Numerical Approximation of the Critical Value of Eikonal Hamilton-Jacobi Equations on Networks

TL;DR

This paper develops numerical methods to approximate the critical value of eikonal Hamilton-Jacobi equations on networks, using large time behavior analysis and providing error estimates and convergence results.

Contribution

It introduces new approximation strategies for the critical value on networks, with proven error bounds and demonstrated numerical performance.

Findings

Algorithms effectively approximate the critical value.

Error estimates ensure reliability of the methods.

Numerical tests confirm convergence and accuracy.

Abstract

The critical value of an eikonal equation is the unique value of a parameter for which the equation admits solutions and is deeply related to the effective Hamiltonian of a corresponding homogenization problem. We study approximation strategies for the critical value of eikonal equations posed on networks. They are based on the large time behavior of corresponding time-dependent Hamilton-Jacobi equations. We provide error estimates and some numerical tests, showing the performance and the convergence properties of the proposed algorithms.