Numerical general relativity has typically been used in studying 1+1 dimensional spherically symmetric spacetimes, and more recently, 3+1 dimensional spacetimes without any symmetries. In this thesis, an intermediate case - 2+1+1-dimensional axisymmetric spacetimes - are studied. In the first part, the equations of general relativity are developed for a general axisymmetric spacetime. This uses the Geroch reduction, and this is generalised to find evolution equations for the matter terms. The conditions for regularity on the symmetry axis are derived for several types of tensor, and this is used to help define appropriate variables for numerical evolution. The characteristic structure of both the geometry and the matter evolution systems is given. Next, the numerical methods used to solve the equations are described. The elliptic constraint equations are solved using multigrid, and the hyperbolic evolution equations are evolved using High Resolution Shock Capturing Methods, using WENO-3 and the Maquina flux solver. The iterative Crank Nicholson method is also described as an alternative method of evolving the geometry equations. The third part of the thesis considers a 1+1-dimensional plane-symmetric spacetime, with perfect fluid matter. Several versions of the equations, with different characteristic structures, are derived. A linearised version of the equations shows that discontinuities may be present in the second derivatives of metric terms, and this is backed up by a variety of numerical results evolved using the non-linear equations. These numerical results also show that the numerical methods used to evolve the geometry equations must be able to deal with discontinuities if the expected order of accuracy is to be maintained. Finally, results from the axisymmetric code are presented. In vacuum, Brill waves are briefly studied, as well as some test problems. With a perfect fluid, the standard shocktube problem is used to test the code, and perturbed, rotating neutron stars are studied.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:596387 |
| Date | January 2004 |
| Creators | Barnes, A. |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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