  Given the compact binary evolution problem of numerical relativity, in the
finite-difference, block-based, adaptive mesh refinement context, choices must
be made on how evolved fields are to be discretized. In GR-Athena++, the
space-time solver was previously fixed to be vertex-centered. Here, our recent
extensions to a cell-centered treatment, are described. Simplifications in the
handling of variables during the treatment of general relativistic
magneto-hydrodynamical (GRMHD) evolution are found. A novelty is that
performance comparison for the two choices of grid sampling is made within a
single code-base. In the case of a binary black hole inspiral-merger problem,
by evolving geometric fields on vertex-centers, an average $\sim 20\%$ speed
increase is observed, when compared against cell-centered sampling. The
opposite occurs in the GRMHD setting. A binary neutron star
inspiral-merger-collapse problem, representative of typical production
simulations is considered. We find that cell-centered sampling for the
space-time solver improves performance, by a similar factor.
