UBC Theses and Dissertations
A study of numerical techniques for radiative problems in general relativity Choptuik, Matthew William
There is continuing interest in the numerical solution of the field equations of general relativity. Many of the most interesting scenarios involve the generation and propagation of gravitational radiation—this thesis is an investigation of numerical techniques which should be useful for such problems. The study is based on a model system of a single massless scalar field coupled to the gravitational field, with the additional requirement of spherical symmetry. The 3+1 formalism for numerical relativity is used and attention is limited to finite difference techniques. As a preliminary step, the properties of some difference schemes are examined in the context of the wave equation in general coordinates. The accuracy, stability, convergence, dissipation, dispersion and efficiency of the schemes are discussed and three of the methods, which are all centred, second order approximations, are identified as being suitable for radiation problems in general relativity. Following work by Piran, the model problem is solved in three different coordinate systems using variations of the schemes devised for the wave equation. Various initial configurations of the scalar field are evolved, ranging from cases where there is a negligible amount of gravitational self-interaction to those where black holes are formed. Although some numerical difficulties remain with the latter calculations, the basic difference techniques developed for the treatment of the radiative degrees of freedom are found to be satisfactory. In the past it has been argued that there is a fundamental inconsistency in finite difference solutions of Einstein's equations which are generated by algorithms that do not continuously impose the constraint equations which arise from the coordinate freedom of general relativity. The deviations of freely evolved quantities from discrete versions of the constraint equations are supposedly of lower order (in the mesh spacing) than the underlying difference scheme. An argument based on Richardson's hypothesis concerning the expansion of the error of a difference solution in powers of the mesh spacing is made which suggests that there should be no such inconsistency. Numerical results from two fully second order algorithms are presented in support of this argument. The remainder of the thesis deals with the application of local mesh refinement techniques to the model problem. A simplified version of a recent algorithm due to Berger and Oliger is implemented for one particular coordinate system and is used to examine the nature of the scalar field's gravitational self-interaction.
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