Integration of stiff mechanical systems by Runge-Kutta methods

Abstract

The numerical integration of stiff mechanical systems is studied in which a strong potential forces the motion to remain close to a manifold. The equations of motion are written as a singular singular perturbation problem with a small stiffness parameter ɛ. Smooth solutions of such systems are characterized, in distinction to highly oscillatory general solutions. Implicit Runge-Kutta methods using step sizes larger than ɛ are shown to approximate smooth solutions, and precise error estimates are derived. As ɛ → 0, Runge-Kutta solutions of the stiff system converge to Runge-Kutta solutions of the associated constrained system formulated as a differential-algebraic equation of index 3. Standard software for stiff initial-value problems does not work satisfactorily on the stiff systems considered here. The reasons for this failure are explained, and remedies are proposed.