Increasingly, numerical simulations are being called upon to answer physical questions that strictly analytic methods have historically either ignored or failed to address. In particular, numerical relativity is beginning to provide answers about dynamical problems lacking the special conditions and symmetries required by the analytic methods. An exceptional example of the success of numerical relativity has been the emergence of the study of nonlinear phenomena occurring at the threshold of black hole formation, black hole critical phenomena.
Here, I study critical phenomena occurring for a self-coupled, complex scalar field in spherical symmetry, namely the harmonic map model. The model contains a free constant $\kappa$ which parameterizes a bifurcation in the stability of the critical solution. The implications of this change in stability are examined in the context of multiply unstable critical solutions. The model also contains two regions in which the non-gravitational, nonlinear effects dominate and the possibility of singularity formation in finite time is discussed. In one of these regions, a new critical solution is found which appears to have a unique structure.
The harmonic map represents the same physical system as a triplet scalar field with a symmetry-breaking (``Mexican hat'') potential in the limit where the coupling goes to infinity. Hence the model contains nontrivial topological issues which are addressed in both the harmonic map model with just the two bosonic degrees of freedom and the triplet scalar field model containing all three degrees of freedom. In particular, the triplet scalar field is modeled in flat space and the dynamics of monopoles and texture collapse are studied. These evolutions show that the collapse of toroidally symmetric textures nucleate monopole-antimonopole pairs, while the collapse of so-called spherically symmetric textures do not.
To answer questions about angular momentum scaling at the threshold of black hole formation and to study the stability of critical solutions to non-spherically symmetric perturbations, a project is undertaken to model axisymmetric gravitational collapse. A significant component of such a project is a method to solve the four elliptic constraint equations, and a solver for that purpose is constructed using multigrid methods. Tests of this solver indicate that it fails to adequately handle the boundary conditions and requires further development. Also: