Analysis of a semi-Lagrangian method for the spherically symmetric Vlasov-Einstein system

¹ Departamento de Matemática Aplicada Facultad de Ciencias, Universidad de Granada, Avda. Fuentenueva s/n, 18071 Granada, Spain. phbe@ugr.es
² Institut de Mathématiques Elie Cartan & Institut Jean Lamour, Département Physique de la Matière et des Matériaux, Nancy-Université, Université Henri Poincaré, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France. besse@iecn.u-nancy.fr

Received: 14 January 2009
Revised: 7 September 2009

Abstract

We consider the spherically symmetric Vlasov-Einstein system in the case of asymptotically flat spacetimes. From the physical point of view this system of equations can model the formation of a spherical black hole by gravitational collapse or describe the evolution of galaxies and globular clusters. We present high-order numerical schemes based on semi-Lagrangian techniques. The convergence of the solution of the discretized problem to the exact solution is proven and high-order error estimates are supplied. More precisely the metric coefficients converge in L^∞ and the statistical distribution function of the matter and its moments converge in L² with a rate of (Δt² + h^m/Δt), when the exact solution belongs to H^m.