Browsing by Subject "Evolution equations"

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Analysis of geometric flows, with applications to optimal homogeneous geometries
(2011-05) Williams, Michael Bradford; Knopf, Dan, 1959-; Freed, Daniel S.; Neitzke, Andrew; Sadun, Lorenzo; Uhlenbeck, Karen
This dissertation considers several problems related to Ricci flow, including the existence and behavior of solutions. The first goal is to obtain explicit, coordinate-based descriptions of Ricci flow solutions--especially those corresponding to Ricci solitons--on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott's blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz. Next, we consider solsolitons involving the nilsolitons in the Heisenberg case above. This uses work of Lauret, which characterizes solsolitons as certain extensions of nilsolitons, and work of Will, which demonstrates that the space of solsolitons extensions of a given nilsoliton is parametrized by the quotient of a Grassmannian by a finite group. We determine these spaces of solsoliton extensions of Heisenberg nilsolitons, and we also explicitly describe many-parameter families of these solsolitons in dimensions greater than three. Finally, we explore Ricci flow coupled with harmonic map flow, both as it arises naturally in certain bundle constructions related to Ricci flow and as a geometric flow in its own right. In the first case, we generalize a theorem of Knopf that demonstrates convergence and stability of certain locally R[superscript N]-invariant Ricci flow solutions. In the second case, we prove a version of Hamilton's compactness theorem for the coupled flow, and then generalize it to the category of etale Riemannian groupoids. We also provide a detailed example of solutions to the flow on the three-dimensional Heisenberg group.
Constrained evolution in numerical relativity
(2004) Anderson, Matthew William; Matzner, Richard A. (Richard Alfred), 1942-
The strongest potential source of gravitational radiation for current and future detectors is the merger of binary black holes. Full numerical simulation of such mergers can provide realistic signal predictions and enhance the probability of detection. Numerical simulation of the Einstein equations, however, is fraught with difficulty. Stability even in static test cases of single black holes has proven elusive. Common to unstable simulations is the growth of constraint violations. This work examines the effect of controlling the growth of constraint violations by solving the constraints periodically during a simulation, an approach called constrained evolution. The effects of constrained evolution are contrasted with the results of unconstrained evolution, evolution where the constraints are not solved during the course of a simulation. Two different formulations of the Einstein equations are examined: the standard ADM formulation and the generalized Frittelli-Reula formulation. In most cases constrained evolution vastly improves the stability of a simulation at minimal computational cost when compared with unconstrained evolution. However, in the more demanding test cases examined, constrained evolution fails to produce simulations with long-term stability in spite of producing improvements in simulation lifetime when compared with unconstrained evolution. Constrained evolution is also examined in conjunction with a wide variety of promising numerical techniques, including mesh refinement and overlapping Cartesian and spherical computational grids. Constrained evolution in boosted black hole spacetimes is investigated using overlapping grids. Constrained evolution proves to be central to the host of innovations required in carrying out such intensive simulations.