  Relativistic shocks accelerate particles by the first order Fermi mechanism.
These particles then emit synchrotron emission in the post shock gas. We have
developed a numerical code which integrates the relativistic Euler equations
for fluid dynamics with a general equation of state, together with the
Liouville equation for the accelerated particles. We present tests of this code
and, in addition, we use it to study the gamma ray burst afterglow predicted by
the fireball model, along with the hydrodynamics of a relativistic blastwave.
  We find that, while, broadly speaking, the behaviour of the emission is
similar to that already predicted with semi-analytic approaches, the detailed
behaviour is somewhat different. The ``breaks'' in the synchrotron spectrum
behave differently with time, and the spectrum above the final break is harder
than previously expected. These effects are due to the incorporation of the
geometry of the (spherical) blastwave, along with relativistic beaming and
adiabatic cooling of the energetic particles leading to a mix, in the observed
spectrum, between recently injected "uncooled" particles and the older "cooled"
population in different parts of the evolving, inhomogeneous flow.
