Numerical GR Overview/Abstract:
From Einstein's theory we know that besides the electromagnetic
spectrum,
objects like quasars, active galactic nuclei, pulsars and black holes
also
generate a physical signal of purely gravitational nature. The actual
form of
the signal is impossible to determine analytically, which lead to the use
of numerical
methods.
Two major approaches emerged. The first one formulates
the gravitational
radiation problem as a standard Cauchy initial value problem, while
the other
approach uses a Characteristic Initial value formulation. In the strong
field
region, where caustics in the wavefronts are likely to form, the Cauchy
formulation is more advantageous. On the other side, the Characteristic
formulation is uniquely suited to study radiation problems because
it describes
space-time in terms of radiation wavefronts.
The fact that the advantages and disadvantages of
these two systems are
complementary suggests that one may want to use the two of them together.
In a
full nonlinear problem it would be advantageous to evolve the
inner (strong
field) region using Cauchy evolution and the outer (radiation) region
with the
Characteristic approach. Cauchy Characteristic Matching enables one
to evolve
the whole space-time matching the boundaries of Cauchy and Characteristic
evolution. The methodology of Cauchy Characteristic Matching has been
successful
in numerical evolution of the spherically symmetric Klein-Gordon-Einstein
field
equations as well as for 3-D non-linear wave equations. Our numerical
relativity group at the University of Pittsburgh is currently investigating
the same methodology in the context of 3-D linearized gravity, using
harmonic
coordinates.
The most involved aspect of the implementation of
a Matching module is that
of numerical stability. One can think of the update algorithm from
one time-level
to the next as a matrix multiplication, where data on the old and the
new
time-levels as well as the update algorithm is represented by matrices.
If the
update-algorithm-matrix has any eigen-value with magnitude larger then
one, an
exponentially growing mode is induced by the numerical evolution. This
can happen
even if the original (analytic) equation does not allow for such solutions.
For
simple cases this matrix takes a simple form with most elements vanishing.
Thus,
in these simple cases, one can carry out the stability analysis of
the
numerical algorithm without ever implementing it. However, in
our case,
the number of coupled PDE-s involved, as well as the cross-grid interpolation
algorithms make this analysis difficult. What remains is to "try and
see," i.e.,
test a number of different approaches. Having in view that such a test
consists
of running the code for up to the order of 10^7 time-steps on 3-D grids
of up to
48^3 points, the need of access to high-performance computing power
becomes clear.
Tempest is now providing us with the computational resources
our research demands
and has reduced our reliance on outside supercomputer facilities.
--Bela Szilagyi
Pittsburgh Numerical Relativity Group -
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