appearing in the state equation and the ratio
of the specific heats 
.
Several different variables can be defined and several sets
can be used. The one-dimensional conservative set of variables 
and its
physical flux function 
are
The physical state of the gas can be described by the previous set of
variables as well as by the primitive set 
:
Let us define a temperature 
from the equation of state 
and the sound speed 
from the isentropic definition
. The state of the gas can still
be defined by the set of variables 
:
The initial discontinuity in Riemann problems for Euler equations will evolve in time breaking into leftward and rightward moving waves separated by a contact surface, see Anderson (1982) [12].
These propagating
waves can be either shocks or rarefaction waves depending on the initial
data. The available combinations can produce four different wave patterns
which are self-similar, i.e. they depend only on 
.
There is also a fifth pattern which is essentially of theoretical interest because it is the limit of the perfect gas equations at zero pressure and temperature and real-gas effects prevent its practical realization. This last case is defined by two rarefaction waves propagating rightward and leftward confined by two contact surfaces separated by a vacuum region in the middle.
A complete solution of the one-dimensional Riemann problem
requires the determination of the types of the waves, their relative
strenghts, and the flow properties in each region between the waves and
the contact surface. In the formulation of Gottlieb and Groth (1978)
[7],
the state of the gas is described by the set of variables 
.
The unknowns are obtained solving the three conservation laws of
mass, momentum and energy and the equation of state, which reduces
to the well-known Rankine-Hugoniot relations across shocks and to
the well-known isentropic characteristic equations across rarefaction
waves. The flow-state is described by the initial left and right states
and by the six state variables 
and
characterizing the solution
on the two sides of the contact surface. The known left state
is connected to the unknown state
by the non-linear algebraic
relations describing the leftward moving wave. In the case of a
compression wave (or a shock), i.e. 
, the
shock-relations are employed; in the case of a rarefaction wave,
i.e. 
, rarefaction-waves relations are employed.
The right state 
is similarly
connected to the unknown state 
to the right of the contact surface. Finally, across the contact surface,
the following relations hold:
Appling this relations, the set of equations describing the Riemann
problem may be reduced to a single non-linear algebraic equation in one
unknown, the common flow velocity in the intermediate states 
,
for any particular wave pattern. This equation is generally
implicit in the unknown 
or 
and an iterative solution
is provided by a standard Newton iterative method.
The initial guess 
and 
is provided by considering
compression waves instead of
shocks. This solution is exactly the same as for the pattern where
two isentropic rarefaction waves separated by a contact discontinuity
are present. The values of 
and 
are given by
the following relations:
where 
is a useful ratio of pressures and sound speeds and
are the Riemann invariants for left- and right-ward
moving waves:
