Algorithms for the Gravitational Characteristic Initial Value Problem

Details Algorithms for the Gravitational Characteristic Initial Value Problem Algorithms for the Gravitational Characteristic Initial Value Problem

Jan 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994phdt........31p&link_type=abstract

Thesis (PH.D.)--UNIVERSITY OF PITTSBURGH, 1994.Source: Dissertation Abstracts International, Volume: 56-01, Section: B, page: 02

Computer Science

2

Initial Value Problem, Discretization

Scientific paper

The existence of gravitational waves has been postulated soon after the formulation of General Relativity (GR). Their detection is uncertain till today, but the subject acquired renewed momentum with the development of laser interferometric detectors. Those experimental advances promise enhanced detection possibilities by the turn of the century. Consequently, the theoretical relativity community embarked into establishing well-calibrated and dependable methods for wave-form prediction. Numerical relativity represents one such method, which is capable of handling the strongly non-linear sector of the theory. In numerical investigations, the Cauchy initial value formulation (CIVP) has typically provided the underlying kinematic framework. The alternative, characteristic, formulation of GR (cIVP) has been used in the past, with great success, for the analytic study of gravitational radiation. It seems, thus, that there are considerable benefits in the development of numerical algorithms for the cIVP. This dissertation presents a set of algorithms that explore, numerically, the axisymmetric vacuum cIVP. As a first crucial step, the linearized problem is investigated, both numerically and analytically. This leads to a rigorous analysis of the algorithmic stability in this regime, and provides the clues for the correct discretization structures. Next, the algorithm for the non-linear Bondi problem is completed. The testing and calibration of the algorithm is extensive, and covers stability and convergence in the case of space -times without black holes. Thc accuracy diagnostics include exact solutions and conservation of energy. The extension of the numerical method to a general three-dimensional cIVP requires careful reexamination of the coordinate choices. In order to satisfy the regularity requirements of the numerical method, a discrete version of the eth formalism is introduced here. This is done in conjunction with the introduction of a complex stereographic coordinate atlas for finite difference methods. This new discretization formalism is tested in various contexts.

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