3 Numerical Schemes2 Equations of General Relativistic 2.2 Covariant approaches

2.3 Going further: Non-ideal hydrodynamics 

Formulations of the equations of non-ideal hydrodynamics in general relativity are also available in the literature. The term ``non-ideal'' accounts for additional physics such as viscosity, magnetic fields, and radiation. These non-adiabatic effects can play a major role in some astrophysical systems as, such as accretion disks or relativistic stars.

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 [192] 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 [224] 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 [143Jump To The Next Citation Point In The Article, 291, 20Jump To The Next Citation Point In The Article]. In particular, Komissarov [143Jump To The Next Citation Point In The Article], and Balsara [20Jump To The Next Citation Point In The Article] 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, 80Jump To The Next Citation Point In The Article]. In [313Jump 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 [80Jump To The Next Citation Point In The Article]. Recently, Koide et al. [141Jump To The Next Citation Point In The Article, 142Jump To The Next Citation Point In The Article] 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 [61Jump To The Next Citation Point In The Article]. 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., [180]; the special relativistic transfer equation is also considered in that reference). Pons et al. [230] 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., [237] and [316] (see also references therein).



3 Numerical Schemes2 Equations of General Relativistic 2.2 Covariant approaches

image Numerical Hydrodynamics in General Relativity
José A. Font
http://www.livingreviews.org/lrr-2003-4
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de