3 High-Resolution Shock-Capturing Methods2 Special Relativistic Hydrodynamics2.2 SRHD as a hyperbolic

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 tex2html_wrap_inline5745 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 tex2html_wrap_inline5747 and R tex2html_wrap_inline5749). 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 [108Jump To The Next Citation Point In The Article], 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 tex2html_wrap_inline5751 and tex2html_wrap_inline5753, where tex2html_wrap_inline5755, and on the ratio tex2html_wrap_inline5757, where tex2html_wrap_inline5759 and tex2html_wrap_inline5761 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 tex2html_wrap_inline5763 and tex2html_wrap_inline5765 (note that tex2html_wrap_inline5767 and tex2html_wrap_inline5769 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 tex2html_wrap_inline5771 (S =L, R) with the corresponding initial states tex2html_wrap_inline5775 . They also allow one to express the fluid flow velocity in the intermediate states tex2html_wrap_inline5777 as a function of the pressure tex2html_wrap_inline5779 in these states. Finally, the steadiness of pressure and velocity across the contact discontinuity implies

  equation177

where tex2html_wrap_inline5781, which closes the system. The functions tex2html_wrap_inline5783 are defined by

equation187

where tex2html_wrap_inline5789  / tex2html_wrap_inline5791 denotes the family of all states which can be connected through a rarefaction / shock with a given state tex2html_wrap_inline5775 ahead of the wave.

  

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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 tex2html_wrap_inline5559, tex2html_wrap_inline5561, and tex2html_wrap_inline5563 separated by a diaphragm at tex2html_wrap_inline5565 . 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 tex2html_wrap_inline5805

  equation207

with

  equation213

the + / - sign of tex2html_wrap_inline5811 corresponding to S =L / S =R. In the above equation, tex2html_wrap_inline5817 is the sound speed of the state tex2html_wrap_inline5775, and c (p) is given by

equation225

The family of all states tex2html_wrap_inline5791, which can be connected through a shock with a given state tex2html_wrap_inline5775 ahead of the wave, is determined by the shock jump conditions. One obtains

  eqnarray232

where the + / - sign corresponds to S =R / S =L. tex2html_wrap_inline5835 and j (p) denote the shock velocity and the modulus of the mass flux across the shock front, respectively. They are given by

  equation244

and

  equation250

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

  equation258

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

  

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Figure 2: Graphical solution in the p- v plane of the Riemann problems defined by the initial states tex2html_wrap_inline5571, tex2html_wrap_inline5573, tex2html_wrap_inline5575, and tex2html_wrap_inline5577, tex2html_wrap_inline5579, tex2html_wrap_inline5581 with i = 1, 2, 3, 4, where tex2html_wrap_inline5585, tex2html_wrap_inline5587, tex2html_wrap_inline5589, and tex2html_wrap_inline5591, 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 tex2html_wrap_inline5871 and tex2html_wrap_inline5873 are displayed in Fig.  2 in a p - v diagram for a particular set of Riemann problems. Once tex2html_wrap_inline5597 has been obtained, the remaining state quantities and the complete Riemann solution,

equation286

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 [108Jump To The Next Citation Point In The Article].



3 High-Resolution Shock-Capturing Methods2 Special Relativistic Hydrodynamics2.2 SRHD as a hyperbolic

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
Problems/Comments to livrev@aei-potsdam.mpg.de