L^1-error estimate for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1.

Olivier Bokanowski (1) , Nicolas Forcadel (2) , Hasnaa Zidani (3, 4)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
4 OC - Optimisation et commande

The goal of this paper is to study some numerical approximations of particular Hamilton-Jacobi-Bellman equations in dimension 1 and with possibly discontinuous initial data. We investigate two anti-diffusive numerical schemes, the first one is based on the Ultra-Bee scheme and the second one is based on the Fast Marching Method. We prove the convergence and derive $L^1$-error estimates for both schemes. We also provide numerical examples to validate their accuracy in solving smooth and discontinuous solutions.

Mots clés: HJB equation; Fast Marching Method; Ultr-Bee scheme; $L^1$- error estimate; antidiffusive scheme

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Olivier Bokanowski, Nicolas Forcadel, Hasnaa Zidani. L^1-error estimate for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1.. Mathematics of Computation, 2010, 79 (271), pp.1395--1426. ⟨inria-00267644⟩