A high order correction to the Lax-Friedrich's method for approximating stationary Hamilton-Jacobi equations
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.
Keywords
Cite
@article{arxiv.2502.03728,
title = {A high order correction to the Lax-Friedrich's method for approximating stationary Hamilton-Jacobi equations},
author = {T. Lewis and X. Xue},
journal= {arXiv preprint arXiv:2502.03728},
year = {2025}
}