3.6 Relativistic HLL method (RHLLE)

Schneider et al. [257Jump To The Next Citation Point] have proposed to use the method of Harten, Lax and van Leer (HLL hereafter [122Jump To The Next Citation Point]) 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
( ) (| u for x < a t, HLL x- { L L u t;uL, uR = |( u∗ for aLt ≤ x ≤ aRt, (47 ) uR for x > aRt,
where a L and a R are lower and upper bounds for the smallest and largest signal velocities, respectively. The intermediate state u ∗ 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
u = aRuR--−-aLuL-−--F(uR-) +-F-(uL)-, (48 ) ∗ aR − aL
and the numerical flux by
HLL a+RF (uL) − a−L F(uR ) + a+Ra−L(uR − uL) ^F = ----------------+----−---------------, (49 ) aR − aL
− + aL = min {0,aL }, aR = max {0,aR }. (50 )

An essential ingredient of the HLL scheme are good estimates for the smallest and largest signal velocities. In the non-relativistic case, Einfeldt [81Jump To The Next Citation Point] proposed calculating them based on the smallest and largest eigenvalues of Roe’s matrix. The HLL scheme with Einfeldt’s recipe (HLLE) is a very robust upwind scheme for the Euler equations and possesses the property of being positively conservative. The HLLE method is exact for single shocks, but it is very dissipative, especially at contact discontinuities.

Schneider et al. [257Jump To The Next Citation Point] have presented results in 1D relativistic hydrodynamics using a relativistic version of the HLL method (RHLLE) with signal velocities given by

v¯+-¯cs- ¯v-−-¯cs- aR = 1 + ¯v¯cs, aL = 1 − ¯v¯cs, (51 )
where cs is the relativistic sound speed, and where the bar denotes the arithmetic mean between the initial left and right states. Duncan and Hughes [78Jump To The Next Citation Point] have generalized this method to 2D SRHD and applied it to the simulation of relativistic extragalactic jets.
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