The construction of boundary conditions for electromagnetic analogues of formulations used in numerical relativity

Abstract

This thesis investigates the construction of maximally dissipative and constraint preserving boundary conditions for electromagnetic analogues of two formulations presently used in numerical relativity. Accurate simulations of astrophysical sit uations either require correct boundaries to be applied or for these boundaries to be pushed out far enough that they are not causally connected to the region of interest. This work looks to tackle the first problem, considering electromag netism in an attempt to construct general concepts that would transfer directly to formulations of the Einstein equations used in numerical relativity. The early sections of the thesis introduce in a general way the requirements for continuum problems and their discretisations and the properties of maximally dissipative and constraint preserving boundary conditions. Consideration is then made of the advantages and limitations of using formulations of the Maxvvell equations as analogues to formulations used in numerical relativity, before the introduction of the two formulations: K\iVB and Zl, that will be considered in this thesis. The basic examples of the first order in space and second order in space wave equation, with and without shift, are used to help construct the boundary condi tions for K\iVB and Zl. Where possible the energy method is used to analytically prove stability for the resulting schemes, which are then tested numerically.

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