  We discuss refinement criteria for the Berger-Rigoutsos (block-based)
refinement algorithm in our numerical relativity code GR-Athena++ in the
context of binary black hole merger simulations. We compare three different
strategies: the "box-in-box" approach, the "sphere-in-sphere" approach and a
local criterion for refinement based on the estimation of truncation error of
the finite difference scheme. We extract and compare gravitational waveforms
using the three different mesh refinement methods and compare their accuracy
against a calibration waveform and demonstrate that the sphere-in-sphere
approach provides the best strategy overall when considering computational cost
and the waveform accuracy. Ultimately, we demonstrate the capability of each
mesh refinement method in accurately simulating gravitational waves from binary
black hole systems -- a crucial aspect for their application in next-generation
detectors. We quantify the mismatch achievable with the different strategies by
extrapolating the gravitational wave mismatch to higher resolution.
