An extension of the theory of Dorsselaer and Spijker

Abstract

We are concerned with the numerical solution of ini- tial value problems (IVPs) for systems of sti¤ ordi- nary di¤erential equations (ODEs). Our emphasis is on implicit linear multistep methods (LMM), partic-
ularly the backward di¤erentiation formulae (BDF). In our quest for saving computations in the time in- tegration of the initial value problems we adopt and extend a theory of Dorsselaer and Spijker The the- ory of Spijker introduces the notion of classifying sti¤ problems into highly nonlinear and mildly nonlinear. In this paper we derive expressions that relate the convergence rate factor for the highly nonlinear and mildly nonlinear problems to variations in the step-size and the Jacobian matrix. The expressions reveal that when the iteration matrix uses the current step- size, hn = h0 then a poor convergence rate is not solely due to the change in the Jacobian (absolute or relative). From these relations it is evident that a vastly changing solution can also lead to poor conver- gence rate. Our expressions further show that for the mildly nonlinear problems it is the absolute change of the Jacobian that can contribute to the convergence rate while for strongly nonlinear problems the relative change in the Jacobian can play a signicant part in the convergence rate.

Key words: Reynolds equation, random roughness, squeeze film, pressure, load carrying capacity