Stability of Computational Methods for Constrained Dynamics Systems

Many methods have been proposed for numerically integrating the differential-algebraic systems arising from the Euler–Lagrange equations for constrained motion. These are based on various problem formulations and discretizations. We offer a critical evaluation of these methods from the standpoint of stability. Considering a linear model, we first give conditions under which the differential-algebraic problem is well conditioned. This involves the concept of an essential underlying ODE. We review a variety of reformulations which have been proposed in the literature and show that most of them preserve the stability of the original problem. Then we consider stiff and nonstiff discretizations of such reformulated models. In some cases, the same implicit discretization may behave in a very different way when applied to different problem formulations, acting as a stiff integrator on some formulations and as a nonstiff integrator on others. We present the approach of projected invariants as a method for yielding problem reformulations which are desirable in this sense.