A NORMAL FORM FOR MULTISTEP COMPANION MATRICES

Abstract

Linear multistep methods, in particular Backward Differentiation Formulas (BDF), are widely used for the numerical integration of differential equations and in particular for stiff systems. Although the classical convergence theory for linear multistep methods is quite well-developed, there is no comprehensive stiff convergence theory. Special classes of stiff problems have been covered so far, see e.g. Hairer, Wanner11 for the analysis of stiff linear systems or Lubich13 for the analysis of problems in standard singular perturbation form. We present a theoretical approach for the analysis of stiff linear systems which will be extended to nonlinear problems in forthcoming papers.