A CONVERGENT QUADRATURE-BASED METHOD FOR THE MONGE–AMPÈRE EQUATION

Document Type

Article

Publication Date

1-1-2023

Abstract

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 the Monge–Ampère equation with either Dirichlet or optimal transport boundary conditions. The use of higher-order quadrature schemes allows for substantial reduction in the component of the error that depends on the angular resolution of the finite difference stencil. This, in turn, allows for significant improvements in both stencil width and formal truncation error. The resulting schemes can achieve a formal accuracy that is arbitrarily close to O(h2), which is the optimal consistency order for monotone approximations of second-order operators. We present three different implementations of this method. The first two exploit the spectral accuracy of the trapezoid rule on uniform angular discretizations to allow for computation on a nearest-neighbors finite difference stencil over a large range of grid refinements. The third uses higher-order quadrature to produce superlinear convergence while simultaneously utilizing narrower stencils than other monotone methods. Computational results are presented in two dimensions for problems of various regularity.

Identifier

85163696578 (Scopus)

Publication Title

SIAM Journal on Scientific Computing

External Full Text Location

https://doi.org/10.1137/22M1494658

e-ISSN

10957197

ISSN

10648275

First Page

A1097

Last Page

A1124

Issue

3

Volume

45

Grant

DMS-1751996

Fund Ref

National Science Foundation

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