Algebraically Stable and Implementable Runge-Kutta Methods of High Order

There are three interesting properties of methods for (stiff) ordinary differential equations: order, stability and efficiency of implementation. This paper constructs Runge-Kutta methods of orders 5 and 6 which possess these properties to a high extent. We further classify all algebraically stable methods of an arbitrary order and give various relationships between contractivity and order of implicit methods