Error estimates for finite difference schemes associated with Hamilton-Jacobi equations on a junction

TL;DR

This paper develops and analyzes finite difference schemes for Hamilton-Jacobi equations on junctions, proving convergence and deriving optimal error estimates for certain junction conditions.

Contribution

It extends existing schemes to general junction conditions and provides rigorous convergence and error estimates for these schemes.

Findings

Proves convergence of schemes to viscosity solutions as mesh size decreases.

Derives optimal error estimates of order (Δx)^{1/2} for specific junction conditions.

Applicable to Hamilton-Jacobi equations with control-type flux limitations.

Abstract

This paper is concerned with monotone (time-explicit) finite difference schemes associated with first order Hamilton-Jacobi equations posed on a junction. They extend the schemes recently introduced by Costeseque, Lebacque and Monneau (2013) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton-Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive optimal error estimates of order $($\Delta$x)^{\frac{1}{2}}$ in $L\_{loc}^{\infty}$ for junction conditions of optimal-control type at least if the flux is "strictly limited".