An approximation scheme for an Hamilton-Jacobi equation defined on a network

TL;DR

This paper develops an approximation scheme for solving Hamilton-Jacobi equations of Eikonal type on networks, proving convergence to the unique viscosity solution using a semi-Lagrangian method.

Contribution

It introduces a new notion of viscosity solution for Hamilton-Jacobi equations on networks and proves convergence of a semi-Lagrangian approximation scheme.

Findings

The scheme converges to the unique viscosity solution.

A new notion of viscosity solution for network-based equations is proposed.

The method provides a reliable numerical approach for Hamilton-Jacobi equations on networks.

Abstract

In this paper we study an approximation scheme for an Hamilton-Jacobi equation of Eikonal type defined on a network. We introduce an appropriate notion of viscosity solution for this class of equations (see \cite{sc}) and we prove that an approximation scheme of semi-Lagrangian type converges to the unique solution of the problem.