where , and where denotes the covariant derivative with respect to coordinate . Furthermore, is the proper rest-mass density of the fluid, its four-velocity, and is the stress-energy tensor, which for a perfect fluid can be written as

Here
is the metric tensor,
*p*
the fluid pressure, and and
*h*
the specific enthalpy of the fluid defined by

where
is the specific internal energy. Note that we use natural units
(i.e., the speed of light
*c*
=1) throughout this review.

In Minkowski spacetime and Cartesian coordinates , the conservation equations (1, 2) can be written in vector form as

where
*i*
= 1,2,3. The state vector
is defined by

and the flux vectors are given by

The five conserved quantities
*D*,
,
,
and
are the rest-mass density, the three components of the momentum
density, and the energy density (measured relative to the rest
mass energy density), respectively. They are all measured in the
laboratory frame, and are related to quantities in the local rest
frame of the fluid (primitive variables) through

where are the components of the three-velocity of the fluid

and
*W*
is the Lorentz factor

The system of equations (5) with definitions (6, 8, 9, 10, 11, 12) is closed by means of an equation of state (EOS), which we shall assume to be given in the form

In the non-relativistic limit (i.e.,
,
)
*D*,
and
approach their Newtonian counterparts
,
and
, and the equations of system (5) reduce to the classical ones. In the relativistic case the
equations of (5) are strongly coupled via the Lorentz factor and the specific
enthalpy, which gives rise to numerical complications (see
Section
2.3).

In classical numerical hydrodynamics it is very easy to obtain from the conserved quantities (i.e., and ). In the relativistic case, however, the task to recover from is much more difficult. Moreover, as state-of-the-art SRHD codes are based on conservative schemes where the conserved quantities are advanced in time, it is necessary to compute the primitive variables from the conserved ones one (or even several) times per numerical cell and time step making this procedure a crucial ingredient of any algorithm (see Section 9.1).

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |