Stepsize Conditions for Boundedness in Numerical Initial Value Problems

For Runge-Kutta methods (RKMs), linear multistep methods (LMMs) and classes of general linear methods (GLMs) much attention has been paid, in the literature, to 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 & 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 well-known methods, 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 focus on stepsize conditions for boundedness directly rather than via the detour of the above special stability properties. We present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual RKMs, LMMs and GLMs whether or not the methods under consideration have the special stability properties mentioned above.