A high order correction to the Lax-Friedrich's method for approximating stationary Hamilton-Jacobi equations

TL;DR

This paper introduces a high-order correction to the Lax-Friedrich's method for stationary Hamilton-Jacobi equations, enhancing accuracy beyond first order while maintaining convergence through a novel cutoff and stability analysis.

Contribution

It proposes a new non-monotone finite difference method with high-order correction and a cutoff to improve accuracy for Hamilton-Jacobi problems, extending beyond traditional monotone schemes.

Findings

The new method achieves higher accuracy than first-order schemes.

Numerical tests demonstrate improved solution quality.

A novel stability analysis framework is developed.

Abstract

A new class of non-monotone finite difference (FD) approximation methods for approximating solutions to non-degenerate stationary Hamilton-Jacobi problems with Dirichlet boundary conditions is proposed and analyzed. The new FD methods add a high order correction to the Lax-Friedrich's method while utilizing a novel cutoff to preserve the convergence properties of the Lax-Friedrich's approximation. Since monotone methods are limited to first order accuracy by the Godunov barrier, the proposed approach provides a template for boosting the accuracy of a monotone method using a modified numerical moment stabilizer with a high-order auxiliary boundary condition. Numerical tests are provided to test the utility of the approach while a novel admissibility and stability analysis technique lays a foundation for analyzing non-monotone methods.