  The theory of viscosity solutions has been effective for representing and
approximating weak solutions to fully nonlinear Partial Differential Equations
(PDEs) such as the elliptic Monge-Amp\`ere equation. The approximation theory
of Barles-Souganidis [Barles and Souganidis, Asymptotic Anal., 4 (1999)
271-283] requires that numerical schemes be monotone (or elliptic in the sense
of [Oberman, SIAM J. Numer. Anal, 44 (2006) 879-895]. But such schemes have
limited accuracy. In this article, we establish a convergence result for nearly
monotone schemes. 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-Amp\`ere
equation and present computational results on smooth and singular solutions.
