The equations of viscous hydrodynamics, the Navier-Stokes-Fourier equations, have been formulated in relativity in terms of causal dissipative relativistic fluids (see the Living Reviews article by Müller  for a review). These extended fluid theories are numerically still almost unexplored in astrophysical systems. The reason may be the lack of an appropriate formulation well-suited for numerical studies. Peitz and Appl  have recently provided a 3+1 coordinate-free representation of different types of dissipative relativistic fluid theories [119, 60, 104], which has the potential of being well adapted to numerical applications.
The inclusion of magnetic fields and the development of formulations for the magneto-hydrodynamic equations, attractive to numerical studies, is still very limited in general relativity. Numerical approaches in special relativity are presented in [110, 220]. 3+1 representations of relativistic magneto-hydrodynamics can be found in [208, 66]. In  the transport of energy and angular momentum in magneto-hydrodynamical accretion onto a rotating black hole was studied adopting Wilson's formulation for the hydrodynamic equations (conveniently modified to account for the magnetic terms), and the magnetic induction equation was solved using the constrained transport method of . Recently  have performed the first magneto-hydrodynamical simulation in general relativity of magnetically driven relativistic jets from an accretion disk around a Schwarzschild black hole.
The interaction between matter and radiation fields, present in different levels of complexity in all astrophysical systems, is described by the equations of radiation hydrodynamics. The Newtonian framework is highly developed (see, e.g., ; the special relativistic transfer equation is also considered in that reference). General relativistic formulations of radiative transfer in curved spacetimes are considered in, e.g.,  and  (see also references therein).
|Numerical Hydrodynamics in General Relativity
José A. Font
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