3 Numerical Schemes2 Equations of General Relativistic 2.2 Covariant approaches

2.3 Going further 

Formulations of the equations of non-ideal hydrodynamics in general relativity are also available in the literature. Here the term ``non-ideal'' includes effects such as viscosity, magnetic fields and radiative transfer. These non-adiabatic effects can play a major role in some astrophysical systems as e.g. accretion disks.

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 [147] 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 [173] 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, 66Jump To The Next Citation Point In The Article]. In [237Jump To The Next Citation Point In The Article] 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 [66Jump To The Next Citation Point In The Article]. Recently [109] 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., [138]; 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., [181] and [240] (see also references therein).

3 Numerical Schemes2 Equations of General Relativistic 2.2 Covariant approaches

image Numerical Hydrodynamics in General Relativity
José A. Font
© Max-Planck-Gesellschaft. ISSN 1433-8351
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