  We introduce a convergent finite difference method for solving the optimal
transportation problem on the sphere. The method applies to both the
traditional squared geodesic cost (arising in mesh generation) and a
logarithmic cost (arising in the reflector antenna design problem). At each
point on the sphere, we replace the surface PDE with a Generated Jacobian
equation posed on the local tangent plane using geodesic normal coordinates.
The discretization is inspired by recent monotone methods for the
Monge-Amp\`ere equation, but requires significant adaptations in order to
correctly handle the mix of gradient and Hessian terms appearing inside the
nonlinear determinant operator, as well as the singular logarithmic cost
function. Numerical results demonstrate the success of this method on a wide
range of challenging problems involving both the squared geodesic and the
logarithmic cost functions.
