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UBC Theses and Dissertations

A study of numerical techniques for the initial value problem of general relativity Choptuik, Matthew William


Numerical relativity is concerned with the generation of solutions to Einstein's equations by numerical means. In general, the construction of such a spacetime is accomplished in two stages: 1) the determination of initial data which is specified on a single spacelike hypersurface and satisfies four initial value equations, and 2) the evolution of the initial data to generate the spacetime or some portion of it. One of the key problems is the development of efficient algorithms for the solutions of these equations, as they are sufficiently complex to tax the fastest present computers. This thesis presents a comparison of various algorithms for the solution of the initial value equations, concentrating on the recently developed multi-grid method. The specific problem examined has been previously studied by Bowen, Piran and York. Their initial data has been interpreted as representing "snapshots" of three new families of black holes. Three of the four initial value equations possess analytic solutions. The remaining 2-dimensional nonlinear partial differential equation is solved numerically in this thesis using finite difference techniques. The performance of the multi-grid method, with respect to three more well-known methods, is evaluated through numerical experiments. The speed and reliability of the multi-grid algorithm are found to be very good. In addition, the results which had been previously calculated numerically by Piran are essentially reproduced, with the correction of some errors in that work. Possible extensions of the work to more complex initial value problems are also discussed.

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