3.7 Marquina's flux formula3 High-Resolution Shock-Capturing Methods3.5 Falle and Komissarov upwind

3.6 Relativistic HLL method 

Schneider et al. [161Jump To The Next Citation Point In The Article] have proposed to use the method of Harten, Lax & van Leer [74Jump To The Next Citation Point In The Article], HLL hereafter, to integrate the equations of SRHD. This method avoids the explicit calculation of the eigenvalues and eigenvectors of the Jacobian matrices and is based on an approximate solution of the original Riemann problems with a single intermediate state

equation640

where tex2html_wrap_inline6047 and tex2html_wrap_inline6049 are lower and upper bounds for the smallest and largest signal velocities, respectively. The intermediate state tex2html_wrap_inline6051 is determined by requiring consistency of the approximate Riemann solution with the integral form of the conservation laws in a grid zone. The resulting integral average of the Riemann solution between the slowest and fastest signals at some time is given by

equation663

and the numerical flux by

equation680

where

equation706

An essential ingredient of the HLL scheme are good estimates for the smallest and largest signal velocities. In the non-relativistic case, Einfeldt [48Jump To The Next Citation Point In The Article] proposed to calculate them based on the smallest and largest eigenvalues of Roe's matrix. This HLL scheme with Einfeldt's recipe is a very robust upwind scheme for the Euler equations and possesses the property of being positively conservative. The method is exact for single shocks, but it is very dissipative, especially at contact discontinuities.

Schneider et al. [161Jump To The Next Citation Point In The Article] have presented results in 1D ultra-relativistic hydrodynamics using a version of the HLL method with signal velocities given by

equation718

equation725

where tex2html_wrap_inline6053 is the relativistic sound speed, and where the bar denotes the arithmetic mean between the initial left and right states. Duncan & Hughes [46Jump To The Next Citation Point In The Article] have generalized this method to 2D SRHD and applied it to the simulation of relativistic extragalactic jets.



3.7 Marquina's flux formula3 High-Resolution Shock-Capturing Methods3.5 Falle and Komissarov upwind

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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