A level set framework for capturing multi-valued solutions of nonlinear first-order equations

Journal of Scientific Computing

Volumn 29, Issue 3, 2006, Pages 353-373

  Liu, Hailiang ^a   Cheng, Li Tien ^b   Osher, Stanley ^c  

^a IOWA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c Level Set Systems Inc   (United States)

Author keywords

Level set method; Multi valued solutions; Nonlinear first order equation

Indexed keywords

FUNCTIONS; NONLINEAR EQUATIONS; PROBLEM SOLVING; SET THEORY;

FIRST ORDER EQUATIONS; HAMILTON-JACOBI EQUATIONS; LEVEL SET METHOD; MULTI-VALUED SOLUTIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 33845497473     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-005-9016-1     Document Type: Article

References (34)

1. Benamou, J. D., Castella, F., Katsaounis T., and Perthame, B. (2001). High-frequency Helmholtz Equation, Geometrical Optics and Particle Methods, Revist. Math. Iberoam.
2. Benamou, J.-D. (1996). Big ray tracing: multivalued travel time field computation using viscosity solution of the eikonal equation. J. Comp. Phys., 128, 463-474.
3. Benamou, J. D. (1999). Direct solution of multivalued phase space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. math., 52, 1443-1475.
4. Brenier, Y. (1984). Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21, 1013-1037.
5. Brenier, Y., and Corrias, L. (1998). A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. Henri Poincare 15, 169-190.
6. Burchard, P., Cheng, L.-T., Merriman, B., and Osher, S. (2001). Motion of curves in three spatial dimensions using a level set approach. J. Comput. Phys. 170, 720-741.
7. Courant, R,. and Hubert, D. (1952-1962). Methods of Mathematical Physics, Vol 2, Interscience Publishers, New York.
8. Cheng, L.-T., Osher, S., Kang, M., Shim, H., and Tsai, Y-H. Reflection in a level set framework for geometric optics. Comp. Methods Eng. Phys. to appear.
9. Cheng, L.-T., Liu, H.-L., and Osher, S. (2003). Computational high-frequency wave propagation in Schödinger equations using the Level Set Method, with applications to the semi-classical limit of Schrödinger equations. Comm. Math. Sci. 1(3), 593-621.
10. Crandall, M., and Lions, P. L. (1984). Viscosity solutions of Hamilton-Jacobi equations. Trans. AM. Math. Soc. 282, 487.
11. Engquist, B., Fatemi, E., and Osher, S. (1995). Numerical solution of the high frequency asymptotic expansion for the scalar wave equation. J. Comput. Phys. 120, 145-155.
12. Engquist, B., and Runborg, O. (1996). Multi-phase computations in geometrical optics J. Comp. Appl. Math. 74, 175-1992.
13. Engquist, B., Runborg, O. and Tornberg, T.-K. (2002). High frequency wave propagation by the segment projection method. J. Compur. Phys. 178, 373-390.
14. Engquist, B., and Runborg, O. (2003). Computational high frequency wave propagation, Acta Numerica, 2003.
15. Evans, L.-C. (1996). A geometric interpretation of the heat equation with multi-valued initial data. SIAM J. Math. Anal. 27, 932-958.
16. Gosse, L. (2002). Using K-branch entropy solutions for multivalued geometric optics computations. J. Comp. Phys., 180, 155-182.
17. Gosse, L., Jin, S., and Li, X. (2003). On two moment systems for computing multi-phase semiclassical limits of the Schrödinger equation. Math. Models Methods Appl. Sci. 13(12), 1689-1723.
18. Gray, S., and May, W. (1994). Kirchhoff migration using eikonal equation travel times. Geophysics, 59, 810-817.
19. Gasser, I., and Markowich, P. A. (1997). Quantum hydrodynamics, Wigner transform and the classical limit. Asymptotic Anal. 14, 97-116.
20. Gerard, P., Markowich, P. A., Mauser, N. J., and Poupaud, F. (1997). Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50, 323-380.
21. Jin, S., and X. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner. Physics D. 182 (2003), no. (1-2), 46-85.
22. Jin, S., and Osher, S. A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations, submitted to Comm. Math. Sci.
23. Liu, H. Geometrical solutions of Hamilton-Jacobi equations, preprint (2004).
24. Min, C. (2003). Simplicial isosurfacing in arbitrary dimension and codimension. J. Camput. Phys. 190, 295-310.
25. Min, C. (2004). Local level set method in high dimension and codimension. J. Comput. Phys. 200, 368-382.
26. Osher, S. (1993). A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations. SIAM J. Math. Anal. 24, 1145-1152.
27. Osher, S., Cheng, L.-T., Kang, M., Shim, H., and Tsai, Y.-H. (2002). Geometric optics in a phase space based level set and eulerian framework. J. Compur. Phys. 179, 622-648.
28. Osher, S., and Fedkiw, R., Level Set Method and Dynamic. Implicit Surfaces. Springer-Verlag, NY.
29. Osher, S., and Sethian, J. (1988). Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12-49.
30. Qian, J., Cheng, L.-T., and Osher, S. (2003). A level set based Eulerian approach for anisotropic wave propagation. Wave Motion 37, 365-379.
31. Runborg, O. (2000). Some new results in multi-phase geometric optics. Math. Model. Num. Anal 34, 1203-1231.
32. Sparber, C., Markowich, P. A., and Mauser, N. J. (2003). Wigner functions versus WKB-methods in multivalued geometrical optics. Asymptot. Anal. 33(2), 153-187.
33. Tsai, Y.-H. R., Giga, Y., and Osher, S. (2003). A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations. Math. Camp. 72, 159-181.
34. Whitham, G. B. (1974). Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York.