In Minkowski spacetime and Cartesian coordinates , the conservation equations (1
, 2
) can
be written in vector form as
In the non-relativistic limit (i.e., ,
)
,
, and
approach their Newtonian
counterparts
,
, and
, and Equations (5
) reduce to the classical ones. In the
relativistic case the equations of system (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 complicated. 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 9.2).
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