Variable stepsize variable formula methods based on predictor-corrector schemes

Abstract

The use of fairly general predictor-corrector (PC) schemes of linear multistep (LM) formulae in the numerical solution of systems of ordinary differential equations (ODE's) is considered. It is assumed that both the stepsize and the PC scheme can be varied during the computational process. The numerical methods obtained under these two assumptions are called predictor-corrector linear multistep variable stepsize variable formula methods (PC LM VSVFM's). The consistency, zero-stability and convergence properties of the PC LM VSVFM's are studied. Several results concerning these fundamental properties of the numerical methods are established. It should be emphasized that all theorems are formulated and proved under very mild assumptions on the stepsize selection strategy. The extension of the results for the so-called one-leg methods is briefly discussed. The use of PC LM VSVFM's leads to a very efficient treatment of many mathematical models describing different phenomena in science and engineering. Such methods have successfully been used in the numerical solution of systems of ODE's arising after the space discretization of some air pollution models.