Ill-conditioned matrices and the integration of stiff ODEs

Abstract

Popular methods for the integration of a stiff initial-value problem for a system of ordinary differential equations (ODEs) require the solution of systems of linear equations. It is shown that the matrices are very ill-conditioned. Implicit linear multistep methods (LMMs) can be evaluated accurately by iteration, even when the matrices are very ill-conditioned. Although semi-implicit methods do not involve iteration, it is observed that codes based on these methods cope with ill-conditioned matrices about as well as codes based on LMMs. An explanation is provided for this fact.