In recent years, the author and his coworkers have suggested a meanwhile so-called contact-stabilized Newmark discretization scheme for dynamical contact problems. In the first part of the talk, this scheme will be discussed, analyzed and compared with existing versions of Newmark schemes. In a second part, an automatic timestep control will be derived on the basis of a non-standard extrapolation method. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact-stabilized Newmark scheme exhibiting surprising new features: The theory necessary to cover the mathematical situation correctly, will have to be based on the space of bounded variation (BV). Both a consistency and a convergence theory will be presented. Finally, an illustrative numerical example will be given.