Geodesic
Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 4, pp. 565-591

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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.

DOI : 10.1051/m2an:2008017
Classification : 70H20, 35F25, 35H20, 49L25, 65M06, 65M15
Keywords: degenerate Hamilton-Jacobi equation, Heisenberg group, finite difference schemes, error estimates 
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     author = {Achdou, Yves and Capuzzo-Dolcetta, Italo},
     title = {Approximation of solutions of {Hamilton-Jacobi} equations on the {Heisenberg} group},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {565--591},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {4},
     year = {2008},
     doi = {10.1051/m2an:2008017},
     mrnumber = {2437774},
     zbl = {1153.65083},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008017/}
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Achdou, Yves; Capuzzo-Dolcetta, Italo. Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 4, pp. 565-591. doi: 10.1051/m2an:2008017

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