## 2.3 Exact solution of the Riemann problem in SRHD

Let us first consider the one dimensional special relativistic flow of an ideal gas with an adiabatic exponent in the absence of a gravitational field. The Riemann problem then consists of computing the breakup of a discontinuity, which initially separates two arbitrary constant states L (left) and R (right) in the gas (see Fig.  1 with L and R ). For classical hydrodynamics the solution can be found, e.g., in [35]. In the case of SRHD, the Riemann problem has been considered by Martí & Müller [108], who derived an exact solution generalizing previous results for particular initial data [173].

The solution to this problem is self-similar, because it only depends on the two constant states defining the discontinuity and , where , and on the ratio , where and are the initial location of the discontinuity and the time of breakup, respectively. Both in relativistic and classical hydrodynamics the discontinuity decays into two elementary nonlinear waves (shocks or rarefactions) which move in opposite directions towards the initial left and right states. Between these waves two new constant states and (note that and in Fig.  1) appear, which are separated from each other through a contact discontinuity moving with the fluid. Across the contact discontinuity the density exhibits a jump, whereas pressure and velocity are continuous (see Fig.  1). As in the classical case, the self-similar character of the flow through rarefaction waves and the Rankine-Hugoniot conditions across shocks provide the relations to link the intermediate states (S =L, R) with the corresponding initial states . They also allow one to express the fluid flow velocity in the intermediate states as a function of the pressure in these states. Finally, the steadiness of pressure and velocity across the contact discontinuity implies

where , which closes the system. The functions are defined by

where  / denotes the family of all states which can be connected through a rarefaction / shock with a given state ahead of the wave.

Figure 1: Schematic solution of a Riemann problem in special relativistic hydrodynamics. The initial state at t = 0 (top figure) consists of two constant states (1) and (5) with , , and separated by a diaphragm at . The evolution of the flow pattern once the diaphragm is removed (middle figure) is illustrated in a spacetime diagram (bottom figure) with a shock wave (solid line) and a contact discontinuity (dashed line) moving to the right. The bundle of solid lines represents a rarefaction wave propagating to the left.

The fact that one Riemann invariant is constant through any rarefaction wave provides the relation needed to derive the function

with

the + / - sign of corresponding to S =L / S =R. In the above equation, is the sound speed of the state , and c (p) is given by

The family of all states , which can be connected through a shock with a given state ahead of the wave, is determined by the shock jump conditions. One obtains

where the + / - sign corresponds to S =R / S =L. and j (p) denote the shock velocity and the modulus of the mass flux across the shock front, respectively. They are given by

and

where the enthalpy h (p) of the state behind the shock is the (unique) positive root of the quadratic equation

which is obtained from the Taub adiabat (the relativistic version of the Hugoniot adiabat) for an ideal gas equation of state.

Figure 2: Graphical solution in the p- v plane of the Riemann problems defined by the initial states , , , and , , with i = 1, 2, 3, 4, where , , , and , respectively. The adiabatic index of the fluid is 5/3 in all cases. Note the asymptotic behavior of the functions as they approach v =1 (i.e., the speed of light).

The functions and are displayed in Fig.  2 in a p - v diagram for a particular set of Riemann problems. Once has been obtained, the remaining state quantities and the complete Riemann solution,

can easily be derived.

In Section  9.3 we provide a FORTRAN program called RIEMANN, which allows one to compute the exact solution of an arbitrary special relativistic Riemann problem using the algorithm just described.

The treatment of multidimensional special relativistic flows is significantly more difficult than that of multidimensional Newtonian flows. In SRHD all components (normal and tangential) of the flow velocity are strongly coupled through the Lorentz factor, which complicates the solution of the Riemann problem severely. For shock waves, this coupling 'only' increases the number of algebraic jump conditions, which must be solved. However, for rarefactions it implies the solution of a system of ordinary differential equations [108].

 Numerical Hydrodynamics in Special Relativity Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de