On the zero-stability of variable stepsize multistep methods: the spectralradius approach

In this paper we illustrate a novel approach for studying the asymptotic be haviour of the solutions of linear difference equations with variable coeff icients. In particular, we deal with the zero-stability of the 3-step BDF-m ethod on grids with variable stepsize for the numerical solution of IVPs fo r ODEs. Our approach is based on the theory of the spectral radios of a fam ily of matrices and yields almost optimal results, which give a slight impr ovement to the best results already known from the literature. The success got on the chosen example suggests that our approach has a good potential f or more general and harder stability analyses of numerical methods.