  In this paper, we present and analyze a new finite difference method for
computing three dimensional wave maps into spheres. By introducing the angular
momentum as an auxiliary variable, we recast the governing equation as a first
order system. For this new system, we propose a discretization that conserves
both the energy and the length constraint. The new method is also fast
requiring only $N\log N$ operations at each time step. Our main result is that
the method converges to a weak solution as discretization parameters go to
zero. The paper is concluded by numerical experiments demonstrating convergence
of the method and its ability to predict finite time blow-up.
