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 & Komissarov [55] have considered two different algorithms to solve the local
Riemann problems in SRHD by extending the methods devised
in [53]. 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 &
Komissarov [55], a linearization of system (31
) 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 (p
runs from 1 to the total number of equations of the system).
In both algorithms, the final step involves the computation of the numerical fluxes for the conservation equations
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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 |