Convergence of a particle-in-cell scheme for the spherically symmetric   Vlasov-Einstein system
  We consider spherically symmetric, asymptotically flat space-times with a collisionless gas as matter model. Many properties of the resulting Vlasov-Einstein system are not yet accessible by purely analytical means. We present a discretized version of this system which is suitable for numerical implementation and is based on the particle-in-cell technique. Convergence of the resulting approximate solutions to the exact solution is proven and error bounds are provided. 
Gerhard Rein, Thomas Rodewis