Regularity properties of multistage integration methods

The numerical method for ordinary differential equations is regular if it has the same set of finite asymptotic values as the underlying differential system. This paper examines the regularity and strong regularity properties of diagonally implicit multistage integration methods (DIMSIMs) introduced recently by J.C. Butcher. A sufficient condition for regularity and strong regularity of such methods of any order is given and it is proved that this condition is also necessary for two-step two-stage DIMSIMs of order greater than or equal to two. It is also demonstrated that there exist regular schemes in the class of explicit DIMSIMs. This is in contrast to explicit Runge-Kutta methods with more than one stage, which are always irregular.