By René Lamour, Roswitha März, Caren Tischendorf
Pt. I. Projector established procedure -- pt. II. Index-1 DAEs : research and numerical therapy -- pt. III. Computational facets -- pt. IV. complicated subject matters
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Extra resources for Differential-algebraic Equations A Projector Based Analysis
We show by induction that the coupling coefficients disappear stepwise with an appropriate choice of admissible projectors. Assume Q0 , . . 2 Projector based decoupling 31 Hk+1 = 0, . . 53) or, equivalently, Qk+1∗ Πμ −1 = 0, . . , Qμ −1∗ Πμ −1 = 0, for a certain k, 0 ≤ k ≤ μ − 2. We build a new sequence by letting Q¯ i := Qi for i = 0, . . , k − 1 (if k ≥ 1) and Q¯ k := Qk∗ . Thus, Qk P¯k = −Q¯ k Pk and the projectors Q¯ 0 , . . , Q¯ k are admissible. The resulting two sequences are related by G¯ i = Gi Zi , i = 0, .
23) and consider the last term in more detail. 24) and therefore, Y j+1 − I = (Y j+1 − I)Π j−1 , j = 1, . . , i. 25) im (Y j − I) ⊆ ker (Y j+1 − I), j = 1, . . , i. 2 Projector based decoupling 21 j Zj −I = ∑ (Yl − I), j = 1, . . , i, l=1 to be satisfied. Consequently, im (I − Zi ) ⊆ N0 + · · · + Ni−1 = N¯ 0 + · · · + N¯ i−1 ⊆ ker Q¯ i . 23), we get G¯ i+1 = Gi Zi + B¯ i Q¯ i Zi , which leads to G¯ i+1 Zi−1 = Gi + B¯ i Q¯ i = Gi + Bi Qi + (B¯ i Q¯ i − Bi Qi ). 22) to find i−1 G¯ i+1 Zi−1 = Gi+1 + Bi (Q¯ i − Qi ) + Gi ∑ Ql Ail Q¯ i .
1, v0 (t) = Q0 v0 (t), which means that the components vi (t), i = 0, . . , μ − 1, belong to the desired subspaces. 46) is the explicit ODE u (t) + Wu(t) = Ld q(t). ) denote the solution fixed by the initial condition uq (t0 ) = 0. We have uq (t) = Πμ −1 uq (t) because of W = Πμ −1 W, Ld = Πμ −1 Ld . ) not belong to im Πμ −1 as we want it to. With the initial condition u(t0 ) = u0 ∈ im Πμ −1 the solution can be kept in the desired subspace, which means that u(t) ∈ im Πμ −1 for all t ∈ I.