4.3 Analytic-Numerical Matching for Waves4 CCM4.1 Computational Boundaries

4.2 Perturbative CCM

In numerous purely analytic applications outside of general relativity, matching techniques have successfully cured pathologies in perturbative expansions [93]. Matching is a strategy for obtaining a global solution by patching together solutions obtained using different coordinate systems for different regions. By adopting each coordinate system to a length scale appropriate to its domain, a globally convergent perturbation expansion is sometimes possible in cases where a single coordinate system would fail. Burke showed that matching could be used to eliminate some of the divergences arising in perturbative calculations of gravitational radiation [94]. Kates and Kegles further showed that the use of a null coordinate system for the exterior region is essential in the perturbation expansions of curved space radiation fields [95]. They investigated the perturbative description of a scalar field on a Schwarzschild background, in which case the asymptotic behavior of the Schwarzschild light cones differs drastically from that of the artificial Minkowski light cones used in the perturbative expansions based upon the flat space Green function. Use of the Minkowski light cones leads to nonuniformities in the expansion of the radiation fields which are eliminated by the use of exterior coordinates based upon the true light cones. Kates, Anderson, Kegles and Madonna extended this work to the fully general relativistic case and reached the same conclusion [96].

Anderson later applied this approach to the slow motion approximation of a binary system and obtained a derivation of the radiation reaction effect on the orbital period which avoided some objections concerning earlier approaches [97]. The use of the true light cones was also essential in formulating as a mathematical theorem that the Bondi news function satisfies the Einstein quadrupole formula to leading order in a Newtonian limit [98]. Although questions of mathematical consistency still remain in the perturbative treatment of gravitational radiation, it is clear that the use of characteristic methods pushes these problems to a higher perturbative order. Characteristic techniques are used in present day codes for the Zerilli equation [99].



4.3 Analytic-Numerical Matching for Waves4 CCM4.1 Computational Boundaries

image Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-1998-5
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