Such a coordinate transformation to locally Minkowskian coordinates at each numerical interface assumes that the solution of the Riemann problem is the one in special relativity and planar symmetry. This last assumption is equivalent to the approach followed in classical fluid dynamics, when using the solution of Riemann problems in slab symmetry for problems in cylindrical or spherical coordinates, which breaks down near the singular points (e.g., the polar axis in cylindrical coordinates). In analogy to classical fluid dynamics, the numerical error depends on the magnitude of the Christoffel symbols, which might be large whenever huge gradients or large temporal variations of the gravitational field are present. Finer grids and improved time advancing methods will be required in those circumstances.
Following [229], we illustrate the procedure for computing the second flux
integral in Equation (45
), which we call
. We begin by expressing the integral on a basis
with
and
forming an orthonormal basis in the plane orthogonal to
with
normal to the surface
and
and
tangent to that surface. The vectors of this basis verify with
the Minkowski metric (in the following, caret subscripts will
refer to vector components in this basis).
Denoting by
the coordinates at the center of the interface at time
t, we introduce the following locally Minkowskian coordinate
system:
, where the matrix
is given by
, calculated at
. In this system of coordinates the equations of general
relativistic hydrodynamics transform into the equations of
special relativistic hydrodynamics in Cartesian coordinates, but
with non-zero sources, and the flux integral reads
(the caret symbol representing the numerical flux in
Equation (45) is now removed to avoid confusion) with
, where we have taken into account that, in the coordinates
,
is described by the equation
(with
), where the metric elements
and
are calculated at
. Therefore, this surface is not at rest but moves with
speed
.
At this point, all the theoretical work developed in recent
years on special relativistic Riemann solvers can be exploited.
The quantity in parentheses in Equation (64) represents the numerical flux across
, which can now be calculated by solving the special relativistic
Riemann problem defined with the values at the two sides of
of two independent thermodynamical variables (namely, the rest
mass density
and the specific internal energy
) and the components of the velocity in the orthonormal spatial
basis
(
).
Once the Riemann problem has been solved, we can take
advantage of the self-similar character of the solution of the
Riemann problem, which makes it constant on the surface
, simplifying the calculation of the above integral
enormously:
where the superscript (*) stands for the value on
obtained from the solution of the Riemann problem. Notice that
the numerical fluxes correspond to the vector fields
,
,
,
,
and linearized Riemann solvers provide the numerical fluxes as
defined in Equation (64
). Thus, the additional relation
has to be used for the momentum equations. The integral in the
right hand side of Equation (65
) is the area of the surface
and can be expressed in terms of the original coordinates as
which can be evaluated for a given metric. The interested reader is addressed to [229] for details on the testing and calibration of this procedure.
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Numerical Hydrodynamics in General Relativity
José A. Font http://www.livingreviews.org/lrr-2003-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |