Error estimates for a finite difference scheme associated with Hamilton–Jacobi equations on a junction

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 (∆x) 1 2 in L ∞ loc for junction conditions of optimal-control type at least if the flux is "strictly limited".

Effective nonlinear Neumann boundary conditions for 1D nonconvex Hamilton–Jacobi equations

“…Lemma 3.13 (Critical slope for sub-solution [17]). Let u be an upper semi-continuous sub-solution of (13) which satisfies (14) and let ϕ be a test function touching u from above at some point (t 0 , 0) where t 0 ∈ (0, T ). Then the critical slope given bȳ [17]).…”

“…In [15], the authors find an error estimate of order ∆x 1 3 of the same scheme as in [9], and prove a convergence result for a general junction function at the junction point. This error estimate has been improved in [14] to order ∆x 1 2 . There are also applications in optimal control, for example in [1] where the authors study problem related to flux-limited functions.…”

“…i) If, for t 0 ∈ (0, T ), u is an upper semi-continuous sub-solution of (13) and satisfies u(t 0 , 0) = lim sup (s,y)→(t 0 ,x),y =0 u(s, y), (14) and if for any test function ϕ touching u from above at (t 0 , 0) with…”

“…Proof of Theorem 3.18. We first prove that relaxed sub-solutions satisfying (14) are fluxlimited sub-solutions. We only do the proof for sub-solutions since it is very similar for super-solutions.…”

Junction Conditions for Hamilton–Jacobi Equations for Solving Real-Time Traffic Flow Problems

“…The goal of this paper is to propose junction conditions for local perturbation, bifurcation, merging and multi-in-multiout nodes on a network. Let us refer to the works [26], [27] for some finite difference schemes to solve this type of equations.…”