We review various approaches to approximating general relativistic effects in hydrodynamic simulations of stellar core collapse and post-bounce evolution. Different formulations of a modified Newtonian gravitational potential are presented. Such an effective relativistic potential can be used in an otherwise standard Newtonian hydrodynamic code. An alternative approximation of general relativity is the assumption of conformal flatness for the three-metric, and its extension by adding second post-Newtonian order terms. Using a code which evolves the coupled system of metric and fluid equations, we apply the various approximation methods to numerically simulate axisymmetric models for the collapse of rotating massive stellar cores. We compare the collapse dynamics and gravitational wave signals (which are extracted using the quadrupole formula), and thereby assess the quality of the individual approximation method. It is shown that while the use of an effective relativistic potential already poses a significant improvement compared to a genuinely Newtonian approach, the two conformal-flatness-based approximation methods yield even more accurate results, which are qualitatively and quantitatively very close to those of a fully general relativistic code even for rotating models which almost collapse to a black hole.
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