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  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  and the relativistic approach of  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
© Max-Planck-Gesellschaft. ISSN 1433-8351
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