4.2.1 Exact Riemann solvers in relativistic hydrodynamics

Analysis of the breakup of an initial discontinuity separating two constant states L (left) and R (right) of an ideal gas (with adiabatic exponent $\Gamma$) in arbitrary (1D) conditions, in the absence of any gravitational field

In relativistic hydrodynamics this problem was first considered by Martí & Müller (1994) who derived an exact solution

The discontinuity between the two constant initial states ${\bf V}_L$ and ${\bf V}_R$ ( ${\bf V} = (p,\rho,v)$) breaks up into two elementary nonlinear waves (shocks or rarefactions), one moving towards the initial left state and the other towards the initial right state

Between the waves two new states appear, ${\bf V}_{L*}$ and ${\bf V}_{R*}$, separated from each other through a contact discontinuity moving along with the fluid. Across the contact discontinuity, pressure and velocity are continuous, while density exhibits a jump.

As in classical hydrodynamics (see, e.g., Courant and Friedrichs 1976) the self-similar character of the flow through rarefaction waves and the Rankine-Hugoniot conditions across shocks provide the conditions for linking the intermediate states ${\bf V}_{S*}$ ($S=L,R$) with their corresponding initial state ${\bf V}_S$

• Martí & Müller (1996) constructed an exact Riemann solver to be implemented in a one-dimensional relativistic extension of the PPM method (Colella & Woodward 1984)

• Another exact solver was also implemented by Wen, Panaitescu & Laguna (1997) in their relativistic (1D) version of Glimm's random choice method (Glimm 1965). Besides being conservative on the average, Glimm's method produces completely sharp shocks and contact discontinuities and it is free of difussion and dispersion