On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups

We use functional calculus methods to investigate qualitative properties of C0-semigroups that are preserved by time-discretization methods. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. Preservation of contractivity (or norm-bound) of the semigroup is investigated in the Banach space setting. The use of the Hille–Phillips (H–P) functional calculus instead of the Dunford–Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. Since the H–P functional calculus is used throughout the paper, we present an elementary introduction to it based on the Riemann–Stieltjes integral.  2004 Published by Elsevier Inc.