Convergence results for multistep Runge - Kuttamethods

Recently Ch. Lubich proved convergence results for Runge-Kutta methods applied to stii mechanical systems. The present paper discusses the new ideas necessary to extend these results to general linear methods, in particular BDF and multistep Runge-Kutta methods. Stii mechanical systems arise in the modelling of mechanical systems containing strong springs and (or) elastic joints. A typical example is the pendulum composed of a mass suspended by a massless stii spring with Hooke's constant 1== 2 , 0 < << 1. Assuming unit mass, length and gravity, the equations of motion are (0:1) For initial values having moderate energy, the strong potential due to the spring, forces the motion of system (0.1) to remain close to that of a mathematical pendulum. This \associated constrained system" is obtained by replacing the stii spring by a rigid bar. Implicit Runge-Kutta and BDF formulas are popular methods for integrating such equations. Multistep collocation methods (for example the multistep Radau method, see 3], 8], 9]) or more generally multistep Runge-Kutta methods are an attractive alternative for solving stii problems. For one step Runge-Kutta methods convergence results are known 6]. The aim of this paper is to generalize Numerische Mathematik Electronic Edition { page numbers may diier from the printed version page 495 of Numer.