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$ L^1$-error estimates for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1


Authors: Olivier Bokanowski, Nicolas Forcadel and Hasnaa Zidani
Journal: Math. Comp. 79 (2010), 1395-1426
MSC (2000): Primary 49L99, 65M15
DOI: https://doi.org/10.1090/S0025-5718-10-02311-2
Published electronically: January 13, 2010
MathSciNet review: 2629998
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Abstract: 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.


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Additional Information

Olivier Bokanowski
Affiliation: Laboratoire Jacques-Louis Lions, Université Paris 6, 75252 Paris Cedex 05, and UFR de Mathématiques, Université Paris Diderot, Case 7012, 75251 Paris Cedex 05, France; and Projet Commands, INRIA Saclay & ENSTA, 32 Bd Victor, 75739 Paris Cedex 15, France
Email: boka@math.jussieu.fr

Nicolas Forcadel
Affiliation: Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris Cedex 16, France
Email: forcadel@ceremade.dauphine.fr

Hasnaa Zidani
Affiliation: Projet Commands, INRIA Saclay & ENSTA, 32 Bd Victor, 75739 Paris Cedex 15, France
Email: Hasnaa.Zidani@ensta.fr

DOI: https://doi.org/10.1090/S0025-5718-10-02311-2
Keywords: Hamilton-Jacobi-Bellman equations, lower semicontinuous viscosity solutions, Fast Marching Method, Ultra-Bee scheme, $L^1$-error estimate, anti-diffusive scheme, comparison principle.
Received by editor(s): April 7, 2008
Received by editor(s) in revised form: February 4, 2009
Published electronically: January 13, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.