A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

TL;DR

This paper introduces a unified framework for designing high-order convergent numerical methods for fully nonlinear second order PDEs, enabling the use of narrow stencils and unstructured grids for improved flexibility.

Contribution

It presents a novel framework based on consistency and g-monotonicity that allows for high-order methods with narrow stencils, expanding beyond traditional monotone schemes.

Findings

Framework ensures convergence and stability.

Methods successfully implemented on unstructured grids.

Numerical experiments validate theoretical results.

Abstract

This paper develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known monotone (finite difference) framework, the proposed new framework allows for the use of narrow stencils and unstructured grids which makes it possible to construct high order methods. The general framework is based on the concepts of consistency and g-monotonicity which are both defined in terms of various numerical derivative operators. Specific methods that satisfy the framework are constructed using numerical moments. Admissibility, stability, and convergence properties are proved, and numerical experiments are provided along with some computer implementation details.