  In this paper we evolve magnetized and unmagnetized circumbinary accretion
disks around supermassive black hole binaries in the relativistic regime. We
use a post-Newtonian expansion to construct an analytical spacetime and
determine how the order of the post-Newtonian (PN) expansion affects the
dynamics of the gas. We find very small differences in the late-time bulk
dynamics of non-magnetized hydrodynamic evolutions between the two spacetimes
down to separations of approximately $40GM/c^2$ where $M$ is the total mass of
the binary. For smaller separations, the differences due to PN-order become
comparable to differences caused by using initial data further from
equilibrium. For magnetized gas, MHD stresses, which drives the accretion
dynamics, tends to mask all higher order PN effects even at separations of
$20GM/c^2$, leading to essentially the same observed electromagnetic
luminosity. This implies that our calculations of the EM signal may be robust
down to small binary separations. Our investigation is the first to demonstrate
how the level of PN accuracy affects a circumbinary disk's evolution and
informs us of the range in separation within which to trust the PN
approximation for this kind of study. We also address the influence the initial
conditions and binary separation have on simulation predictions.
