The Z-transform and linear multistep stability

摘要

This paper makes systematic use of control-theoretic methods such as the Z-transform, small-gain theorems and frequency-domain stability criteria in the analysis of the stability behaviour of linear multistep methods. Some of the results in Nevanlinna's work are recovered and a number of new boundedness and asymptotic properties of solutions of numerical schemes are obtained. In particular, we give a careful and detailed analysis of the nonlinear stability properties of strictly zero-stable methods.