[I am not the right person to answer your question but this might help]
Two kind of CFD methods are more popular in Computational Astrophysics, compared to Mechanical/ Chemical/Civil engineering CFD, and there are solid reasons for that:
1) Smooth Particle Hydrodynamics (SPH) method. It is developed in first place for astrophysics and due to its Lagrangian nature it can tackle questions of astrophysics very nice.
2) Spectral Methods (I do not mean spectral FEM, I mean methods such as Chebyshev polynomials spectral method): Those methods are of very high order and they are computationally effective and suitable for "large" domains of astrophysics, on one hand; On the other hand, in astrophysics we are not dealing with odd and dynamic geometries of the domain and this fact eliminates one of the main limitation of spectral methods.
Università degli Studi della Campania "Luigi Vanvitelli
Actually, the equations are not exactly the same, you need to start from the relativistic fluid dynamics topics, as reported in the textbook Landau Lifshitz, FluidMechanics, -3ed.
Have also a look to:
Durante, D.; Riccardi, G. An accurate and efficient Riemann solver with tangent velocities for Godunov schemes in Special Relativistic Hydrodynamics, Int. J. of Numerical Methods in Fluids 64-1 pp. 1 − 22 (2010). DOI: 10.1002/fld.2133. IF: 1.06.
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