A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations

Mathematics of Computation

Volumn 71, Issue 241, 2003, Pages 159-181

  Tsai, Yen Hsi Richard ^a   Giga, Yoshikazu ^b   Osher, Stanley ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b HOKKAIDO UNIVERSITY   (Japan)

Author keywords

Hamilton Jacobi equations; Level sets; Singular diffusion

Indexed keywords

EID: 0037235382     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (34)

1. Martino Bardi and Italo Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser Boston Inc., Boston, MA, 1997, With appendices by Maurizio Falcone and Pierpaolo Soravia. MR 99e:14001
2. Guy Barles, Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit, Nonlinear Anal. 20 (1993), no. 9, 1123-1134. MR 94d:49047
3. Guy Barles, Solution de viscosité des équations de Hamilton-Jacobi, Springer-Verlag, 1994. MR 2000b:49054
4. E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations 15 (1990), no. 12, 1713-1742. MR 91b:35069
5. C Caratheodory, Calculus of varieties of partial differential equations of the first order, Chelsea, 1982. MR 33:597; MR 38:590 (earlier ed.)
6. M.G. Crandall and P.L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation 43 (1984), 1-19. MR 86j:65121
7. Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1-42. MR 85g:35029
8. Lawrence C. Evans, A geometric interpretation of the heat equation with multivalued initial data, SIAM J. Math. Anal. 27 (1996), no. 4, 932-958. MR 98g:35092
9. Mi-Ho Giga and Yoshikazu Giga, Crystalline and level set flow-convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free boundary problems: theory and applications, I (Chiba, 1999), Gakkōtosho, Tokyo, 2000. pp. 64-79. MR 2002f:53117
10. Mi-Ho Giga, Yoshikazu Giga, and Ryo Kobayashi, Very singular diffusion equations, Advanced Studies in Pure Mathematics 31, 2001, pp. 93-125.
11. Y. Giga, S. Goto, H. Ishii, and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 (1991). no. 2, 443-470. MR 92h:35010
12. Yoshikazu Giga. Shocks and very strong vertical diffusion, To appear, Proc. of international conference on partial differential equations in celebration of the seventy fifth birthday of Professor Louis Nirenberg, Taiwan, 2001.
13. _, Viscosity solutions with shocks, Hokkaido Univ. Preprint Series in Math. (2001), no. 519.
14. Yoshikazu Giga and Moto-Hiko Sato, A level set approach to semicontinuous viscosity solutions for Cauchy problems, Comm. Partial Differential Equations 26 (2001). no. 5-6, 813-839.
15. Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357-393. MR 84g:65115
16. Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys. 71 (1987), no. 2, 231-303. MR 90a:65199
17. J. Heimsen, E. Puckett, P. Colella, and M. Dorr, Two new methods for simulating photolithography development in 3d, SPIE 2726, 1996, pp. 253-261.
18. Hitoshi Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ. 28 (1985), 33-77. MR 87k:35055
19. _, Existence and uniqueness of solutions of Hamilton-Jacobi equations, Funkcial. Ekvac. 29 (1986), no. 2, 167-188. MR 88c:35037
20. _, Perron's method for Hamilton-Jacobi equations. Duke Math. J. 55 (1987), no. 2, 369-384. MR 89a:35053
21. Guang-Shan Jiang and Danping Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (2000), no. 6, 2126-2143 (electronic). MR 2001e:65124
22. R. Kobayashi and Y. Giga. Equations with singular diffusivity, J. Statist. Phys. 95 (1999), no. 5-6, 1187-1220. MR 2001f:82077
23. Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR 50:2709
24. Chi-Tien Lin and Eitan Tadmor, High-resolution nonoscillatory central schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (2000), no. 6, 2163-2186 (electronic). MR 2001e:65125
25. Stanley Osher, A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations, SIAM J Math Anal 24 (1993), no. 5, 1145-1152. MR 94h:35039
26. Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12-49. MR 89h:80012
27. Stanley Osher and Chi-Wang Shu. High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 907-922. MR 92e:65118
28. Danping Peng, Barry Merriman, Stanley Osher, Hongkai Zhao, and Myungjoo Kang. A PDE-based fast local level set method, J. Comput. Phys. 155 (1999), no. 2, 410-438. MR 2000j:65104
29. J.A. Sethian, Fast marching level set methods for three dimensional photolithography development, SPIE 2726, 1996. pp. 261-272.
30. Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1989), no. 1, 32-78. MR 90i:65167
31. Panagiotis E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations 59 (1985), no. 1, 1-43. MR 86k:35028
32. Yen-Hsi Richard Tsai, Rapid and accurate computation of the distance function using grids, UCLA CAM Report 00 (2000), no. 36.
33. John Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Transactions on Automatic Control 40 (1995), no. 9, 1528-1538. MR 96d:49039
34. Bram van Leer, Towards the ultimate conservative difference schemes V, J. Comput. Phys. 32 (1979), 102-136. MR 90h:70003