4.2.5 Extensions of classical solvers to relativistic fluid dynamics

• Eulderink solver (A&ASS 110, 587-623, 1995):
  • Eulderink and Mellema extended the Roe solver to compute numerical approximations of the general relativistic hydrodynamic equations

  • They look for a local linearization of the Jacobian matrices of the system which fulfills the properties demanded by Roe's matrix

  • It is done in terms of the average state
    

    $$\widetilde{\bf w} = \frac{ {\bf w}_{\rm L} + {\bf w}_{\rm R}} {k_{\rm L} + k_{\rm R}},$$ (113)

    
    
    
    

    $${\bf w} = (ku^0,ku^1,ku^2,ku^3,k\frac{p}{\rho h}),$$ (114)

    
    
    
    

    $$k^2 = \sqrt{-g} \rho h.$$ (115)

    
    

  • The role played by ${\rho}$ in the non relativistic Roe solver as weight for averaging, is now played by $k$, which apart from geometrical factors, tends to $\rho$ in the non relativistic limit

    Relaxing Condition 1 in Roe's linearization, the Roe/Eulderink solver is no longer exact for shocks but still produces accurate solutions. The advantage is that the reduced set of conditions is fulfilled by a large number of averages, a typical choice being the arithmetic mean

• Falle & Komissarov (MNRAS 278, 586, 1996)
  • proposed several approximate special relativistic Riemann solvers relying on a primitive-variable formulation of the relativistic hydrodynamic equations in quasilinear form:
    

    $$\frac{\partial {\bf V}}{\partial t} + {\bf B} \frac{\partial {\bf V}} {\partial x} = 0,$$ (116)

    
    
    with ${\bf B} = {\bf B}({\bf V})$ and ${\bf V}$ being any set of primitive variables, e.g., $(\rho, v_x, v_y, v_z, p)$

  • A local linearization of the system above allows to obtain the solution of the Riemann problem

• Schneider et al (JCP 105, 92, 1993)
  • proposed the use of the method of Harten, Lax and van Leer (1983) to integrate the equations of special relativistic fluid dynamics

  • This method avoids the explicit calculation of the eigenvectors of the Jacobian matrices

  • The numerical flux function is given by
    

    $${\widehat{\bf f}}^{\rm HLL} = \frac{a_{\rm R}^{+}{\bf f}({\b... ...{\bf u}_{\rm R}-{\bf u}_{\rm L})}{a_{\rm R}^{+}-a_{\rm L}^{-}}$$ (117)

    
    
    where
    

    $$a_{\rm L}^{-} = {\rm min}\{0,a_{\rm L}\}, a_{\rm R}^{+} = {\rm max}\{0,a_{\rm R}\}.$$ (118)

    
    
    and $a_{\rm L}$ and $a_{\rm R}$ are lower and upper bounds for the smallest and largest signals velocities

  • Einfeldt proposed calculating them based on the smallest and largest eigenvalues of the system

• All-shock approximation (Colella, SIAM J. Sci. Stat. Comput. 3, 76, 1982)
  • extended to relativistic hydrodynamics by
    1. Balsara (JCP 114, 284, 1994)
    2. Dai & Woodward (SIAM J. Sci. Stat. Comput. 18, 982, 1997)

  • simplification to the Riemann solver: ignore the possibility of rarefaction waves, finding Riemann solutions in which each pair of states is connected along the Hugoniot locus

  • the solution cosnsists only of discontinuities satisfying the algebraic Rankine-Hugoniot conditions