Doctor of Philosophy (PhD)
Physics and Astronomy
The successful construction of long time convergent finite difference schemes approximating highly gravitating systems in general relativity remains an elusive task. The presence of constraints and the introduction of artificial time-like boundaries contribute significantly to the difficulty of this problem. Whereas in the absence of boundaries the Bianchi identities ensure that the constraints vanish during evolution provided that they are satisfied initially, this is no longer true when time-like boundaries are introduced. In this work we consider the linearization around the Minkowski space-time in Cartesian coordinates of the generalized Einstein-Christoffel system and analyze different kinds of boundary conditions that are designed to ensure that the constraints vanish throughout the computational domain: the Neumann, Dirichlet, and Sommerfeld cases. In addition to the situation in which the boundary is aligned with a coordinate surface, we examine the presence of corners in the computational domain. We find that, at a corner, there are compatibility conditions which the boundary data and its derivatives must satisfy and that, in general, achieving consistency of a finite difference scheme can be troublesome. We present several numerical experiments aimed at establishing or confirming the well-posedness or ill-posedness of a problem and the consistency of the numerical boundary conditions at the corners. In the case of a smooth boundary we are able to find stable discretizations for all three cases. However, when a corner is present no stable discretization was found for the Sommerfeld case. Finally, we propose an alternative implementation of the Sommerfeld boundary conditions that would preserve the constraints, offer a good approximation for absorbing boundary conditions, and eliminate the problem of the corners.
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Calabrese, Gioel, "Constraint preserving boundary conditions for the linearized Einstein equations" (2003). LSU Doctoral Dissertations. 346.