Regularity properties of general linear methods for initial value problemsof ordinary differential equations

The main purpose of the present paper is to examine the asymptotic states o f general linear methods for initial value problems of ordinary differentia l equations, and to extend the existing relevant results of Runge-Kutta met hods, linear multistep methods by Humphries (1993) and Stuart and Humphries (1993). In particular, the existence of spurious steady solutions and peri od-2 solutions in the time step is studied, and the concepts of (strong and weak) R-[1]-regularity and (weak) R-[2]-regularity of general linear metho ds are introduced and studied. Some sufficient conditions for (strong and w eak) R-[1]-regularity and (weak) R-[2]-regularity of such methods applied t o initial value problems of ordinary differential equations with a globally Lipschitz condition or contractive or monotone condition are given. (C) 20 00 IMACS. Published by Elsevier Science B.V. All rights reserved.