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Convergence of a semi-discretization scheme for the Hamilton--Jacobi equation: a new approach with the adjoint method
journal contribution
posted on 2023-06-08, 18:44 authored by Filippo Cagnetti, H V Tran, D GomesWe 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(sqrt{h}) convergence rate in terms of the L^infty norm and O(h) in terms of the L^1 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.
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Publication status
- Published
File Version
- Accepted version
Journal
Applied Numerical MathematicsISSN
0168-9274Publisher
ElsevierExternal DOI
Volume
73Page range
2-15Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2014-10-20First Open Access (FOA) Date
2014-10-20First Compliant Deposit (FCD) Date
2014-10-20Usage metrics
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