Numerical Treatment of Unstructured. Differential-Algebraic Equations. with Arbitrary Index

Numerical Treatment of Unstructured Differential-Algebraic Equations with Arbitrary Index Peter Kunkel (Leipzig) SDS2003, Bari-Monopoli, 22. 25.06.2003

Outline Numerical Treatment of Unstructured Differential-Algebraic Equations with Arbitrary Index I Theoretical Basis 1 Problem Description 2 Linear Problems with Constant Coefficients 3 Linear Problems with Variable Coefficients 4 Nonlinear Problems II Numerical Procedures 5 Discretizations 6 Linear Initial Value Problems 7 Nonlinear Initial Value Problems 8 Boundary Value Problems 9 Literature/Software Note The slides may be more detailed than one would usually expect and prefer. However, we hope that in this way they will provide a kind of terse manuscript for the interested reader. The latest version of the slides can be found at http://www.math.uni-leipzig.de/~kunkel/bari.html

1.1 Application 1 Electronic circuits

1.2 Application 2 Multibody systems

1.3 Problem Description The equations for the timely development of a (discrete) system in general have the form F (t, x, ẋ) = 0, so-called differential-algebraic equation (DAE), with F : I D x Dẋ R m, sufficiently smooth. I R compact interval, D x, Dẋ R n open, This includes over- and underdetermined problems arising, e. g., from 1. control problems, 2. redundancies in the model generation. In addition, we typically have initial conditions x(t 0 ) = x 0 with t 0 I, x 0 D x, or boundary conditions r(x(t), x(t)) = 0 with r : D x D x R d sufficiently smooth for some problem specific d and I = [t, t].

1.4 Problem Description (cont.) Linearization of F along a given x C 1 (I, R n ) yields E(t)ẋ = A(t)x + f(t) with E(t) = Fẋ(t, x (t), ẋ (t)), A(t) = F x (t, x (t), ẋ (t)), f(t) = F (t, x (t), ẋ (t)). If F does not depend explicitly on t and if x is constant, then E and A are constant, yielding a problem of the form Eẋ = Ax + f(t) with E, A R m,n.

1.5 Basic Notions We use the following basic definitions. Definition 1. A function x C 1 (I, R n ) is called a solution of the DAE if x satisfies the DAE pointwise. 2. The function x is called a solution of the initial or boundary value problem if x furthermore satisfies the initial or boundary condition. 3. A DAE is called solvable if it has at least one solution. 4. An initial condition is called consistent with the DAE if the associated IVP has at least one solution. 5. In the linear case, an inhomogeneity f is called consistent with (E, A) if the associated DAE is solvable. Remark In the discussion of linear DAEs we use C instead of R.

2.1 Linear DAEs with Constant Coefficients In this section, we consider Eẋ = Ax + f(t) with E, A C m,n and f C(I, C m ) sufficiently smooth. Rescaling of the equation and the unknown by nonsingular matrices does not change the solution behaviour. Definition Two matrix pairs (E 1, A 1 ) and (E 2, A 2 ) are called (strongly) equivalent if there are nonsingular matrices P C m,m and Q C n,n such that E 2 = P E 1 Q, A 2 = P A 1 Q. A corresponding normal form is the so-called Kronecker canonical form (KCF). It exhibits all properties of a linear DAE with constant coefficients.

2.2 Kronecker Canonical Form Theorem Let E, A C m,n. There are nonsingular matrices P C m,m and Q C n,n such that (for all λ C) P (λe A)Q = diag (L ε1,..., L εp, M η1,..., M ηq, J ϱ1,..., J ϱr, N σ1,..., N σs ), where 1. L εj C ε j,ε j +1, ε j N 0, with 0 1 1 0 λ............, 0 1 1 0 2. M ηj C η j+1,η j, η j N 0, with 0 1 λ 1...... 0 1 0...... 1 0 3. J ϱj C ϱ j,ϱ j, ϱ j N, λ j C, with 1... λ... 1 4. N σj C σ j,σ j, σ j N, with 0 1 1......... λ... 1 0, λ j 1......... 1 λ j,.... 1

2.3 Regularity Definition Let E, A C m,n. The matrix pair (E, A) is called regular if p(λ) = det(λe A) 0. Theorem Let (E, A) be regular. Then the KCF has the form ([ ] [ ]) I 0 J 0,, 0 N 0 I where N is nilpotent and J and N are in Jordan canonical form. Definition The nilpotency index ν of N is called the index of (E, A). Remark The solution of the DAE is given by Nẋ = x + f(t) x(t) = ν 1 i=0 N i f (i) (t). Theorem Let (E, A) be regular of index ν and f C ν (I, C n ). Then every associated IVP with consistent initial condition is uniquely solvable.

3.1 Linear DAEs with Variable Coefficients In this section, we consider E(t)ẋ = A(t)x + f(t) with E, A C(I, C m,n ), f C(I, C m ) sufficiently smooth. Rescaling of the equation and the unknown can now be done by pointwise nonsingular matrix functions. Definition Two pairs (E 1, A 1 ) and (E 2, A 2 ) of matrix functions are called (globally) equivalent if there are pointwise nonsingular matrix functions P C 0 (I, C m,m ) and Q C 1 (I, C n,n ) such that E 2 = P E 1 Q, A 2 = P A 1 Q P E 1 Q. For fixed ˆt I and given P (ˆt), Q(ˆt) and Q(ˆt), we can always find suitable functions P and Q that assume these values. This gives the following local version of global equivalence. Definition Two matrix pairs (E 1, A 1 ) and (E 2, A 2 ) are called (locally) equivalent if there are nonsingular P C m,m, Q C n,n and an arbitrary R C n,n such that E 2 = P E 1 Q, A 2 = P A 1 Q P E 1 R.

3.2 Local Canonical Form Theorem Let E, A C m,n and T basis von kernel E, Z basis von corange E = kernel E H, T basis von cokernel E = range E H, V basis von corange(z H AT ). Then the quantities r = rank E, a = rank(z H AT ), s = rank(v H Z H AT ), d = r s, u = n r a, v = m r a s (rank) (algebraic part) (strangeness) (differential part) (undefined part) (vanishing part) are invariant under local equivalence and (E, A) is locally equivalent to I s 0 0 0 0 I d 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0 0 0 0 0 0 0 0 0 0 I a 0 I s 0 0 0 0 0 0 0.

3.3 Global Canonical Form Turning back to pairs of matrix functions, we get functions of characteristic values. Theorem r, a, s: I N 0 Let E, A C(I, C m,n ) be sufficiently smooth and suppose that r(t) r, a(t) a, s(t) s for the local invariants of (E(t), A(t)). Then (E, A) is globally equivalent to I s 0 0 0 0 I d 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0 A 12 0 A 14 0 0 0 A 24 0 0 I a 0 I s 0 0 0 0 0 0 0.

3.4 Reduction Process The DAE is thus transformed to ẋ 1 = A 12 (t)x 2 + A 14 (t)x 4 + f 1 (t), ẋ 2 = A 24 (t)x 4 + f 2 (t), 0 = x 3 + f 3 (t), 0 = x 1 + f 4 (t), 0 = f 5 (t). Differentiating the fourth equation and eliminating ẋ 1 gives a modified DAE with the same solution set. The corresponding pair of matrix functions reads 0 0 0 0 0 I d 0 0 0 0 0 0 0 0 0 0 0 0 0 0, 0 A 12 0 A 14 0 0 0 A 24 0 0 I a 0 I s 0 0 0 0 0 0 0. Thus we can define a sequence (E i, A i ) starting with (E 0, A 0 ) = (E, A) by iteratively transforming to the above canonical form (assuming constant ranks) and performing the elimination. Theorem The invariants r i, a i, s i of (E i, A i ) are characteristic for (E, A). Because of r i+1 = r i s i, the iterative process becomes stationary when the index µ = min{i N 0 s i = 0}, the so-called strangeness index, is reached.

3.5 Existence and Uniqueness Theorem Let µ be well-defined for (E, A) and let f C µ (I, C m ). Then the solutions of the associated DAE correspond one-to-one (via a pointwise nonsingular matrix function) to the solutions of a DAE of the form ẋ 1 = A 13 (t)x 3 + f 1 (t), 0 = x 2 + f 2 (t), 0 = f 3 (t), (d µ differential equations) (a µ algebraic equations) (v µ consistency conditions) where the f i depend on f, f,..., f (µ). If f C µ+1 (I, C m ), we have: 1. The given DAE is solvable iff f 3 = 0. 2. An initial condition is consistent iff in addition it implies x 2 (t 0 ) = f 2 (t 0 ). 3. The corresponding IVP is uniquely solvable iff in addition u µ = 0.

3.6 Global Canonical Form (cont.) Theorem Let µ be well-defined for (E, A) with characteristic values r i, a i, s i, i = 0,..., µ. Furthermore, let w 0 = v 0, w i+1 = v i+1 v i and c 0 = a 0 + s 0, c i+1 = s i w i+1. Then (E, A) is globally equivalent to I d µ 0 0 0 F, 0 0 0 0 0 0 0 G 0 0 I aµ with F = 0 F µ......... F 1 0, G = 0 G µ......... G 1 0. The entries F i and G i are of the size (w i, c i 1 ) and (c i, c i 1 ) and satisfy [ ] Fi rank = c i + w i = s i 1 c i 1. G i In particular, the matrix functions F i and G i together have full row rank.

3.7 Exceptional Points Theorem Let M C(I, C m,n ). Then there are open intervals I j I, j N, with j N I j = I, I i I j = for i j, and integers r j N 0, j N, with rank M(t) = r j for all t I j. Applying this property to the above construction of the sequence (E i, A i ), we immediately obtain the following result. Theorem Let E, A C(I, C m,n ) be sufficiently smooth. Then there are open intervals I j, j N, as above such that the strangeness index is well-defined for (E, A) restricted to I j for every j N.

3.8 Derivative Arrays For higher index problems the solution depends on derivatives not only of f but also of E and A. In addition to the original DAE we therefore use derivatives of it yielding augmented DAEs M l (t)ż l = N l (t)z l + g l (t), l N 0 for z l = (x, ẋ,..., x (l+1) ) with the so-called derivative arrays N l = E Ė A E M l =....... E (l) la (l 1) lė A E A 0 0 A 0 0... A (l) 0 0., Theorem Let the sufficiently smooth (E, A) and (Ẽ, Ã) be globally equivalent with well-defined strangeness index µ. Then the corresponding matrix pairs (M l (t), N l (t)) and ( M l (t), Ñl(t)) are locally equivalent for every t I.

3.9 Differentiation Index The idea leading to the differentiation index is based on the question whether it is possible to extract an ODE for x from some augmented DAE. Obviously we must restrict ourselves to the case m = n. Definition A block matrix M C kn,ln of (n, n)-blocks is called 1-full if there is a nonsingular R C kn,kn such that RM = [ ] In 0. 0 H Definition Let (E, A) be sufficiently smooth with derivative arrays (M l, N l ). The smallest number ν N 0, if it exists, for which M ν is pointwise 1-full and has constant rank, is called the differentiation index of (E, A).

3.10 Differentiation Index (cont.) Theorem The differentiation index is invariant under global equivalence transformations. Remark If the differentiation index ν is well-defined, then there is a pointwise nonsingular smooth R C(I, C (ν+1)n,(ν+1)n ) with RM ν = [ ] In 0 0 H implying the so-called underlying ODE ẋ = [ I n 0 ]R(t)N ν (t)[ I n 0 ] H x + [ I n 0 ]R(t)g ν (t). Theorem Let E, A C(I, C n,n ) be sufficiently smooth and suppose 1. for every sufficiently smooth f C(I, C n ) the corresponding DAE is solvable, 2. the solution is unique for every consistent initial condition, and 3. depends smoothly on the data. Then the differentiation index of (E, A) is well-defined.

3.11 Hypothesis Hypothesis Let (E, A) be sufficiently smooth. There are integers µ, a and d with a + d = n such that 1. rank M µ (t) = (µ + 1)n a for all t I = Ẑ 2 smooth, max. rank a, orth. columns, Ẑ H 2 M µ = 0 on I, 2. rank ẐH 2 (t)n µ(t)[ I n 0 0 ] H = a for all t I = ˆT 2 smooth, max. rank d, orth. columns, Ẑ T 2 N µ(t)[ I n 0 0 ] H ˆT2 = 0 on I, 3. rank E(t) ˆT 2 (t) = d for all t I = Ẑ 1 smooth, max. rank d, orth. columns, Ẑ T 1 E ˆT 2 nonsingular on I. Theorem The above hypothesis is invariant under global equivalence transformations.

3.12 Properties Theorem Let (E, A) be sufficiently smooth with well-defined differentiation index ν. Then (E, A) satisfies the above hypothesis with d = n a and µ = { 0 for ν = 0, ν 1 otherwise, a = { 0 for ν = 0, corank M ν 1 otherwise. Remark Let (E, A) satisfy the above hypothesis. Setting Ê 1 = ẐH 1 E, Â 1 = ẐH 1 A, Â 2 = ẐH 2 N µ[ I n 0 0 ] H, ˆf 1 = ẐH 1 f, ˆf 2 = ẐH 2 g µ, the original DAE implies the reduced DAE Ê 1 (t)ẋ = Â1(t)x + ˆf 1 (t), 0 = Â2(t)x + ˆf 2 (t), (d differential equations) (a algebraic equations) which is strangeness-free, i. e., has vanishing strangeness index. Theorem Let (E, A) satisfy the above hypothesis. Then the reduced DAE has the same solutions as the original DAE.

3.13 Properties (cont.) Remark The reduced DAE has differentiation index at most one due to the fact that there is a splitting x = (x 1, x 2 ) such that 0 = Â21(t)x 1 + Â22(t)x 2 + ˆf 2 (t) can be solved for x 2, i. e., x 2 = Â22(t) 1 (Â21(t)x 1 + ˆf 2 (t)), and the remaining Ê 11 (t)ẋ 1 + Ê12(t)ẋ 2 = Â11(t)x 1 + Â12(t)x 2 + ˆf 1 (t) with eliminated x 2, ẋ 2 can be solved for ẋ 1.

3.14 Additional Topics Further research included (will include) DAE operators, generalized inverses, weak (distributional) solutions, global canonical form for regular singular problems, control problems, regularization by feedback.

4.1 Nonlinear DAEs In this section, we consider F (t, x, ẋ) = 0 with F C(I D x Dẋ, R n ), D x, Dẋ R n open, sufficiently smooth. Generalizing the linear case, we aim for a hypothesis that is invariant under (parametrized) diffeomorphisms in the domain and image of F. The corresponding augmented DAEs are given by F l (t, x, ẋ,..., x (l+1) ) = 0 with F l (t, x, ẋ,..., x (l+1) ) = F (t, x, ẋ) F (t, x, ẋ) d dt. ( d dt )l F (t, x, ẋ).

4.2 Hypothesis Hypothesis Let F be sufficiently smooth. There are integers µ, a and d with a + d = n such that and L µ = F 1 µ ({0}) 1. rank F µ;ẋ,...,x (µ+1) = (µ + 1)n a on L µ = Ẑ 2 smooth, max. rank a, orth. columns, Ẑ T 2 F µ;ẋ,...,x (µ+1) = 0 on L µ, 2. rank ẐT 2 F µ;x = a on L µ = ˆT 2 smooth, max. rank d, orth. columns, Ẑ T 2 F µ;x ˆT 2 = 0 on L µ, 3. rank Fẋ ˆT2 = d on L µ = Ẑ 1 smooth, max. rank d, orth. columns, Ẑ T 1 F ẋ ˆT 2 nonsingular on L µ.

4.3 Invariances Theorem Let F satisfy the above hypothesis with µ, a and d, and let F be given by F (t, x, x) = F (t, Q(t, x), Q t (t, x) + Q x (t, x) x) with sufficiently smooth Q C(I R n, R n ), where Q(t, ) is bijective for every t I and Q x (t, x) is nonsingular for every (t, x) I R n. Then F satisfies the above hypothesis with µ, a and d. Theorem Let F satisfy the above hypothesis with µ, a and d, and let F be given by F (t, x, ẋ) = P (t, x, ẋ, F (t, x, ẋ)) with sufficiently smooth P C(I R n R n R n, R n ), where P (t, x, ẋ, ) is bijective with P (t, x, ẋ, 0) = 0 and P w (t, x, ẋ, 0) nonsingular for every (t, x, ẋ) I R n R n. Then F satisfies the above hypothesis with µ, a and d.

4.4 Reduced Problem Let (t 0, x 0, y 0 ) L µ and T 2,0 = ˆT 2 (t 0, x 0, y 0 ), Z 1,0 = Ẑ1(t 0, x 0, y 0 ), Z 2,0 = Ẑ2(t 0, x 0, y 0 ). Let [Z 2,0, Z 2,0] be orthogonal and T 1,0 basis of kernel Z T 2,0 F µ;ẋ,...,x (µ+1)(t 0, x 0, y 0 ). The nonlinear equation F µ (t, x, y) Z 2,0 α = 0, T T 1,0 (y y 0) = 0 for (t, x, y, α) is locally solvable for (y, α). defines a function ˆF 2 according to In particular, it α = ˆF (t, x). Furthermore, set ˆF 1 (t, x, ẋ) = Z1,0 T F (t, x, ẋ). The DAE ˆF 1 (t, x, ẋ) = 0, ˆF 2 (t, x) = 0 (d differential equations) (a algebraic equations) is called the associated reduced DAE.

4.5 Solvability Remark The reduced DAE has differentiation index at most one in the sense that there is a splitting x = (x 1, x 2 ) such that ˆF 2 (t, x 1, x 2 ) = 0 can be solved for x 2, say by x 2 = G 2 (t, x 1 ), and the remaining ˆF 1 (t, x 1, G 2 (t, x 1 ), ẋ 1, G 2;t (t, x 1 ) + G 2;x1 (t, x 1 )ẋ 1 ) = 0 can be solved for ẋ 1, say by ẋ 1 = G 1 (t, x 1 ). Remark Together with x 1 (t 0 ) = x 0,1, x 0 = (x 0,1, x 0,2 ) the reduced DAE yields a (locally defined) unique solution. This solution can be extended until the boundary of L µ is reached.

4.6 Solvability (cont.) Remark An initial condition x(t 0 ) = x 0 is consistent with the reduced DAE if there is a y 0 with F µ (t 0, x 0, y 0 ) = 0. Theorem Let F satisfy the above hypothesis with µ, a and d. Then every solution of the original DAE also solves the reduced DAE. If F furthermore satisfies the hypothesis with µ + 1, a and d, then every solution of the reduced DAE also solves the original DAE. Remark In the latter case, for a given solution x C 1 (I, R n ) there exists a function P C(I, R (µ+1)n ) that coincides with ẋ in the first n components and satisfies F µ (t, x(t), P (t)) = 0 for all t I.

4.7 Example Let F (t, x, ẋ) = 0 be given by ẋ 1 = x 4, ẋ 2 = x 5, ẋ 3 = x 6, ẋ 4 = 2x 1 x 7, ẋ 5 = 2x 2 x 7, ẋ 6 = 1 x 7, 0 = x 3 x 2 1 x2 2, describing the motion of a mass point on a parabola under the influence of gravity. It satisfies the above hypothesis with µ = 2, d = 4, and a = 3. The equation F µ = 0 implies the constraints 0 = x 3 x 2 1 x2 2, 0 = x 6 2x 1 x 4 2x 2 x 5, 0 = 1 x 7 2x 2 4 4x 2 1x 7 2x 2 5 4x 2 2x 7. A possible reduced DAE reads ẋ 1 = x 4, ẋ 2 = x 5, 0 = x 6 2x 1 x 4 2x 2 x 5, ẋ 4 = 2x 1 x 7, ẋ 5 = 2x 2 x 7, 0 = 1 x 7 2x 2 4 4x 2 1x 7 2x 2 5 4x 2 2x 7, 0 = x 3 x 2 1 x2 2.

4.8 Additional Topics Further research included (will include) over- and underdetermined systems, control problems, regularization by (piecewise linear) feedback, structured problems, conservation laws.

5.1 Discretizations Direct discretization of higher index problems (i. e., problems with µ 1) that have no special structure usually does not lead to convergent methods. A DAE that satisfies one of the above hypotheses can (at least theoretically) be transformed to a DAE with µ = 0 having the same solutions. Definition The k-step BDF discretization of F (t, x, ẋ) = 0 has the form F (t i, x i, D h x i ) = 0, where t i = t 0 + ih with given stepsize h > 0 and D h x i = 1 k α l x i l. h l=0 The coefficients α i are fixed by the condition ẋ(t i ) 1 k α l x(t i l ) Ch k h l=0 for sufficiently smooth x, C independent of h. Theorem Given a DAE with µ = 0 and a consistent initial condition x(t 0 ) = x 0 with corresponding sufficiently smooth solution x C 1 (I, R n ). Then the k-step BDF method with 1 k 6 is convergent of order k, i. e., for x N with N = (t t 0 )/h, t I fixed, we have x(t) x N Ch k with C independent of h.

5.2 Discretizations (cont.) Remark Runge-Kutta methods and their derivates in general require the reformulation of F (t, x, ẋ) = 0 with µ = 0 into ẋ = y, 0 = F (t, x, y), which then has µ = 1 but special structure. Remark There is a class of partitioned Runge-Kutta methods which use the special structure of the reduced DAEs, see the treatment of boundary value problems. In the numerical treatment of initial value problems we shall concentrate on the BDF discretization. Idea Discretize the reduced DAE.

6.1 Linear Initial Value Problems Let E(t)ẋ = A(t)x + f(t) (with real data) satisfy the corresponding hypothesis. The reduced DAE reads Ê 1 (t)ẋ = Â1(t)x + ˆf 1 (t), 0 = Â2(t)x + ˆf 2 (t) (d differential equations) (a algebraic equations) with Ê 1 = ẐT 1 E, Â 1 = ẐT 1 A, Â 2 = ẐT 2 N µ[ I n 0 0 ] T, ˆf 1 = ẐT 1 f, ˆf 2 = ẐT 2 g µ.

6.2 Linear Initial Value Problems (cont.) Consistency of initial values Let x 0 R n be an estimate for an initial value at t 0 I. Then x 0 R n fixed by i. e. x 0 x 0 2 = min! s. t. 0 = Â2(t 0 )x 0 + ˆf 2 (t 0 ), x 0 = (I Â2(t 0 ) + A 2 (t 0 )) x 0 Â2(t 0 ) + ˆf2 (t 0 ) is consistent at t 0. Integration step BDF-discretization of the reduced DAE yields Ê 1 (t i )D h x i = Â1(t i )x i + ˆf 1 (t i ), 0 = Â2(t i )x i + ˆf 2 (t i ), which has to be solved for x i. The corresponding coefficient matrix reads [ α0 h Ê 1 (t i ) Â1(t ] i ) Â2(t i ) and is nonsingular for sufficiently small h. Remark We cannot produce (globally) smooth functions Ẑ1 and Ẑ2 by numerical techniques. The methods presented here, however, are not influenced by possibly non-smooth choices of orthogonal bases.

7.1 Nonlinear Initial Value Problems Let F (t, x, ẋ) = 0 satisfy the corresponding hypothesis. The (locally defined) reduced DAE reads ˆF 1 (t, x, ẋ) = 0, ˆF 2 (t, x) = 0, (d differential equations) (a algebraic equations) with ˆF 1 (t, x, ẋ) = Z T 1,0F (t, x, ẋ) and F µ (t, x, y) Z 2,0 ˆF2 (t, x) = 0.

7.2 Nonlinear Initial Value Problems (cont.) Consistency of initial values Let (t 0, x 0, ỹ 0 ) R (µ+2)n+1 be an estimate for a point in L µ. Solve F µ (t 0, x 0, y 0 ) = 0 for (x 0, y 0 ), say by the Gauß-Newton method, starting with ( x 0, ỹ 0 ). Integration step BDF-discretization of the reduced DAE yields Z T 1,0F (t i, x i, D h x i ) = 0, F µ (t i, x i, y i ) = 0, which has to be solved for (x i, y i ), say by the Gauß-Newton method. Remark In both cases it is known that the Jacobian at the solution has full row rank (provided h is sufficiently small in the second case) implying quadratic convergence of the Gauß-Newton method. Moreover, in the second case the part x i of the solution is uniquely determined.

8.1 Boundary Value Problems In this section, we consider F (t, x, ẋ) = 0, r(x(t), x(t)) = 0, where F satisfies the corresponding hypothesis. Let x C 1 ([t, t], R n ) be a solution of the BVP, i. e., let F (t, x (t), ẋ (t)) = 0 on [t, t], F µ (t, x (t), P (t)) = 0 on [t, t], r(x (t), x (t)) = 0, with some P : I R (µ+1)n.

8.2 Regularity We can globally define a reduced BVP ˆF 1 (t, x, ẋ) = 0, ˆF2 (t, x) = 0, r(x(t), x(t)) = 0. Linearization around x C 1 ([t, t], R n ) yields Ê 1 (t)ẋ = Â1(t)x, 0 = Â2(t)x, Ĉx(t) + ˆDx(t) = 0, where Ê 1 (t) = ˆF 1;ẋ (t, x (t), ẋ (t)), Â 1 (t) = ˆF 1;x (t, x (t), ẋ (t)), Â 2 (t) = ˆF 2;x (t, x (t)), Ĉ = r xa (x (t), x (t)), ˆD = r xb (x (t), x (t)). Definition The solution x C 1 ([t, t], R n ) is called regular if the above linearized BVP only admits the trivial solution.

8.3 Multiple Shooting Given a grid t = t 0 < t 1 < < t N 1 < t N = t, N N together with (sufficient good) estimates (x i, y i ) R (µ+2)n, i = 0,..., N. The nonlinear systems F µ (t i, ˆx i, ŷ i ) = 0, T T 2,i (ˆx i x i ) = 0, T T 1,i(ŷ i y i ) = 0 (d equations) (a equations) for (x i, y i, ˆx i, ŷ i ) with appropriately chosen T 1,i, T 2,i define local projections S i, i = 0,..., N, with (t i, ˆx i, ŷ i ) L µ, (ˆx i, ŷ i ) = S i (x i, y i ). Moreover, because of the unique solvability of initial value problems F (t, x, ẋ) = 0, x(t i ) = ˆx i with ˆx i sufficiently close to x (t i ) we have transfer functions Φ i : (ˆx i, ŷ i ) x(t i+1 ).

8.4 Multiple Shooting (cont.) The multiple shooting system then reads F µ (t i, x i, y i ) = 0, T T 2,i+1 (x i+1 Φ(S i (x i, y i ))) = 0, r(x 0, x N ) = 0. i = 0,..., N, i = 0,..., N 1, This system can be solved by a Gauß-Newton-like method such that 1. we only must integrate d trajectories for the approximation of the Jacobian, 2. the only arising global linear system is cyclic with size (N + 1)d. If the solution is regular, the convergence rate is superlinear.

8.5 Collocation Given knots 0 < ϱ 1 < < ϱ k < 1, 0 = σ 0 < < σ k = 1, k N, and collocation points t ij = t i + h i ϱ j, s ij = t i + h i σ j, j = 1,..., k, j = 0,..., k, i = 0,..., N 1, the collocation system reads Z T 1,ijF (t ij, x π (t ij ), ẋ π (t ij )) = 0, j = 1,..., k, i = 0,..., N 1, F µ (s ij, x π (s ij ), y ij ) = 0, j = 1,..., k, i = 0,..., N 1, j = 0, i = 0, r(x π (t), x π (t)) = 0 for (x π, y ij ) P k+1,π C 0 ([t, t], R n ) R (kn+1)(µ+1)n. This system can be solved by a Gauß-Newton-like method such that the main part of an iteration consists of the solution of a linear BVP where no quantities y ij are needed. If the solution is regular, the convergence rate is superlinear.

8.6 Collocation (cont.) Existence and uniqueness of collocation solutions Let x be a regular solution of the BVP. Then for sufficiently small h = min i=0,...,n 1 (t i+1 t i ) the collocation system has a locally unique solution x π with x x π C 0 Chk. Superconvergence For Gauß knots ϱ j and Lobatto knots σ j we even have and x x π C 0 Ch k+1 max i=0,...,n x (t i ) x π (t i) Ch 2k.

9.1 Literature/Software Related literature [1] S. L. Campbell: A general form for solvable linear time varying singular systems of differential equations. SIAM J. Sci. Stat. Comput. 6, 334 348 (1985) [2] S. L. Campbell, E. Griepentrog: Solvability of general differential algebraic equations. SIAM J. Sci. Comput. 16, 257 270 (1995) [3] P. Kunkel, V. Mehrmann: Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comput. Appl. Math. 56, 225 251 (1994) [4] P. Kunkel, V. Mehrmann: A new class of discretization methods for the solution of linear differential-algebraic equations. SIAM J. Numer. Anal. 33, 1941 1961 (1996) [5] P. Kunkel, V. Mehrmann: Local and global invariants of linear differential algebraic equations and their relation. Electr. Trans. Numer. Anal. 4, 138 157 (1996) [6] P. Kunkel, V. Mehrmann: Regular solutions of nonlinear differential-algebraic equations and their numerical determination. Numer. Math. 79, 581 600 (1998) [7] P. Kunkel, V. Mehrmann: Analysis of over- and underdetermined nonlinear differential-algebraic systems with application to nonlinear control problems. Math. Contr. Sign. Syst. 14, 233 256 (2001)

9.2 Literature/Software (cont.) [8] P. Kunkel, V. Mehrmann, R. Stöver: Multiple shooting for unstructured nonlinear differential-algebraic equations of arbitrary index. Institut für Mathematik, TU Berlin, Techn. Report 751-02 (2002) [9] P. Kunkel, V. Mehrmann, R. Stöver: Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index. Zentrum für Technomathematik, Uni Bremen, Techn. Report 02-12 (2002) Software [1] P. Kunkel, V. Mehrmann, W. Rath, J. Weickert: GELDA: A software package for the solution of GEneral Linear Differential Algebraic equations. Fachbereich Mathematik, TU Chemnitz, Techn. Report SPC 95 8 (1995) [2] P. Kunkel, V. Mehrmann, I. Seufer: GENDA: A software package for the solution of GEneral Nonlinear Differential-Algebraic equations, Institut für Mathematik, TU Berlin, Techn. Report 730-02 (2002)