We introduce a new class of "filtered" schemes for some first order
non-linear Hamilton-Jacobi-Bellman equations. The work follows recent ideas of
Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013). The
proposed schemes are not monotone but still satisfy some $\epsilon$-monotone
property. Convergence results and precise error estimates are given, of the
order of $\sqrt{\Delta x}$ where $\Delta x$ is the mesh size. The framework
allows to construct finite difference discretizations that are easy to
implement, high--order in the domains where the solution is smooth, and
provably convergent, together with error estimates. Numerical tests on several
examples are given to validate the approach, also showing how the filtered
technique can be applied to stabilize an otherwise unstable high--order scheme.Comment: 20 pages (including references), 26 figure

Bokanowski, Olivier

Falcone, Maurizio

Sahu, Smita

English

arXiv

Portsmouth University Research Portal (Pure)

An efficient filtered scheme for some first order time-dependent Hamilton--Jacobi equations

We introduce a new class of "filtered" schemes for some first order non-linear Hamilton-Jacobi-Bellman equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013). The proposed schemes are not monotone but still satisfy some -monotone property. Convergence results and precise error estimates are given, of the order of √ ∆x where ∆x is the mesh size. The framework allows to construct finite difference discretizations that are easy to implement, high–order in the domains where the solution is smooth, and provably convergent, together with error estimates. Numerical tests on several examples are given to validate the approach, also showing how the filtered technique can be applied to stabilize an otherwise unstable high–order scheme

Archive Ouverte en Sciences de l'Information et de la Communication

AN EFFICIENT FILTERED SCHEME FOR SOME FIRST ORDER HAMILTON-JACOBI-BELLMAN EQUATIONS

Hal-Diderot

We introduce a new class of “filtered” schemes for some first order nonlinear
Hamilton–Jacobi equations. The work follows recent ideas of Froese and Oberman [SIAM J. Numer.
Anal., 51 (2013), pp. 423–444] and Oberman and Salvador [J. Comput. Phys., 284 (2015),
pp. 367–388] for steady equations. Here we mainly study the time-dependent setting and focus on
fully explicit schemes. Furthermore, specific corrections to the filtering idea are also needed in order
to obtain high-order accuracy. The proposed schemes are not monotone but still satisfy some
ε-monotone property. A general convergence result together with a precise √ error estimate of order
Δx are given (Δx is the mesh size). The framework allows us to construct finite difference discretizations
that are easy to implement and high-order in the domain where the solution is smooth. A
novel error estimate is also given in the case of the approximation of steady equations. Numerical
tests including evolutive convex and nonconvex Hamiltonians, and obstacle problems are presented
to validate the approach. We show with several examples how the filter technique can be applied to
stabilize an otherwise unstable high-order scheme

FALCONE, Maurizio

Archivio della ricerca- Università di Roma La Sapienza

An efficient filtered scheme for some first order time-dependent Hamilton-Jacobi equations

An efficient filtered scheme for some first order
  Hamilton-Jacobi-Bellman equations

An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations

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Portsmouth University Research Portal (Pure)

Archive Ouverte en Sciences de l'Information et de la Communication

Hal-Diderot

Archivio della ricerca- Università di Roma La Sapienza