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. © 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

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.

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

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