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

Filippo Cagnetti; Diogo A. Gomes; Hung Vinh Tran

doi:10.1016/j.apnum.2013.05.004

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

Filippo Cagnetti, Diogo A. Gomes, Hung Vinh Tran

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

4 Scopus citations

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.

Original language	English (US)
Pages (from-to)	2-15
Number of pages	14
Journal	Applied Numerical Mathematics
Volume	73
DOIs	https://doi.org/10.1016/j.apnum.2013.05.004
State	Published - Nov 2013

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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.

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics
Numerical Analysis

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10.1016/j.apnum.2013.05.004

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

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. {\textcopyright} 2013 IMACS.",

author = "Filippo Cagnetti and Gomes, {Diogo A.} and Tran, {Hung Vinh}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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.",

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AU - Gomes, Diogo A.

AU - Tran, Hung Vinh

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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.

PY - 2013/11

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N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/563062

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DO - 10.1016/j.apnum.2013.05.004

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VL - 73

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JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -

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

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