Splitting methods for Hamilton-Jacobi equations

Numerical Methods for Partial Differential Equations

Volumn 22, Issue 2, 2006, Pages 381-396

  Tourin, Agnès ^a  

^a MCMASTER UNIVERSITY   (Canada)

Author keywords

Alternate Directions method; Finite difference, monotone schemes; Hamilton Jacobi equations; Nonlinear elliptic or parabolic partial differential equations; Operator splitting; Viscosity solutions

Indexed keywords

EID: 33644995184     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20100     Document Type: Article

References (29)

1. 1 K. Morton and D. Mayers, Numerical solution of partial differential equation, an introduction, Cambridge University Press, 1994.
2. 2 P.‐L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J Numer Anal 16 (1979), 964–979.
3. 3 P.‐L. Lions and B. Mercier, Approximation numeriques des equations de Hamilton‐Jacobi‐Bellman, RAIRO Anal Numer 14 (1980), 369–393.
4. 4 P.‐L. Lions, Formule de trotter et équations de Hamilton‐Jacobi‐Bellman, Calcolo 17 (1980), 321–331.
5. 5 P. E. Souganidis, Max‐min representations and product formulas for the viscosity solutions of Hamilton‐Jacobi equations with applications to differential games, Nonlinear Analysis 9 (1985), 217–257.
6. 6 G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptot Anal 4 (1991), 271–283.
7. 7 M. Falcone, P. Lanucara, and A. Seghini, A splitting algorithm for hamilton‐jacobi‐bellman equations, Appl Numer Math 15 (1994), 207–218.
8. 8 M. Sun, Alternating directions algorithms for solving hamilton‐jacobi‐bellman equations, Applied Math Optim 34 (1996), 267–277.
9. 9 A. Tourin and T. Zariphopoulou, Investment‐consumption models with transaction fees and markov‐chain parameters, Comput Econom 7 (1994), 287–307.
10. 10 A. Tourin and T. Zariphopoulou, Viscosity solutions and numerical schemes for investment/consumption models with transaction costs, L. C. G. Rogers and D. Talay, editors, Numerical methods in finance, Cambridge University Press, 1997, pp 245–269.
11. 11 G. Barles, C. Daher, and M. Romano, Convergence of approximation schemes for parabolic equations arising in finance theory, Math Model Numer Anal 5 (1995), 125–143.
12. 12 G. Barles, The convergence of approximation schemes for parabolic equations arising in finance theory, Numerical methods in finance, Cambridge University Press, 1997, 1–21.
13. 13 E. R. Jakobsen, K. H. Karlsen, and N. H. Risebro, On the convergence rate of operator splitting for hamilton‐jacobi equations with source terms, SIAM J Numer Anal 39 (2001), 499–518.
14. 14 K. H. Karlsen and N. H. Risebro, An operator splitting method for nonlinear convection‐diffusion equations, Numer Math 77 (1997), 365–382.
15. 15 K. H. Karlsen and N. H. Risebro, An operator splitting method for nonlinear parabolic equations, SIAM J Numer Anal 37 (2000), 980–1003.
16. 16 G. Barles and A. Tourin, Commutation properties of semigroups for first‐order Hamilton‐Jacobi equations and application to multi‐time equations, Indiana U Math J 50 (2001), 1523–1544.
17. 17 M. Bardi and I. Capuzzo‐Dolcetta, Optimal control and viscosity solutions of Hamilton‐Jacobi‐Bellman equations, Systems & control: foundations & applications, Birkhäuser, Boston, 1997.
18. 18 G. Barles, Solutions de viscosité des équations de Hamilton‐Jacobi, Mathématiques Applications Springer‐Verlag, New York, 1994.
19. 19 I. Capuzzo Dolcetta and P. L. Lions, editors. Viscosity solutions and applications (Montecatini Terme 1995), Springer‐Verlag, 1997.
20. 20 M. G. Crandall, H. Ishii, and P.‐L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull Amer Math Soc 27 (1992), 1–67.
21. 21 M. G. Crandall and P.‐L. Lions, Viscosity solutions of Hamilton‐Jacobi equations, Trans Amer Math Soc 277 (1983), 1–42.
22. 22 P.‐L. Lions, Generalized solutions of Hamilton‐Jacobi equations, Research Notes in Mathematics, Pitman, Boston, 1982.
23. 23 H. J. Kushner and P. G. Dupuis, Numerical methods for stochastic control problems in continuous time, Springer‐Verlag, 1992.
24. 24 G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton‐Jacobi‐Bellman equations, M2AN Math Model Numer Anal 36 (2002), 33–54.
25. 25 G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton‐Jacobi‐Bellman equations, (to appear in SINUM).
26. 26 C.‐T. Lin and E. Tadmor, l 1 ‐stability and error estimates for approximate hamilton‐jacobi solutions. Numer Math 87 (2001), 701–735.
27. 27 A. Oberman, Convergent difference schemes for nonlinear elliptic and parabolicequations: Hamilton‐Jacobi equations and free boundary problems, (to appear in SINUM).
28. 28 P.‐L. Lions and J. Rochet, Hopf formula and multi‐time hamilton‐jacobi equations, Proc Amer Math Soc 96 (1986), 79–84.
29. 29 E. Rouy and A. Tourin, A viscosity solutions approach to shape‐from‐shading, Siam J Numer Anal 29 (3) (1992), 867–884.