  The large dynamic range in some astrophysical N-body problems led to the use
of adaptive multi-time-steps; however, the search for optimal strategies is
still challenging. We numerically quantify the performance of the hierarchical
Hamiltonian Splitting (HHS) integrator for collisionless simulations using a
direct summation code. We compare HHS with the constant global time-step
leapfrog integrator, and with the adaptive one (AKDK). We find that HHS is
approximately reversible, whereas AKDK not. Therefore, it is possible to find a
combination of parameters where the energy drift is considerably milder for
HHS, resulting in a better performance. We conclude that HHS is an attractive
alternative to AKDK, and it is certainly advantageous for direct summation and
P3M codes. Also, we find advantages with GADGET4 (Tree/FMM) HHS implementation
that are worth exploring further.
