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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.

A New Angular Momentum Method for Computing Wave Maps into Spheres