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.

翻译：我们引入了一种集中的有限差异方法来解决球体上的最佳运输问题。 这种方法既适用于传统的平方大地测量成本( 在网状生成中产生),也适用于对数成本( 在反射天线设计问题中产生 ) 。 在球体上的每一点,我们用使用测深正方座标在当地近距离平面上绘制的生成的雅各式方程式取代地表PDE 。 离散受蒙古- 安培埃雷方程式最近采用单调方法的启发,但需要进行重大调整,以便正确处理非线性决定因素操作员内部的梯度和赫森术语的混合,以及单线性对数成本功能。 数值结果显示这种方法在涉及正方大地和对数成本功能的众多挑战性问题上取得了成功。