A Convergent Quadrature-Based Method for the Monge–Ampère Equation

We introduce an integral representation of the Monge–Ampère equation, which leads to a new finite difference method based upon numerical quadrature. The resulting scheme is monotone and fits immediately into existing convergence proofs for equation with either Dirichlet or optimal transport boundary conditions. use higher-order quadrature schemes allows substantial reduction in component error that depends on angular resolution stencil. This, turn, significant improvements both stencil width formal truncation error. can achieve accuracy arbitrarily close , consistency order approximations second-order operators. present three different implementations this method. first two exploit spectral trapezoid rule uniform discretizations allow computation nearest-neighbors over large range grid refinements. third uses produce superlinear while simultaneously utilizing narrower stencils than other methods. Computational results are presented dimensions problems various regularity.