On the computation of accurate initial conditions for linear higher-index differential-algebraic equations and its application in initial value solvers

TL;DR

This paper introduces a numerical method for accurately computing initial conditions in linear higher-index differential-algebraic equations, enabling the development of fully numerical initial-value solvers for these complex systems.

Contribution

It develops a novel approach based on canonical subspaces and geometric reduction, tailored for higher-index DAE initial conditions, and integrates it into a time-stepping solver.

Findings

First fully numerical solver for higher-index initial-value problems.

Effective transfer conditions for time-stepping in DAE systems.

Improved accuracy in initial condition computation for complex DAEs.

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.