Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinous initial data

Olivier Bokanowski; Nadia Megdich; Hasnaa Zidani

doi:10.1007/s00211-009-0271-1

Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinous initial data

Olivier Bokanowski
, Nadia Megdich
, Hasnaa Zidani

École Polytechnique

UPMC Université de Paris VI
UFR de Mathématiques
Laboratoire de Probabilités et Modèles Aléatoires
ISECS
ENet'com Sfax

Research output: Contribution to journal › Article › peer-review

14 Citations (Scopus)

Abstract

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton-Jacobi-Bellman equation of the form v_t + max_α(f(x, α)v_x) = 0, v(0, x) = v₀(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1-33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L¹-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.

Original language	English
Pages (from-to)	1-44
Number of pages	44
Journal	Numerische Mathematik
Volume	115
Issue number	1
DOIs	https://doi.org/10.1007/s00211-009-0271-1
Publication status	Published - 1 Feb 2010

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abstract = "We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton-Jacobi-Bellman equation of the form vt + maxα(f(x, α)vx) = 0, v(0, x) = v0(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1-33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L1-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.",

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AU - Bokanowski, Olivier

AU - Megdich, Nadia

AU - Zidani, Hasnaa

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N2 - We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton-Jacobi-Bellman equation of the form vt + maxα(f(x, α)vx) = 0, v(0, x) = v0(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1-33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L1-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.

AB - We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton-Jacobi-Bellman equation of the form vt + maxα(f(x, α)vx) = 0, v(0, x) = v0(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1-33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L1-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.

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Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinous initial data

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