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).
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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 |