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  and references therein). These extended fluid theories, however, remain unexplored, numerically, in astrophysical systems. The reason may be the lack of an appropriate formulation well-suited for numerical studies. Work in this direction was done by Peitz and Appl  who provided a 3+1 coordinate-free representation of different types of dissipative relativistic fluid theories which possess, in principle, the potentiality of being well adapted to numerical applications.
The inclusion of magnetic fields and the development of formulations for the MHD equations, attractive to numerical studies, is still very limited in general relativity. Numerical approaches in special relativity are presented in [143, 291, 20]. In particular, Komissarov , and Balsara  developed two different upwind HRSC (or Godunov-type) schemes, providing the characteristic information of the corresponding system of equations, and proposed a battery of tests to validate numerical MHD codes. 3+1 representations of relativistic MHD can be found in [272, 80]. 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, Koide et al. [141, 142] performed the first MHD simulation, in general relativity, of magnetically driven relativistic jets from an accretion disk around a Schwarzschild black hole (see Section 4.2.2). These authors used a second-order finite difference central scheme with nonlinear dissipation developed by Davis . Even though astrophysical applications of Godunov-type schemes (see Section 3.1.2) in general relativistic MHD are still absent, it is realistic to believe this situation may change in the near future.
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). Pons et al.  discuss a hyperbolic formulation of the radiative transfer equations, paying particular attention to the closure relations and to extend HRSC schemes to those equations. 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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