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

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    Type
    Article
    Authors
    Cagnetti, Filippo
    Gomes, Diogo A. cc
    Tran, Hung Vinh
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Date
    2013-11
    Preprint Posting Date
    2011-06-02
    Permanent link to this record
    http://hdl.handle.net/10754/563062
    
    Metadata
    Show full item record
    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.
    Citation
    Cagnetti, F., Gomes, D., & Tran, H. V. (2013). Convergence of a semi-discretization scheme for the Hamilton–Jacobi equation: A new approach with the adjoint method. Applied Numerical Mathematics, 73, 2–15. doi:10.1016/j.apnum.2013.05.004
    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.
    Publisher
    Elsevier BV
    Journal
    Applied Numerical Mathematics
    DOI
    10.1016/j.apnum.2013.05.004
    arXiv
    1106.0444
    Additional Links
    http://arxiv.org/abs/arXiv:1106.0444v2
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.apnum.2013.05.004
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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