where and are approximations of the state vector at the left and right side of a zone interface obtained by a second-order accurate interpolation in space and time, and is the solution of the Riemann problem defined by the two interpolated states at the position of the initial discontinuity.

The PPM interpolation algorithm described in [33] 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
*j*, the quartic polynomial with zone-averaged values
,
,
,
, and
(where
) is used to interpolate the structure inside the zone. In
particular, the values of
*a*
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.

Both, the non relativitic PPM scheme described in [33] and the relativistic approach of [109] follow the same procedure to compute the time-averaged fluxes
at an interface
*j*
+1/2 separating zones
*j*
and
*j*
+1. They 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. The relativistic version of PPM uses the
characteristic speeds and Riemann invariants of the equations of
relativistic hydrodynamics in this procedure.

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 |