Monotone Mixed Narrow/Wide Stencil Finite Difference Scheme for Monge-Amp\`ere Equation

In this paper, we propose a monotone mixed narrow/wide stencil finite difference scheme for solving the two-dimensional Monge-Amp\`ere equation. In order to accomplish this, we convert the Monge-Amp\`ere equation to an equivalent Hamilton-Jacobi-Bellman (HJB) equation, which is numerically more manageable. Based on the HJB formulation, we apply narrow stencil discretization, which is second order accurate, to the grid points wherever monotonicity holds, and apply wide stencil discretization elsewhere to ensure monotonicity on the entire computational domain. To solve the discretized system, policy iteration is implemented. By dividing the admissible control set into six regions and optimizing the sub-problem in each region, the computational cost of the optimization problem at each grid point in one policy iteration is reduced from $O(M^2)$ to $O(M)$, where the control grid is $M\times M$. We prove that our numerical scheme satisfies consistency, stability and monotonicity, and hence is convergent to the viscosity solution of the Monge-Amp\`ere equation. In the numerical results, second order convergence rate is achieved for smooth solutions and up to order one convergence is achieved for non-smooth solutions.