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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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posted on 2023-06-08, 18:44 authored by Filippo Cagnetti, H V Tran, D Gomes
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(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.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Applied Numerical Mathematics

ISSN

0168-9274

Publisher

Elsevier

Volume

73

Page range

2-15

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2014-10-20

First Open Access (FOA) Date

2014-10-20

First Compliant Deposit (FCD) Date

2014-10-20

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