3.1 One+One Dimensional CodesCharacteristic Evolution and Matching2 The Characteristic Initial Value

3 Characteristic Evolution Codes

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 with the particular choice. However, all characteristic evolution codes have certain common advantages:

(i) There are no elliptic constraint equations. This eliminates the need for time consuming iterative methods to enforce constraints or to otherwise test the challenge of constraint free evolution.

(ii) No second time derivatives appear so that the number of basic variables is half the number for the corresponding version of the Cauchy problem.

(iii) The main Einstein equations form a system of coupled ordinary differential equations with respect to the parameter tex2html_wrap_inline1523 varying along the characteristics. This allows construction of an evolution algorithm in terms of a simple march along the characteristics.

(iv) 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.

(v) 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) with no need to propagate it across the grid.

(vi) In black hole space-times, 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 space-time.

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, 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 Pade approximants. The structural stability of the singularities should in principle make this possible, and algorithms to evolve the elementary caustics have been proposed [19Jump To The Next Citation Point In The Article, 20]. 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 [21, 22].

3.1 One+One Dimensional CodesCharacteristic Evolution and Matching2 The Characteristic Initial Value

image Characteristic Evolution and Matching
Jeffrey Winicour
© Max-Planck-Gesellschaft. ISSN 1433-8351
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