Convergence of a semi-discretization scheme for the Hamilton-Jacobi equation: A new approach with the adjoint method

Handle URI:
http://hdl.handle.net/10754/563062
Title:
Convergence of a semi-discretization scheme for the Hamilton-Jacobi equation: A new approach with the adjoint method
Authors:
Cagnetti, Filippo; Gomes, Diogo A. ( 0000-0002-3129-3956 ) ; Tran, Hung Vinh
Abstract:
We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L.C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(h) convergence rate in terms of the L∞ norm and O(h) in terms of the L1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper. © 2013 IMACS.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Applied Mathematics and Computational Science Program
Publisher:
Elsevier BV
Journal:
Applied Numerical Mathematics
Issue Date:
Nov-2013
DOI:
10.1016/j.apnum.2013.05.004
ARXIV:
arXiv:1106.0444
Type:
Article
ISSN:
01689274
Sponsors:
F. Cagnetti was supported by the UTAustin vertical bar Portugal partnership through the FCT post-doctoral fellowship SFRH/BPD/51349/2011, CAMGSD-LARSys through FCT Program POCTI-FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009. D. Gomes was partially supported by CAMGSD-LARSys through FCT Program POCTI-FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009. H. Tran was supported in part by VEF fellowship.
Additional Links:
http://arxiv.org/abs/arXiv:1106.0444v2
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorCagnetti, Filippoen
dc.contributor.authorGomes, Diogo A.en
dc.contributor.authorTran, Hung Vinhen
dc.date.accessioned2015-08-03T11:34:54Zen
dc.date.available2015-08-03T11:34:54Zen
dc.date.issued2013-11en
dc.identifier.issn01689274en
dc.identifier.doi10.1016/j.apnum.2013.05.004en
dc.identifier.urihttp://hdl.handle.net/10754/563062en
dc.description.abstractWe consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L.C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(h) convergence rate in terms of the L∞ norm and O(h) in terms of the L1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper. © 2013 IMACS.en
dc.description.sponsorshipF. Cagnetti was supported by the UTAustin vertical bar Portugal partnership through the FCT post-doctoral fellowship SFRH/BPD/51349/2011, CAMGSD-LARSys through FCT Program POCTI-FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009. D. Gomes was partially supported by CAMGSD-LARSys through FCT Program POCTI-FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/2009. H. Tran was supported in part by VEF fellowship.en
dc.publisherElsevier BVen
dc.relation.urlhttp://arxiv.org/abs/arXiv:1106.0444v2en
dc.subjectAdjoint methoden
dc.subjectHamilton-Jacobi equationen
dc.subjectNumerical schemeen
dc.titleConvergence of a semi-discretization scheme for the Hamilton-Jacobi equation: A new approach with the adjoint methoden
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.identifier.journalApplied Numerical Mathematicsen
dc.contributor.institutionDepartamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugalen
dc.contributor.institutionDepartment of Mathematics, University of California, Berkeley, CA 94720-3840, United Statesen
dc.identifier.arxividarXiv:1106.0444en
kaust.authorGomes, Diogo A.en
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