Representations of Runge-Kutta Methods and Strong Stability Preserving Methods

Over the last few years a great effort has been made to develop monotone high order explicit Runge--Kutta methods by means of their Shu--Osher representations. In this context, the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation. In this paper we extend the Shu--Osher representations for any Runge--Kutta method giving sufficient conditions for monotonicity. We show how optimal Shu--Osher representations can be constructed from the Butcher tableau of a Runge--Kutta method. The optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the Runge--Kutta method [L. Ferracina and M. N. Spijker, SIAM J. Numer. Anal., 42 (2004), pp. 1073--1093], and hence if this radius is zero, the method is not monotone. In the Shu--Osher representation, methods with zero radius require negative coefficients, and to deal with them, an extra associate problem is considered. In this paper we interpret these schemes as representations of perturbed Runge--Kutta methods. We extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity. Optimal representations can be constructed from the Butcher tableau of a perturbed Runge--Kutta method.