Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group

Yves Achdou; Italo Capuzzo-Dolcetta

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 4, page 565-591
  • ISSN: 0764-583X

Abstract

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We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like h where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting. The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems.

How to cite

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Achdou, Yves, and Capuzzo-Dolcetta, Italo. "Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group." ESAIM: Mathematical Modelling and Numerical Analysis 42.4 (2008): 565-591. <http://eudml.org/doc/250427>.

@article{Achdou2008,
abstract = { We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt\{h\}$ where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting. The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems. },
author = {Achdou, Yves, Capuzzo-Dolcetta, Italo},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Degenerate Hamilton-Jacobi equation; Heisenberg group; finite difference schemes; error estimates.; degenerate Hamilton Jacobi equation; error estimates; numerical examples; viscosity solution; geodesics},
language = {eng},
month = {5},
number = {4},
pages = {565-591},
publisher = {EDP Sciences},
title = {Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group},
url = {http://eudml.org/doc/250427},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Achdou, Yves
AU - Capuzzo-Dolcetta, Italo
TI - Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/5//
PB - EDP Sciences
VL - 42
IS - 4
SP - 565
EP - 591
AB - We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt{h}$ where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting. The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems.
LA - eng
KW - Degenerate Hamilton-Jacobi equation; Heisenberg group; finite difference schemes; error estimates.; degenerate Hamilton Jacobi equation; error estimates; numerical examples; viscosity solution; geodesics
UR - http://eudml.org/doc/250427
ER -

References

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