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 Figure 1
) appear, which are
separated from each other by a contact discontinuity moving with
the fluid. Accordingly, the time evolution of a Riemann problem can
be represented as
As in the Newtonian case, the compressive
character of shock waves (density and pressure rise across the
shock) allows us to discriminate between shocks () and rarefaction waves (
):
Across the contact discontinuity the density
exhibits a jump, whereas pressure and normal velocity are
continuous (see Figure 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
(
) with the corresponding initial states
. They also allow one to express the normal fluid
flow velocity in the intermediate states (
for the case of
an initial discontinuity normal to the
axis) as a function
of the pressure
in these states.
The solution of the Riemann problem consists in
finding the intermediate states and
, as well as the positions of the waves separating
the four states (which only depend on
,
,
, and
). The functions
and
allow one to determine the
functions
and
, respectively.
The pressure
and the flow velocity
in the intermediate states are then given by the
condition
In the case of relativistic hydrodynamics, the major difference to classical hydrodynamics stems from the role of tangential velocities. While in the classical case the decay of the initial discontinuity does not depend on the tangential velocity (which is constant across shock waves and rarefactions), in relativistic calculations the components of the flow velocity are coupled by the presence of the Lorentz factor in the equations. In addition, the specific enthalpy also couples with the tangential velocities, which becomes important in the thermodynamically ultrarelativistic regime.
The functions are defined
by
Considering that in a Riemann problem the state
ahead of the rarefaction wave is known, the integration of
Equation (19) allows one to connect
the states ahead (
) and behind the rarefaction wave.
Moreover, using Equation (21
), the EOS, and the
following relation obtained from the constraint
, that holds across the rarefaction
wave,
In the limit of zero tangential velocities, , the function
does not contribute. In this
limit and in the case of an ideal gas EOS one has
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
Finally, the tangential velocities in the
post-shock states can be obtained from [234]
Figure 2 shows the solution of
a particular mildly relativistic Riemann problem for different
values of the tangential velocity. The crossing point of any two
lines in the upper panel gives the pressure and the normal velocity
in the intermediate states. The range of possible solutions in the
(
)-plane is marked by the shaded region. While the
pressure in the intermediate state can take any value between
and
, the normal flow velocity can be
arbitrarily close to zero in the case of an extremely relativistic
tangential flow. The values of the tangential velocity in the
states
and
are obtained from the value of the
corresponding functions at
in the lower panel of
Figure 2
. The influence of
initial left and right tangential velocities on the solution of a
Riemann problem is enhanced in highly relativistic problems. We
have computed the solution of one such problem (see Section
6.2.2 below, Problem 2) for different
combinations of
and
. The initial data
are
,
,
;
,
,
, and the 9 possible combinations of
. The results are given in Figure
3
and Table 1, and a complete
discussion can be found in [234
].
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Finally, let us note that the procedure to obtain
the pressure in the intermediate states is valid for
general EOS. Once
has been obtained, the remaining state
quantities and the complete Riemann solution,
Solving a Riemann problem involves the solution
of an algebraic equation for the pressure (Equation (17)). Moreover, the
functional form of this equation depends on the wave pattern under
consideration (see expressions (16
). In a recent
paper [240
], Rezzolla and
Zanotti have presented a procedure, suitable for implementation
into an exact Riemann solver in one dimension, which removes the
ambiguity arising from the wave pattern. That method exploits the
fact that the expression for the relative velocity between the two
initial states is a (monotonic) function of the unknown pressure,
, which determines the wave pattern. Hence, comparing
the value of the (special relativistic) relative velocity between
the initial left and right states with the values of the limiting
relative velocities for the occurrence of the wave
patterns (16
), one can determine a
priori which of the three wave patterns will actually result (see
Figure 4
). In [241] the authors extend the
above procedure to multi-dimensional flows.
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