How AD can help solve differential-algebraic equations.

*(English)*Zbl 1409.65050This paper is concerned with the numerical solution to differential-algebraic equations (DAEs) in their most general form
\[
f_i(t, \text{the } x_j(t) \text{ and derivatives of them}) = 0, \;i=1,2,\dots,n,
\]
where \(x_j(t)\), \(i=1,2,\dots,n\), are unknown variables.

A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of the so-called index reduction or regularization. The aim of this step is to reformulate a given DAE to a better form for solving it merically. This is commonly done with the help of a computer algebra system. This paper explores two significant cases where it can be done efficiently and simple by pure algorithmic differentiation. The first is the dummy derivatives (DD) method and the second is the Lagrangian formulation of mechanical systems. Both cases are described algorithmically and are illustrated by numerical simulations. The main contribution of this research is to show that: (1) for the DD method, one can combine index and order reduction in one simple framework, so one does not need to reformulate a given system in a first-order form; (2) for Lagrangian formulation of mechanical systems, one can combine all phases by automatic differentiation instead of symbolic algebra.

A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of the so-called index reduction or regularization. The aim of this step is to reformulate a given DAE to a better form for solving it merically. This is commonly done with the help of a computer algebra system. This paper explores two significant cases where it can be done efficiently and simple by pure algorithmic differentiation. The first is the dummy derivatives (DD) method and the second is the Lagrangian formulation of mechanical systems. Both cases are described algorithmically and are illustrated by numerical simulations. The main contribution of this research is to show that: (1) for the DD method, one can combine index and order reduction in one simple framework, so one does not need to reformulate a given system in a first-order form; (2) for Lagrangian formulation of mechanical systems, one can combine all phases by automatic differentiation instead of symbolic algebra.

Reviewer: Phi Ha (Hanoi)

##### MSC:

65L80 | Numerical methods for differential-algebraic equations |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

68W30 | Symbolic computation and algebraic computation |

##### Keywords:

algorithmic differentiation; differential-algebraic equations; dummy derivatives; Lagrangians##### References:

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