Least-squares collocation for higher-index linear differential-algebraic equations: Estimating the instability threshold

Abstract

In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than the absolute dimension of the described process, and the actual lower-dimensional configuration space of the system is deeply implicit. For linear higher-index differential-algebraic equations, we develop an appropriate numerical method based on properties of canonical subspaces and on the so-called geometric reduction. Taking into account the fact that higher-index differential-algebraic equations lead to ill-posed problems in naturally given norms, we modify this approach to serve as transfer conditions from one time-window to the next in a time stepping procedure and combine it with window-wise overdetermined least-squares collocation to construct the first fully numerical solvers for higher-index initial-value problems.

Least-squares collocation for linear higher-index differential–algebraic equations

“…The theoretical justification appears to be quite challenging. So far, only sufficient convergence conditions are obtained for linear problems [7,6,8]. In the present paper we provide a first proof for nonlinear problems.…”

“…Following the ideas of [8,7] concerning linear problems, we investigate also the nonlinear problem (1),(2) described in Section 1 as operator equation F x = 0 in a Hilbert space setting, which is most comfortable for treating ill-posed problems. Besides standard function spaces such as L 2 , H 1 , C, etc., equipped with usual inner products and norms, we use the space [14,Lemma 6.9].…”

“…At this place we emphasize again, that higher-index DAEs lead to ill-posed problems. In the context here this means that im F ′ (x * ) and im F ′ (x * ) are nonclosed subsets in L 2 and L 2 × R l , respectively, see [7,Theorem 2.4], and the inverse F ′ (x * ) −1 is unbounded.…”

“…Proof. The existence of c γ > 0 as well as the inequality (33) are ensured by [7,Theorem 4.1] concerning the instability threshold. c γ may depend on the ratio r. The injectivity of F ′ (x * ) immediately implies ker F ′ (x * )U π = ker U π .…”

“…A thorough discussion as well as numerical experiments of the version linearized in the solution x * is given in [7]. In order to stimulate discussions of the least-squares method for nonlinear problems, also results for the original nonlinear version have been provided in this reference.…”