After the appropriate choice of the state vector variables,
the conservation laws, Eqs. (7) and (8
), are re-written in flux-conservative form. The flow variables
are expressed in terms of a parameter vector
as
where
,
and
. The vector
represents the state vector (the unknowns), and each vector
is the corresponding flux in the coordinate direction
.
Eulderink and Mellema computed the appropriate ``Roe
matrix'' [183] for the vector (41
) and obtained the corresponding spectral decomposition. The
characteristic information is used to solve the system
numerically using Roe's generalized approximate Riemann solver.
Roe's linearization can be expressed in terms of the average
state
, where L and R denote the left and right states in a Riemann
problem (see Section
3.2). Further technical details can be found in [62,
64
].
The performance of this general relativistic Roe solver was tested in a number of one-dimensional problems for which an exact solution is known, including non-relativistic shock tubes, special relativistic shock tubes and spherical accretion of dust and a perfect fluid onto a (static) Schwarzschild black hole. In its special relativistic version it has been used in the study of the confinement properties of relativistic jets [63]. No astrophysical applications in strong-field general relativistic flows have yet been attempted with this formulation.
Note that these variables slightly differ from previous
choices (see, e.g., Eqs. (24), (33
), (34
), (35
) and (41
)). With those definitions the equations take the standard
conservation law form,
with
A
=(0,
i,4). The flux vectors
and the source terms
(which depend only on the metric, its derivatives and the
undifferentiated stress energy tensor), are given by
The state of the fluid is uniquely described using either
vector of variables, i.e. either
or
, and each one can be obtained from the other via the
definitions (42
,
43
,
44
) and the use of the normalization condition for the 4-velocity,
.
The local characteristic structure of these equations has been
presented in [171]. The formulation has proved well suited for the numerical
implementation of HRSC schemes. A comprehensive numerical study
of this approach was also presented in [171
], where it was applied to simulate one-dimensional relativistic
flows on null spacetime foliations. The demonstrations performed
include standard shock tube tests in Minkowski spacetime, perfect
fluid accretion onto a Schwarzschild black hole using ingoing
null Eddington-Finkelstein coordinates, and dynamical spacetime
evolutions of polytropes (i.e. stellar models satisfying the
Tolman-Oppenheimer-Volkoff equilibrium equations) sliced along
the radial null cones, and accretion of self-gravitating matter
onto a central black hole.
Procedures for integrating various forms of the hydrodynamic
equations on null hypersurfaces have been presented before
in [103] (see [28] for a recent implementation). This approach is geared towards
smooth isentropic flows. A Lagrangian method, applicable in
spherical symmetry, has been presented by [139]. Recent work in [58] includes a Eulerian non-conservative formulation for general
fluids in null hypersurfaces and spherical symmetry, including
their
matching
to a spacelike section.
A technical remark must be included here: In all conservative
formulations reviewed in Sections
2.1.3,
2.2.1,
2.2.2, the time-update of the numerical algorithm is applied to the
conserved quantities
. After the update the vector of primitive quantities must be
reevaluated, as those are needed in the Riemann solver (see
Section
3.1.2). The relation between the two sets of variables is not in
closed form and, hence, the update of the primitive variables is
done using a root-finding procedure, typically a Newton-Raphson
algorithm. This feature may lead to accuracy losses in regions of
low density and small speeds, apart from being computationally
inefficient. Specific details on this issue can be found
in [17
,
64
,
171
]. We note that the covariant formulation discussed in this
section, when applied to null spacetime foliations, allows for an
explicit recovery of the primitive variables, as a consequence of
the particular form of the Bondi-Sachs metric. We end by pointing
out that the formulation presented in this section has been
developed for a perfect fluid EOS. Extensions to account for
generic EOS, as well as a comprehensive analysis of general
relativistic hydrodynamics in conservation form, have been
recently presented in [168].
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Numerical Hydrodynamics in General Relativity
José A. Font http://www.livingreviews.org/lrr-2000-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |