Convergent filtered schemes for the Monge-Amp\`ere partial differential equation

TL;DR

This paper introduces a convergent filtered scheme for the Monge-Ampère PDE that achieves higher accuracy for smooth solutions and detects singularities, improving upon traditional monotone schemes.

Contribution

It develops a nearly monotone scheme framework allowing high-order finite difference discretizations for the Monge-Ampère equation, with convergence guarantees.

Findings

Higher accuracy for smooth solutions.

Effective detection of singularities.

Successful computational experiments.

Abstract

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.