Stability of Computational Methods for Constrained Dynamics Systems

Abstract

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. \nConsidering 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 well-conditioning 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.