Generalizations of the Stage Order of Runge-Kutta Methods

Runge-Kutta methods are used to solve stiff systems of ordinary differential equations and differential-algebraic equations. The solution of such problems often exhibits order reduction, when, for prescribed accuracy, the actual order of a method is lower than its classical order, which inevitably increases computational costs. To avoid order reduction, the method has to have a sufficiently high stage order. However, methods with the most convenient and efficient implementation have a low stage order. Accordingly, a task of importance is to construct methods of low stage order that have properties of methods with a higher stage order. The construction of methods of this type is addressed in the present paper. Singly diagonally implicit and explicit methods and methods inverse to explicit ones are considered. Results of solving test problems are presented.