, and m = 2 needed 103 seconds computing time. Although the given problem has a high strangeness index ^ = 4, there were no problems in the necessary rank checks due to well scaled Jacobians. 6. Conclusions. We de ned a class of nonlinear DAEs for which we showed existence and uniqueness of solutions. In particular, the de nition of this class also included the de nition of an index. For this class it is also obvious what should be understood by consistency of initial values. The used techniques were similar to those used for nonlinear problems in nite dimensional spaces, namely to show regularity of the solution in the sense of Newton's method which implies existence and (local) uniqueness. The results obtained in this way were used to design numerical methods for the determination of consistent initial values and the integration of the DAE so that we obtained a procedure for the numerical treatment of nonlinear DAEs with max
13
. Vertical motion of the driver seat in the truck model of Example 5.3
Fig. 5.2
Table 5.3
Rank checks during the determination of the characteristic values for Example 5.4
trial 1 2 3 4 5 6 7 8 9 10
characteristic values violated rank check ^ = 0, d^ = 8, a^ = 0 (35.a) ^ ^ = 0, d = 6, a^ = 2 (35.c) ^ = 1, d^ = 6, a^ = 2 (35.a) ^ = 1, d^ = 4, a^ = 4 (35.c) ^ ^ = 2, d = 4, a^ = 4 (35.a) ^ = 2, d^ = 2, a^ = 6 (35.c) ^ = 3, d^ = 2, a^ = 6 (35.a) ^ = 3, d^ = 1, a^ = 7 (35.c) ^ ^ = 4, d = 1, a^ = 7 (35.a) ^ = 4, d^ = 0, a^ = 8 none 14
arbitrary index. The numerical examples show that these methods indeed work. A generalization to DAEs with undetermined solution components as treated in [10, 11] for linear problems seems to be possible and is currently under investigation. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
, Numerical Solution of Initial-Value Problems in Di erential Algebraic Equations, Elsevier, North Holland, New York, 1989. S. L. Campbell, A general form for solvable linear time varying singular systems of di erential equations, SIAM J. Math. Anal., 18 (1987), pp. 1101{1115. S. L. Campbell and C. W. Gear, The index of general nonlinear DAEs, Numer. Math., 72 (1995), pp. 173{196. S. L. Campbell and E. Griepentrog, Solvability of general di erential algebraic equations, SIAM J. Sci. Comput., 16 (1995), pp. 257{270. S. L. Campbell and E. Moore, Constraint preserving integrators for general nonlinear higher index DAEs, Numer. Math., 69 (1995), pp. 383{399. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. E. Griepentrog, Index reduction methods for di erential algebraic equations, Tech. Report 91-12, Sektion Mathematik, Humboldt-Universitat Berlin, 1991. rz, Di erential-Algebraic Equations and Their Numerical TreatE. Griepentrog and R. Ma ment, Teubner Verlag, Leipzig, 1986. W. Kampowski, P. Rentrop, and W. Schmidt, Classi cation and numerical simulation of electric circuits, tech. report, Mathematisches Institut, Technische Universitat Munchen, 1991. P. Kunkel and V. Mehrmann, A new class of discretization methods for the solution of linear di erential-algebraic equations. Materialien LXII, FSP Mathematisierung, Universitat Bielefeld, 1992. To appear in SIAM J. Numer. Anal. , Canonical forms for linear di erential-algebraic equations with variable coecients, J. Comput. Appl. Math., 56 (1994), pp. 225{251. , Local and global invariants of linear di erential algebraic equations and their relation, Tech. Report SPC 95 25, Fachbereich Mathematik, TU Chemnitz, D-09107 Chemnitz, FRG, 1995. , Generalized inverses of di erential-algebraic operators, SIAM J. Matrix Anal., 17 (1996), pp. 426{442. P. Kunkel, V. Mehrmann, W. Rath, and J. Weickert, GELDA: A software package for the solution of GEneral Linear Di erential Algebraic equations, Tech. Report SPC 95 8, Fachbereich Mathematik, TU Chemnitz, D-09107 Chemnitz, FRG, 1995. To appear in SIAM J. Sci. Comput. rz, Some new results concerning index-3 di erential-algebraic equations, J. Math. Anal. R. Ma Appl., 140 (1989), pp. 177{199. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, San Diego, 1970. P. J. Rabier and W. C. Rheinboldt, A general existence and uniqueness theory for implicit di erential-algebraic equations, Di . Integ. Equ., 4 (1991), pp. 563{582. , A geometric treatment of implicit di erential-algebraic equations, J. Di . Equ., 109 (1994), pp. 110{146. S. Reich, On an existence and uniqueness theory for nonlinear di erential-algebraic equations, Circuit Systems Signal Processing, 10 (1991), pp. 343{359. W. C. Rheinboldt, Di erential-algebraic systems as di erential equations on manifolds, Math. Comput., 43 (1984), pp. 473{482. hrer, and P. Rentrop, A nonlinear truck model and its treatment B. Simeon, F. Grupp, C. Fu as a multibody system, J. Comput. Appl. Math., 50 (1994), pp. 523{532. K. E. Brenan, S. L. Campbell, and L. R. Petzold
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