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.
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.
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].
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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 |