A Note on Unconditional Maximum Norm Contractivity of Diagonally Split Runge–Kutta Methods

Abstract

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 $p \leq 1$ 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 $p \leq 4$,$r \leq 1$.