Boundedness and strong stability of Runge-Kutta methods

In the literature, much attention has been paid to Runge-Kutta methods (RKMs) satisfying special nonlinear stability requirements indicated by the terms total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize conditions, guaranteeing these properties, were derived by Shu and Osher [J. Comput. Phys., 77 (1988) pp. 439-471] and in numerous subsequent papers. These special stability requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded. Unfortunately, for many RKMs, the above special requirements are violated, so that one cannot conclude in this way that the methods are (total-variation) bounded. In this paper, we study stepsize-conditions for boundedness directly, rather than via the detour of the above special stability properties. We focus on stepsize-conditions which are optimal, in that they are not unnecessarily restrictive. We find that, in situations where the special stability properties mentioned above are violated, boundedness can be present only within a class of very special RKMs. As a by-product, our analysis sheds new light on the known theory of monotonicity for RKMs. We obtain separate results for internal and external monotonicity, as well as a new proof of the fundamental relation between monotonicity and Kraaijevanger’s coefficient. This proof distinguishes itself from older ones in that it is shorter and more transparent, while it requires simpler assumptions on the RKMs under consideration.