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 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.
@article{M2AN_2008__42_4_565_0,
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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {42},
year = {2008},
pages = {565-591},
doi = {10.1051/m2an:2008017},
mrnumber = {2437774},
zbl = {1153.65083},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2008__42_4_565_0}
}
Achdou, Yves; Capuzzo-Dolcetta, Italo. Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 565-591. doi : 10.1051/m2an:2008017. http://gdmltest.u-ga.fr/item/M2AN_2008__42_4_565_0/
[1] and , A finite difference scheme on a non commutative group. Numer. Math. 89 (2001) 401-424. | MR 1864424 | Zbl 1008.65074
[2] , A boundary value problem for the minimum-time function. SIAM J. Control Optim. 27 (1989) 776-785. | MR 1001919 | Zbl 0682.49034
[3] and , Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1997). With appendices by M. Falcone and P. Soravia. | MR 1484411 | Zbl 0890.49011
[4] and , Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal. 43 (2005) 540-558 (electronic). | MR 2177879 | Zbl 1092.65077
[5] , and , Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. 79 (2000) 633-689. | MR 1776501 | Zbl 0959.35035
[6] and , Eds., Sub-Riemannian Geometry, Progress in Mathematics 144. Birkhäuser Verlag, Basel (1996). | MR 1421821
[7] and , Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Commun. Pure Appl. Anal. 2 (2003) 461-479. | MR 2019062 | Zbl 1043.35025
[8] , Control theory and singular Riemannian geometry, in New directions in applied mathematics (Cleveland, Ohio, 1980), Springer, New York (1982) 11-27. | MR 661282 | Zbl 0483.49035
[9] , On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optim. 10 (1983) 367-377. | MR 713483 | Zbl 0582.49019
[10] , The Hopf-Lax solution for state dependent Hamilton-Jacobi equations (Viscosity solutions of differential equations and related topics) (Japanese). Sūrikaisekikenkyūsho Kōkyūroku 1287 (2002) 143-154. | MR 1959717
[11] , The Hopf solution of Hamilton-Jacobi equations, in Elliptic and parabolic problems (Rolduc/Gaeta, 2001), World Sci. Publishing, River Edge, NJ (2002) 343-351. | MR 1937554 | Zbl 1033.35021
[12] , A generalized Hopf-Lax formula: analytical and approximations aspects, in Geometric Control and Nonsmooth Analysis, F. Ancona, A. Bressan, P. Cannarsa, F. Clarkeă and P.R. Wolenski Eds., Series on Advances in Mathematics for Applied Sciences 76, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008). | MR 2487751
[13] and , Approximate solutions of the Bellman equation of deterministic control theory. Appl. Math. Optim. 11 (1984) 161-181. | MR 743925 | Zbl 0553.49024
[14] and , Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp. 43 (1984) 1-19. | MR 744921 | Zbl 0556.65076
[15] and , Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations. ESAIM: COCV 13 (2007) 484-502. | Numdam | MR 2329172 | Zbl 1125.70013
[16] and , One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321-351. | MR 606500 | Zbl 0469.65067
[17] , A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optim. 15 (1987) 1-13. | MR 866164 | Zbl 0715.49023
[18] and , Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations. Numer. Math. 67 (1994) 315-344. | MR 1269500 | Zbl 0791.65046
[19] , and , Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89-112 (electronic). | MR 1854647 | Zbl 0967.65098
[20] , , and , Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys. 71 (1987) 231-303. | MR 897244 | Zbl 0652.65067
[21] and , Quasiconformal mappings on the Heisenberg group. Invent. Math. 80 (1985) 309-338. | MR 788413 | Zbl 0567.30017
[22] , On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients. Probab. Theory Relat. Fields 117 (2000) 1-16. | MR 1759507 | Zbl 0971.65081
[23] , The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients. Appl. Math. Optim. 52 (2005) 365-399. | MR 2174020 | Zbl 1087.65100
[24] and , A version of the Hopf-Lax formula in the Heisenberg group. Comm. Partial Diff. Eq. 27 (2002) 1139-1159. | MR 1916559 | Zbl 1080.49023
[25] and , Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | MR 965860 | Zbl 0659.65132
[26] and , High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907-922. | MR 1111446 | Zbl 0736.65066
[27] , Level set methods and fast marching methods, Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge Monographs on Applied and Computational Mathematics 3. Cambridge University Press, Cambridge, 2nd edition (1999). | MR 1700751 | Zbl 0973.76003