Homogenization of Hamilton-Jacobi equations: Numerical methods

Mathematical Models and Methods in Applied Sciences

Volumn 18, Issue 7, 2008, Pages 1115-1143

  Achdou, Yves ^a,b   Camilli, Fabio ^c   Dolcetta, Italo Capuzzo ^d  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^c UNIVERSITY OF L'AQUILA   (Italy)

^d SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

Hamilton Jacobi equation; Homogenization; Numerical approximation; Viscosity solution

Indexed keywords

ERROR ANALYSIS; HAMILTONIANS; NUMERICAL METHODS; PARAMETER ESTIMATION; POLYNOMIAL APPROXIMATION;

DISCRETIZATION; EFFECTIVE HAMILTONIAN (EH); ERROR ESTIMATES; HAMILTON-JACOBI (HJ) EQUATIONS; LIMIT PROBLEM;

MECHANICS;

EID: 47749089404     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202508002978     Document Type: Article

References (28)

1. O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result, Arch. Rational. Mech. Anal. 170 (2003) 17-61.
2. N. Bacaër, Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel - Kontorova models and homogenization of Hamilton -Jacobi equations, Math. Mod. Numer. Anal. 35 (2001) 1185-1195.
3. G. Barles and P. Souganidis, Convergence of approximation scheme for fully nonlinear second order equations, Asympt. Anal. 4 (1991) 271-283.
4. M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton - Jacobi - Bellman Equations (Birkhäuser, 1997).
5. F. Camilli, I. Capuzzo Dolcetta and D. Gomes, Error estimates for the approximation of the Effective Hamiltonian, Appl. Math. Optim., to appear.
6. I. Capuzzo Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of Deterministic Control Theory, Appl. Math. Optim. 11 (1984) 161-181.
7. I. Capuzzo Dolcetta and H. Ishii, On the rate of convergence in Homogenization of Hamilton - Jacobi equations, Indiana Univ. Math. J. 50 (2001) 1113-1129.
8. M. C. Concordel, Periodic homogenization of Hamilton - Jacobi equations: Additive eigenvalues and variational formula, Indiana Univ. Math. J. 45 (1996) 1095-1117.
9. G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal. 8 (1998) 788-809.
10. B. Cockburn and J. Qian, Continuous dependence results for Hamilton - Jacobi equations, Collected lectures on the Preservation of Stability Under Discretization, Fort Collins, CO, 2001, (SIAM, 2002), pp. 67-90.
11. B. Craciun and K. Bhattacharya, Homogenization of a Hamilton - Jacobi equation associated with the geometric motion of an interface, Proc. R. Soc. Edinburgh Sect. A 133 (2003) 773-805.
12. M. G. Crandall and P. L. Lions, Two approximations of solutions of Hamilton - Jacobi equations, Math. Comp. 43 (1984) 1-19.
13. J. W. Demmel, J. R. Gilbert and X. S. Li, SuperLU Users' Guide, LBNL -44289 (Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA (US) 1999).
14. B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981) 321-351.
15. M. Falcone, Appendix A in Ref. 4.
16. M. Falcone and R. Ferretti, Semi-Lagrangian schemes for Hamilton - Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys. 175 (2002) 559-575.
17. M. Falcone and T. Giorgi, An approximation scheme for evolutive Hamilton - Jacobi equations, in Stochastic Analysis, Control, Optimization and Applications, Systems Control Found. Appl. (Birkhäuser, 1999), pp. 289-303.
18. D. Gomes and A. Oberman, Computing the effective Hamiltonian using a variational approach, SIAM J. Control Optim. 43 (2004) 792-812.
19. A. Greenbaum, Iterative Methods for Solving Linear Systems, Frontiers in Applied Mathematics, Vol. 17 (SIAM, 1997).
20. P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton -Jacobi equations, unpublished.
21. A. J. Majda and T. Souganidis, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity 7 (1994) 1-30.
22. G. Namah and J.-M. Roquejoffre, The "hump" effect in solid propellant combustion, Interfaces Free Bound. 2 (2000) 449-467.
23. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, Vol. 153 (Springer-Verlag, 2003).
24. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton - Jacobi formulations, J. Comput. Phys. 79 (1988) 12-49.
25. J. Qian, Two approximations for effective Hamiltonians arising from homogenization of Hamilton - Jacobi equations, Department of Mathematics, UCLA, preprint, 2003.
26. M. Rorro, An approximation scheme for the effective Hamiltonian and applications, Appl. Numer. Math. 56 (2006) 1238-1254.
27. J. A. Sethian, Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (Cambridge Univ. Press, 1999).
28. P. Souganidis, Approximation schemes for viscosity solutions of Hamilton - Jacobi equations, J. Diff. Eqns. 57 (1985) 1-43.