

Computational implementation of characteristic evolution may be
based upon different versions of the formalism (i.e. metric or
tetrad) and different versions of the initial value problem (i.e.
double null or worldtube-nullcone). The performance and
computational requirements of the resulting evolution codes can
vary drastically. However, most characteristic evolution codes
share certain common advantages:
- There are no elliptic constraint equations. This eliminates
the need for time consuming iterative methods to enforce
constraints.
- No second time derivatives appear so that the number of
basic variables is at least half the number for the
corresponding version of the Cauchy problem.
- The main Einstein equations form a system of coupled
ordinary differential equations with respect to the parameter
varying along the characteristics. This allows construction of
an evolution algorithm in terms of a simple march along the
characteristics.
- In problems with isolated sources, the radiation zone can
be compactified into a finite grid boundary using Penrose's
conformal technique. Because the Penrose boundary is a null
hypersurface, no extraneous outgoing radiation condition or
other artificial boundary condition is required.
- The grid domain is exactly the region in which waves
propagate, which is ideally efficient for radiation studies.
Since each null hypersurface of the foliation extends to
infinity, the radiation is calculated immediately (in retarded
time).
- In black hole spacetimes, a large redshift at null infinity
relative to internal sources is an indication of the formation
of an event horizon and can be used to limit the evolution to
the exterior region of spacetime.
Characteristic schemes also share as a common disadvantage the
necessity either to deal with caustics or to avoid them
altogether. The scheme to tackle the caustics head on by
including their development as part of the evolution is perhaps a
great idea still ahead of its time but one that should not be
forgotten. There are only a handful of structurally stable
caustics, and they have well known algebraic properties. This
makes it possible to model their singular structure in terms of
Padé approximants. The structural stability of the singularities
should in principle make this possible, and algorithms to evolve
the elementary caustics have been proposed [47
,
134]. In the axisymmetric case, cusps and folds are the only stable
caustics, and they have already been identified in the horizon
formation occurring in simulations of head-on collisions of black
holes and in the temporarily toroidal horizons occurring in
collapse of rotating matter [104,
127]. In a generic binary black hole horizon, where axisymmetry is
broken, there is a closed curve of cusps which bounds the
two-dimensional region on the horizon where the black holes
initially form and merge [100
,
89
].


|
Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-2001-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to
livrev@aei-potsdam.mpg.de
|