Stability analysis of linear multistep methods via polynomial type variation

The linear stability analysis for linear multistep methods leads to study the location of the roots of the associated characteristic polynomial with respect to the unit circle in the complex plane. It is known that if the discrete problem is an initial value one, it is sufficient to determine when all the roots are inside the unit disk. This requirement is, however, conflicting with the order conditions, as established by the Dahlquist barrier. The conflict disappears if one uses a linear multistep method coupled with boundary conditions (BVMs). In this paper, a rigorous analysis of the linear stability for some classes of BVMs is presented. The study is carried out by using the notion of type of a polynomial.