  This paper introduces a new effort to study the collision of plane-fronted
gravitational waves in four dimensional, asymptotically flat spacetime, using
numerical solutions of the Einstein equations. The pure vacuum problem requires
singular, Aichelburg-Sexl type sources to achieve finite energy solutions,
which are problematic to treat both mathematically and numerically. Instead
then, we use null (massless) particles to source non-trivial geometry within
the initial wave fronts. The main purposes of this paper are to (a) motivate
the problem, (b) introduce methods for numerically solving the Einstein
equations coupled to distributions of collisionless massless or massive
particles, and (c) present a first result on the formation of black holes in
the head-on collision of axisymmetric distributions of null particles.
Regarding the last-named, initial conditions are chosen so that a black hole
forms promptly, with essentially no matter escaping the collision. This can be
interpreted as approaching the ultra-relativistic collision problem from within
an infinite boost limit, but where the matter distribution is spread out, and
thus non-singular. We find results that are consistent with earlier
perturbative calculations of the collision of Aichelburg-Sexl singularities, as
well as numerical studies of the high-speed collision of boson stars, black
holes, and fluid stars: a black hole is formed containing most of the energy of
the spacetime, with the remaining $15\pm1\%$ of the initial energy radiated
away as gravitational waves. The methods developed here could be relevant for
other problems in strong field gravity and cosmology that involve particle
distributions of matter.
