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

¹ UFR Mathématiques, Université Paris 7, Case 7012, 75251 Paris Cedex 05, France.
² Laboratoire Jacques-Louis Lions, Université Paris 6, 75252 Paris Cedex 05, France. achdou@math.jussieu.fr
³ Dipartimento di Matematica, Università Roma “La Sapienza”, Piazzale A. Moro 2, 00185 Roma, Italy. capuzzo@mat.uniroma1.it

Received: 17 April 2007

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 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.