Properties and implementation of ρ-Adams methods based on mixed-type interpolation

We investigate the properties of the coefficients of modified r-Adams methods for the integration of ODEs. The derivation of these methods is, in contrast with the classical Adams methods, not based on a polynomial interpolation theory, but rather starts from a mixed interpolation theory in which a parameter κ is involved. It will be shown that the coefficients of the modified methods possess properties which make these methods very attractive. Further, we will discuss the role of so-called over-implicit modified r-Adams schemes in the construction of more general linear multistep methods. Our second goal is to show that the modified Adams-Bashforth/Adams-Moulton methods are very well suited to be implemented as a predictor-corrector pair. In particular, we will discuss the choice of the interpolation parameter when such a method is applied to general systems of equations. Numerical tests are performed to support the theory.