Gravitational Entropy in Cosmological Models

We discuss whether an appropriately defined dimensionless scalar function might be an acceptable candidate for the gravitational entropy, by explicitly considering Szekeres and Bianchi type VI _h models that admit an isotropic singularity. We also briefly discuss other possible gravitational entropy functions, including an appropriate measure of the velocity dependent Bel-Robinson tensor. PACS: 04.20.-q 95.30.Sf 98.80.Jk     

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers. Received August 5, 2000 / Published online June 20, 2001     

Yang-Mills cosmologies and collapsing gravitational sphalerons

Cosmological solutions with a homogeneous Yang-Mills field which oscillates and passes between topologically distinct vacua are discussed. These solutions are used to model the collapsing Bartnik-McKinnon gravitational sphaleron and the associated anomalous production of fermions. The Dirac equation is analyzed in these backgrounds. It is shown explicity that a fermion energy level crosses from the negative to positive energy spectrum as the gauge field evolves between the topologically distinct vacua. The cosmological solutions are also generalized to include an axion field.