Relativistic PPM

3.1 Relativistic PPM

Martí and Müller [181

] have used the procedure discussed in Section 2.3 to construct an exact Riemann solver, which they then incorporated in an extension of the PPM method [60

] for 1D SRHD. In their relativistic PPM method, numerical fluxes are calculated according to

where $uL$ and $uR$ 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 $u(0;uL, uR)$ 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 [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 $v = (p,ρ, v)$ , which are obtained from zone averaged values of the conserved quantities $u$ . For each zone j, the quartic polynomial with zone averaged values $aj−2$ , $aj− 1$ , $aj$ , $aj+1$ , and $aj+2$ (where $a = ρ, p,v$ ) is used to interpolate the structure inside the zone. In particular, the values of $a$ at the left and right interface of the zone, $aL,j$ and $aR,j$ , are obtained this way. These reconstructed values are then modified such that the parabolic profile, which is uniquely determined by $a L,j$ , $a R,j$ and $a j$ , is monotonic inside the zone.

The time-averaged fluxes at an interface j + 1/2 separating zones j and j + 1 are computed from two spatially averaged states $vj+1,L 2$ and $vj+ 1,R 2$ 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 ].