  We consider a PDE approach to numerically solving the optimal transportation
problem on the sphere. We focus on both the traditional squared geodesic cost
and a logarithmic cost, which arises in the reflector antenna design problem.
At each point on the sphere, we replace the surface PDE with a generalized
Monge-Amp\`ere type equation posed on the tangent plane using normal
coordinates. The resulting nonlinear PDE can then be approximated by any
consistent, monotone scheme for generalized Monge-Amp\`ere type equations on
the plane. Existing techniques for proving convergence do not immediately apply
because the PDE lacks both a comparison principle and a unique solution, which
makes it difficult to produce a stable, well-posed scheme. By augmenting this
discretization with an additional term that constrains the solution gradient,
we obtain a strong form of stability. A modification of the Barles-Souganidis
convergence framework then establishes convergence to the mean-zero solution of
the original PDE.
