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.