3.2 The relativistic Glimm method3 High-Resolution Shock-Capturing Methods3 High-Resolution Shock-Capturing Methods

3.1 Relativistic PPM 

Martí & Müller [109Jump To The Next Citation Point In The Article] have used the procedure discussed in Section  2.3 to construct an exact Riemann solver, which they then incorporated in an extension of the piecewise parabolic method (PPM) [33Jump To The Next Citation Point In The Article] for 1D SRHD. In their relativistic PPM method numerical fluxes are calculated according to

equation314

where tex2html_wrap_inline5881 and tex2html_wrap_inline5883 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 tex2html_wrap_inline5885 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 [33Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline5755, which are obtained from zone averaged values of the conserved quantities tex2html_wrap_inline5699 . For each zone j, the quartic polynomial with zone-averaged values tex2html_wrap_inline5893, tex2html_wrap_inline5895, tex2html_wrap_inline5897, tex2html_wrap_inline5899, and tex2html_wrap_inline5901 (where tex2html_wrap_inline5903) is used to interpolate the structure inside the zone. In particular, the values of a at the left and right interface of the zone, tex2html_wrap_inline5907 and tex2html_wrap_inline5909, are obtained this way. These reconstructed values are then modified such that the parabolic profile, which is uniquely determined by tex2html_wrap_inline5907, tex2html_wrap_inline5909, and tex2html_wrap_inline5897, is monotonic inside the zone.

Both, the non relativitic PPM scheme described in [33Jump To The Next Citation Point In The Article] and the relativistic approach of [109Jump To The Next Citation Point In The Article] 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.



3.2 The relativistic Glimm method3 High-Resolution Shock-Capturing Methods3 High-Resolution Shock-Capturing Methods

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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