The PPM interpolation algorithm described
in [60] gives monotonic
conservative parabolic profiles of variables within a numerical
zone. In the relativistic version of PPM, the original
interpolation algorithm is applied to zone averaged values of the
primitive variables
, which are obtained from
zone averaged values of the conserved quantities
. For each zone
, the quartic polynomial with
zone averaged values
,
,
,
, and
(where
) is used to interpolate the structure inside the
zone. In particular, the values of
at the left and right
interface of the zone,
and
, are obtained this way. These reconstructed values
are then modified such that the parabolic profile, which is
uniquely determined by
,
and
, is monotonic inside the zone.
The time-averaged fluxes at an interface separating zones
and
are computed from two spatially averaged states
and
at the left and right side
of the interface, respectively. These left and right states are
constructed taking into account the characteristic information
reaching the interface from both sides during the time step. In the
relativistic version of PPM the same procedure as in [60
] has been followed,
using the characteristic speeds and Riemann invariants of the
equations of relativistic hydrodynamics. The results presented
in [181
] were obtained with
an Eulerian code (rPPM) based on this method. The corresponding
FORTRAN program rPPM is provided in
Section 9.4.3. A relativistic Lagrangian version
of the original PPM method in spherical coordinates and spherical
symmetry has been developed by Daigne and Mochkovich [66
].