The detection of relativistic corrections in cosmological N-body
  simulations
  Cosmological N-body simulations are done on massively parallel computers.
This necessitates the use of simple time integrators, and, additionally, of
mesh-grid approximations of the potentials. Recently, Adamek et al. (2015);
Barrera-Hinojosa et al. (2019) have developed general relativistic N-body
simulations to capture relativistic effects mainly for cosmological purposes.
We therefore ask whether, with the available technology, relativistic effects
like perihelion advance can be detected numerically to a relevant precision. We
first study the spurious perihelion shift in the Kepler problem, as a function
of the integration method used, and then as a function of an additional
interpolation of forces on a 2-dimensional lattice. This is done for several
choices of eccentricities and semi-major axes. Using these results, we can
predict which precisions and lattice constants allow for a detection of the
relativistic perihelion advance in N-body simulation. We find that there are
only small windows of parameters---such as eccentricity, distance from the
central object and the Schwarzschild radius---for which the corrections can be
detected in the numerics.
