A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods

In this paper we consider diagonally split Runge-Kutta methods for the numerical solution of initial value problems for ordinary differential equations. This class of numerical methods was recently introduced by Bellen, Jackiewicz, and Zennaro [SIAM J. Numer. Anal., 31 (1994), pp. 499-523], and comprises the well-known class of Runge-Kutta methods. Their results strongly indicate that diagonally split Runge-Kutta methods break the order barrier p less than or equal to 1 for unconditional contractivity in the maximum norm. In this paper we investigate the effect of the requirement of unconditional contractivity in the maximum norm on the accuracy of a diagonally split Runge-Kutta method. Besides the classical order p, we deal with an order of accuracy r which is relevant to the case where the method is applied to dissipative initial value problems that are arbitrarily stiff We show that if a diagonally split Runge-Kutta method is unconditionally contractive in the maximum norm, then it has orders p, r which satisfy p less than or equal to 4, r less than or equal to 1.