Conservation of polynomial type and the stability problem for Linear Multistep Methods

The study of the linear stability for linear multistep methods leads to consider the location of the zeros 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. The choice to fix all the conditions at the beginning of the interval of integration leads, however, to severe restrictions on the order of the methods (Dahlquist barriers). To overcome this drawback one can use a linear multistep method coupled with boundary conditions (BVMs). In the BVMs approach the classical stability conditions are generalized. In this talk, a rigorous analysis of the linear stability for some classes of BVMs is presented. The necessary information on the coefficients of the methods are obtained by an extensive use of properties of the Pascal matrix.