Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations

TL;DR

This paper introduces a new framework for designing convergent finite difference methods for 1-D fully nonlinear second order PDEs, extending concepts from first order equations and validating with numerical results.

Contribution

It develops a novel framework with new concepts of numerical operator and numerical moment for second order PDEs, enabling the design of monotone, consistent, and stable finite difference methods.

Findings

Proposes two classes of finite difference methods: Lax-Friedrichs-like and Godunov-like.

Proves stability and convergence of the proposed methods.

Numerical experiments confirm the theoretical properties and effectiveness.

Abstract

This paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations (PDEs) in 1-D. The goal of the paper is to extend the successful framework of monotone, consistent, and stable finite difference methods for first order fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs such as Monge-Amp\`ere and Bellman type equations. New concepts of consistency, generalized monotonicity, and stability are introduced; among them, the generalized monotonicity and consistency, which are easier to verify in practice, are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear Hamilton-Jacobi equations. The main component of the proposed framework is the concept of "numerical operator", and the main idea used to design consistent, monotone and stable finite difference methods is the concept of "numerical moment". These two new concepts play the same roles as the "numerical Hamiltonian" and the "numerical viscosity" play in the finite difference framework for first order fully nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also are proved to be stable and the second class contains Godunov-like methods. Numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper.