

3.5 Falle and Komissarov
upwind scheme
Instead of starting from the conservative form of the hydrodynamic
equations, one can use a primitive variable formulation in
quasi-linear form,
where
is any set of primitive variables. A local
linearization of the above system allows one to obtain the solution
of the Riemann problem, and from this the numerical fluxes needed
to advance a conserved version of the equations in time.
Falle and Komissarov [89
] have considered two
different algorithms to solve the local Riemann problems in SRHD by
extending the methods devised in [87]. In a first algorithm,
the intermediate states of the Riemann problem at both sides of the
contact discontinuity,
and
, are obtained by solving the system
where
is the right eigenvector of
associated with sound waves moving upstream, and
is the right eigenvector of
of sound waves moving downstream. The continuity of
pressure and of the normal component of the velocity across the
contact discontinuity allows one to obtain the wave strengths
and
from the above expressions, and hence
the linear approximation to the intermediate state
.
In the second algorithm proposed by Falle and
Komissarov [89
], a linearization of
system (41) is obtained by
constructing a constant matrix
. The solution of the corresponding Riemann problem
is that of a linear system with matrix
, i.e.,
or, equivalently,
with
where
,
, and
are the
eigenvalues and the right and left eigenvectors of
, respectively (
runs from 1 to the number of
equations of the system).
In both algorithms, the final step involves the
computation of the numerical fluxes for the conservation
equations,

