Numerical Integrators that Preserve Symmetries and Reversing Symmetries

We consider properties of flows, the relationships between them, and whether numerical integrators can be made to preserve these properties. This is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields. This new framework unifies several known constructions. We also use the concept of "covariance" of a numerical method with respect to a group of coordinate transformations. The main application is to explore the relationship between spatial symmetries, reversing symmetries, and time symmetry of flows and numerical integrators.