Numerical integration of the equations of relativistic hydrodynamics

The gravitational field equations are solved by starting with the '3+1' approach. The freedom to choose coordinates is used to reduce Einstein's field equations to a set of elliptic type quations which have no time derivatives and a set of first order (hyperbolic) time evolution equations for the true gravitational degrees of freedom only. In a spherically symmetric system there are no gravitational degrees of gravitational freedom. In an axially symmetric (nonrotating) there is one degree of gravitational freedom. Computer programs have only been written for spherical and axial symmetry so far, but the same approach should easily work for the full non symmetric systems. The axymmetriy so far, but the same approach should easily work for the full non symmetric systems. The axially symmetric case will be discussed in detail below since it best illustrates the method. The equations for spherical symmetry will be given along with some results of computer calcualtions. Some results will be given for the axial symmetric program, but these results are very preliminary at this point and since the computer program has not been well checked and probably still has errors, these problems only illustrate that the method seems to work well. (orig.)