Departamento de Gravitación y Teoría de Campos

                          Seminarios 2012

  

                          Instituto de Ciencias Nucleares, UNAM
[ ICN-UNAM]

Jueves 17 de mayo, a las 17:00 hrs en la sala de seminarios del ICN

Olivier Sarbach (UMSNH Morelia)

Continuum and Discrete Initial-Boundary-Value Problems and Einstein's Field Equations

Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black hole problem within Einstein's theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity. This talk reviews some of the theory necessary to understand the continuum and discrete initial-boundary value problems arising from hyperbolic partial differential equations and discusses its applications to numerical relativity. In particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, briefly discuss multi-domain high-order discretization methods to solve them, and mention open problems in the field.