Convergent Filtered Schemes for the Monge--Ampère Partial Differential Equation

The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear partial differential equations such as the elliptic Monge--Ampère equation. The approximation theory of Barles and Souganidis [Asymptotic Anal., 4 (1991), pp. 271--283] requires that numerical schemes be monotone (or elliptic in the sense of [A. M. Oberman, SIAM J. Numer. Anal., 44 (2006), pp. 879--895]). But such schemes have limited accuracy. In this article, we establish a convergence result for filtered schemes, which are nearly monotone. This allows us to construct finite difference discretizations of arbitrarily high-order. We demonstrate that the higher accuracy is achieved when solutions are sufficiently smooth. In addition, the filtered scheme provides a natural detection principle for singularities. We employ this framework to construct a formally second-order scheme for the Monge--Ampère equation and present computational results on smooth and singular solutions.