Abstract
The smoothed particle hydrodynamics (SPH) method has been extensively applied to predicting the behaviour of compressible Newtonian fluids for astrophysical problems. In this paper it is extended to the post-Newtonian (PN) approximation to the fully general relativistic equations retaining perturbations up to 1/c^2. An obvious advantage of this approach is that standard numerical methods for Newtonian hydrodynamics can be extended easily to the post-Newtonian equations unlike the situation for the equations governing general relativity. The post-Newtonian SPH method has been validated against a number of test cases including relativistic polytropes and general relativistic hydrodynamical collapse calculations. Preliminary results have been obtained for the collapse of rotating stellar cores. In general, the method is applicable to general relativistic astrophysical problems such as stellar core collapse and the coalescence of neutron stars in binaries. Both of these problems have previously been treated mainly by Newtonian methods.