We present the results of fully three dimensional, post-Newtonian hydrodynamical simulations of the dynamical evolution of mergers between compact stellar remnants (neutron stars and black holes). Although the code is essentially Newtonian, we simulate gravitational wave emission and the corresponding effect on the fluid flow via a post-Newtonian correction. Also, we use a modified Newtonian potential which reproduces certain aspects of the Schwarzschild and Kerr solutions to improve the physics in the vicinity of the black hole. Changes to the energy by neutrino/antineutrino emission are accounted for by an extensive neutrino leakage scheme. The hydrodynamical equations are integrated using the piecewise parabolic method (PPM) and the neutron star matter is described by a tabulated equation of state (EoS). Since the physics of matter at the extreme densities found in neutron stars is not yet certain, we compare results computed using two such tables to ascertain whether this uncertainty in the micro-physics extends to an uncertainty in the energy available to power a short-period gamma-ray burst. With an aim to including magnetic field physics to these simulations, we present a survey of approximate Riemann solvers which may be more easily extended to the system of equations of magnetohydrodynamics (MHD) than the exact or iterative Riemann solver used in the PPM scheme. Tests are performed using the linearised solver of Roe and the approximate Harten, Lax, van Leer and Einfeldt Riemann solvers (HLLE and HLLEM) with the PPM reconstruction scheme. Finally, we discuss the effectiveness of these approximate Riemann solvers in the simulation of mergers between compact stellar remnants.