Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems

SIAM Journal on Numerical Analysis

Volumn 44, Issue 2, 2006, Pages 879-895

  Oberman, Adam M ^a  

^a SIMON FRASER UNIVERSITY   (Canada)

Author keywords

Finite difference schemes; Free boundary problems; Hamilton Jabobi equation; Monotone schemes; Partial differential equations; Viscosity solution

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FINITE DIFFERENCE METHOD; GAME THEORY; HAMILTONIANS; JACOBIAN MATRICES; RANDOM PROCESSES;

FREE BOUNDARY PROBLEMS; HAMILTON JABOBI EQUATIONS; MONOTONE SCHEMES; VISCOSITY SOLUTIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 34247269678     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903435235     Document Type: Article

References (35)

1. D. G. ARONSON, The stability of finite difference approximations to second order linear parabolic differential equations, Duke Math. J., 30 (1963), pp. 117-127.
2. M. BARDI AND I. CAPUZZO-DOLCETTA, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Boston, MA, 1997.
3. M. BARDI, M. G. CRANDALL, L. C. EVANS, H. M. SONER, AND E. P. SOUGANIDIS, Viscosity Solutions and Applications, Lecture Notes in Math. 1660, Springer-Verlag, Berlin, 1997.
4. G. BARLES, Solutions de viscosité des équations de Hamilton-Jacobi, Math. Appl. 17, Springer-Verlag, Paris, 1994.
5. G. BARLES AND P. E. SOUGANIDIS, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), pp. 271-283.
6. G. BARLES AND P. E. SOUGANIDIS, On the large time behavior of solutions of Hamilton- Jacobi equations, SIAM J. Math. Anal., 31 (2000), pp. 925-939.
7. J. H. BRAMBLE, B. E. HUBBARD, AND V. THOMÉE, Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comp., 23 (1969), pp. 695-709.
8. L. A. CAFFARELLI AND J. L. VAZQUEZ, Viscosity solutions for the porous medium equation, in Differential Equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math. 65, AMS, Providence, RI, 1999, pp. 13-26.
9. R. COURANT, K. O. FRIEDRICHS, AND H. LEWY, On the partial difference equations of mathematical physics, IBM J. Res. Develop., 11 (1967), pp. 215-234.
10. M. G. CRANDALL AND P.-L. LIONS, Two approximations of solutions of Hamilton- Jacobi equations, Math. Comp., 43 (1984), pp. 1-19.
11. 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. (N.S.), 27 (1992), pp. 1-67.
12. M. G. CRANDALL AND P.-L. LIONS, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numer. Math., 75 (1996), pp. 17-41.
13. M. G. CRANDALL AND A. MAJDA, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), pp. 1-21.
14. M. G. CRANDALL AND L. TARTAR, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), pp. 385-390.
15. L. C. EVANS, Partial Differential Equations, Graduate Studies in Math. 19, AMS, Providence, RI, 1998.
16. M. FALCONE AND R. FERRETTI, Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys., 175 (2002), pp. 559-575.
17. S. GOTTLIEB, C.-W. SHU, AND E. TADMOR, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89-112.
18. A. HARTEN, J. M. HYMAN, AND P. D. LAX, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math., 29 (1976), pp. 297-322.
19. A. HARTEN AND S. OSHER, Uniformly high-order accurate nonoscillatory schemes, I, SIAM J. Numer. Anal., 24 (1987), pp. 279-309.
20. R. JENSEN, P.-L. LIONS, AND P. E. SOUGANIDIS, A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. Amer. Math. Soc., 102 (1988), pp. 975-978.
21. F. JOHN, On integration of parabolic equations by difference methods, I. Linear and quasi-linear equations for the infinite interval, Comm. Pure Appl. Math., 5 (1952), pp. 155-211.
22. I. C. KIM, Uniqueness and existence results on the Hele-Shaw and the Stefan problems, Arch. Ration. Mech. Anal., 168 (2003), pp. 299-328.
23. H. J. KUO AND N. S. TRUDINGER, On the discrete maximum principle for parabolic difference operators, RAIRO Modél. Math. Anal. Numér., 27 (1993), pp. 719-737.
24. H. J. KUO AND N. S. TRUDINGER, Discrete methods for fully nonlinear elliptic equations, SIAM J. Numer. Anal., 29 (1992), pp. 123-135.
25. H.-J. KUO AND N. S. TRUDINGER, Maximum principles for difference operators, in Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math. 177, Dekker, New York, 1996, pp. 209-219.
26. H.-J. KUO AND N. S. TRUDINGER, Positive difference operators on general meshes, Duke Math. J., 83 (1996), pp. 415-433.
27. C.-T. LIN AND E. TADMOR, High-resolution nonoscillatory central schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), pp. 2163-2186.
28. T. S. MOTZKIN AND W. WASOW, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys., 31 (1953), pp. 253-259.
29. A. M. OBERMAN, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), pp. 365-379.
30. A. M. OBERMAN, A convergent difference scheme for the infinity Laplacian: Construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), pp. 1217-1230.
31. S. OSHER AND C.-W. SHU, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), pp. 907-922.
32. J. A. SETHIAN, Fast marching methods, SIAM Rev., 41 (1999), pp. 199-235.
33. P. E. SOUGANIDIS, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations, 59 (1985), pp. 1-43.
34. Y.-H. R. TSAI, L.-T. CHENG, S. OSHER, AND H.-K. ZHAO, Fast sweeping algorithms for a class of Hamilton-Jacobi equations, SIAM J. Numer. Anal., 41 (2003), pp. 673-694.
35. O. B. WIDLUND, Stability of parabolic difference schemes in the maximum norm, Numer. Math., 8 (1966), pp. 186-202.