Stability Issues in Multivalue Numerical Methods for Ordinary Differential Equations

We describe the derivation of highly stable general linear methods for the numerical solution of initial value problems for systems of ordinary differential equations. In particular we describe the construction of explicit Nordsiek methods and implicit two step Runge Kutta methods with stability properties determined by quadratic stability functions. We aim for methods which have wide stability regions in the explicit case and which are A- and L-stable in the implicit one case. We moreover describe the construction of algebraically stable and G-stable two step Runge Kutta methods. Examples of methods are then provided