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<p>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. Furthermore, 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 with window-wise overdetermined least-squares collocation. This results in the first fully numerical time-stepping procedures for general linear higher-index initial-value problems.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月4日

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  • Michael Hanke,
  • Roswitha März

摘要

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. Furthermore, 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 with window-wise overdetermined least-squares collocation. This results in the first fully numerical time-stepping procedures for general linear higher-index initial-value problems.