How AD Can Help Solve Differential-Algebraic Equations
For researchers and engineers solving differential-algebraic equations, this work provides an efficient alternative to computer algebra for index reduction, though it is incremental as it applies existing AD techniques to known problems.
The paper shows that algorithmic differentiation can efficiently replace computer algebra systems for index reduction in differential-algebraic equations, demonstrated through the Dummy Derivatives method and a Lagrangian-based mechanical system solver, with successful long-time integration of a solar system model.
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 so called index reduction or regularisation, to prepare them for numerical solution. This is often done with the help of a computer algebra system. We show in two significant cases that it can be done efficiently by pure algorithmic differentiation. The first is the Dummy Derivatives method, here we give a mainly theoretical description, with tutorial examples. The second is the solution of a mechanical system directly from its Lagrangian formulation. Here we outline the theory and show several non-trivial examples of using the "Lagrangian facility" of the Nedialkov-Pryce initial-value solver DAETS, namely: a spring-mass-multipendulum system, a prescribed-trajectory control problem, and long-time integration of a model of the outer planets of the solar system, taken from the DETEST testing package for ODE solvers.