For Runge-Kutta methods, linear multistep methods and other classes of
general linear methods much attention has been paid in the literature
to important nonlinear stability properties known as
total-variation-diminishing (TVD), strong stability preserving (SSP)
and monotonicity. Stepsize conditions guaranteeing these properties
were studied by Shu \& Osher (1988) and in numerous subsequent papers.
Unfortunately, for many useful methods it has turned out that these
properties do not hold. For this reason attention has been paid
in the recent literature to the related and more general properties
called total-variation-bounded (TVB) and boundedness.
In the present paper we focus on stepsize conditions guaranteeing
boundedness properties of a special type. These boundedness
properties are optimal, and distinguish themselves
also from earlier boundedness results by being relevant to sublinear
functionals, discrete maximum principles and preservation of nonnegativity.
Moreover, the corresponding stepsize conditions are more easily verified
in practical situations than the conditions for general boundedness
given thus far in the literature.
The theoretical results are illustrated by application to the two-step
Adams-Bashforth method and a class of two-stage multistep methods