  In this thesis different numerical methods, as well as applications of the
methods to a number of current problems in relativistic astrophysics, are
presented. In the first part the theoretical foundation and numerical
implementation of a new general relativistic magnetohydrodynamics code is
discussed. A new form of the equations of motion using global coordinates, but
evolving the dynamical variables from the point of view of a local observer is
presented. No assumptions are made about the background metric and the design
is ready to be coupled with methods solving the full Einstein equations. In the
second part of the thesis important results concerning the understanding of
collisionless shocks, obtained from experiments with a relativistic charged
particle code, are presented. Relativistic collisionless shocks are important
in a range of astrophysical objects; in particular in gamma ray burst
afterglows and other relativistic jets. It is shown that a strong small scale,
fluctuating, and predominantly transversal magnetic field is unavoidably
generated by a two-stream instability. The magnetic energy density reaches a
few percent of equipartition. A new acceleration mechanism for electrons in
ion-electron collisionless shocks is proposed. The mechanism is capable of
creating a powerlaw electron distribution in a collisionless shocked region.
The non-thermal acceleration of the electrons is directly related to the ion
current channels generated by the two-stream instability and is local in
nature. Thus the observed radiation field may be tied directly to the local
conditions of the plasma and could be a strong handle on the physical
processes. (abridged)
