Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation

Journal of Computational Physics

Volumn 205, Issue 1, 2005, Pages 222-241

  Jin, Shi ^a   Liu, Hailiang ^b   Osher, Stanley ^c   Tsai, Yen Hsi Richard ^d  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b IOWA STATE UNIVERSITY   (United States)

^c Level Set Systems Inc   (United States)

^d PRINCETON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DELTA FUNCTIONS; DROP BREAKUP; FUNCTIONAL ANALYSIS; INITIAL VALUE PROBLEMS; LEVEL MEASUREMENT; LIOUVILLE EQUATION; PHASE SPACE METHODS; QUANTUM OPTICS; ROTATIONAL FLOW; SET THEORY;

ARBITRARY NUMBER; COMMON ZEROS; EULERIAN FORMULATIONS; LEVEL SET METHOD; LEVEL-SET FUNCTION; LIOUVILLE'S EQUATION; MULTI-VALUED; PHASE SPACES; SEMICLASSICAL LIMIT; SPACE DIMENSIONS;

NUMERICAL METHODS;

EID: 16844373042     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jcp.2004.11.008     Document Type: Article

References (40)

1. W.Z. Bao S. Jin P.A. Markowich On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regimes J. Comput. Phys. 175 2002 487-524
2. Y. Brenier L. Corrias A kinetic formulation for multi-branch entropy solutions of scalar conservation laws Ann. Inst. H. Poincaré Anal. Non Linéaire 15 2 1998 169-190
3. Y. Brenier E. Grenier Sticky particles and scalar conservation laws SIAM J. Numer. Anal. 35 6 1998 2317-2328
4. Li-Tien Cheng Hailiang Liu Stanley Osher Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations Comm. Math. Sci. 1 3 2003 593-621
5. Li-Tien Cheng Stanley Osher Myungjoo Kang Hyeseon Shim Yen-Hsi Tsai Reflection in a level set framework for geometric optics Comp. Meth. Eng. Phys. 5 2004 347-360
6. Michael G. Crandall Pierre-Louis Lions Viscosity solutions of Hamilton-Jacobi equations Trans. Am. Math. Soc. 277 1 1983 1-42
7. J.J. Duistermaat Oscillatory integrals, lagrange immersions and unfolding of singularitites Comm. Pure Appl. Math. 27 1974 207-281
8. J.J. Duistermaat Fourier Integral Operators 1995 Birkhäuser Basel
9. Björn Engquist Olof Runborg Multi-phase computations in geometrical optics J. Comput. Appl. Math. 74 1-2 1996 175-192
10. B. Engquist, O. Runborg Computational high frequency wave propagation Acta Numerica, 2003, Acta Numer 2003 Cambridge University Press Cambridge 181-266
11. B. Engquist, A.-K. Tornberg, Y.-H. Tsai, Dirac-δ functions in level set methods, CAM Report 04-16, UCLA, 2003, Under review
12. Ingenuin Gasser Peter A. Markowich Quantum hydrodynamics, Wigner transforms and the classical limit Asymptot. Anal. 14 2 1997 97-116
13. Patrick Gérard Peter A. Markowich Norbert J. Mauser Frédéric Poupaud Homogenization limits and Wigner transforms Comm. Pure Appl. Math. 50 4 1997 323-379
14. Laurent Gosse Using K-branch entropy solutions for multivalued geometric optics computations J. Comput. Phys. 180 1 2002 155-182
15. Laurent Gosse Shi Jin Xiantao Li On two moment systems for computing multiphase semiclassical limits of the Schrödinger equation Math. Model Meth. Appl. Sci. 13 2003 1689-1723
16. M.F. Herman E. Kluk J. Chem. Phys. 91 1984 27
17. Guang-Shan Jiang Danping Peng Weighted ENO schemes for Hamilton-Jacobi equations SIAM J. Sci. Comput. 21 6 2000 2126-2143
18. Shi Jin Xiantao Li Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner Physica D 182 2003 46-85
19. Shi. Jin Stanley Osher A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations Comm. Math. Sci. 1 6 2003 575-591
20. J.B. Keller Corrected Bohr-Sommerfeld quantum conditions for non-separable systems Ann. Phys. 4 1958 180-188
21. X.T. Li J.G. Wöhlbier S. Jin J.H. Booske An Eulerian method for computing multivalued solutions of the Euler-Poisson Equations and applications to wave breaking in klystrons Phys. Rev. E 70 2004 016502
22. Pierre-Louis Lions Thierry Paul Sur les mesures de Wigner Rev. Mat. Iberoamericana 9 3 1993 553-618
23. Hailiang Liu, Li-Tien Cheng, Stanley Osher, A level set framework for tracking multivalued solutions to nonlinear first-order equations, 2003, Preprint
24. D. Ludwig Uniform asymptotic expansions at a caustic Comm. Pure Appl. Math. 19 1966 215-250
25. P.A. Markowich P. Pietra C. Pohl Numerical approximations of quadratic observables of Schrödinger-type equations in the semiclassical limit Numer. Math. 81 1999 595-630
26. V.P. Maslov Perturbation theory and asymptotic methods 1965 Moskov. Gov. Univ.
27. V.P. Maslov Semiclassical Approximation in Quantum Mechanics 1981 Reidel Dordrecht
28. Chohong Min Simplicial isosurfacing in arbitrary dimension and codimension J. Comput. Phys. 190 2003 295-310
29. Stanley Osher Li-Tien Cheng Myungjoo Kang Hyeseon Shim Yen-Hsi Tsai Geometric optics in a phase-space-based level set and Eulerian framework J. Comput. Phys. 179 2 2002 622-648
30. Danping Peng Barry Merriman Stanley Osher Hongkai Zhao Myungjoo Kang A PDE-based fast local level set method J. Comput. Phys. 155 2 1999 410-438
31. Jianliang Qian Li-Tien Cheng Stanley Osher A level set-based Eulerian approach for anisotropic wave propagation Wave Motion 37 4 2003 365-379
32. Chi-Wang Shu Stanley Osher Efficient implementation of essentially nonoscillatory shock-capturing schemes II J. Comput. Phys. 83 1 1989 32-78
33. C. Sparber P.A. Markowich N.J. Mauser Multivalued geometrical optics: Wigner functions versus WKB-methods Asymptot. Anal. 33 2003 153-187
34. R.J. Spiteri, S.J. Ruuth, A new class of optimal high-order strong-stability-preserving time discretization methods, Preprint
35. John Strain Fast tree-based redistancing for level set computations J. Comput. Phys. 152 2 1999 664-686
36. John Strain Semi-Lagrangian methods for level set equations J. Comput. Phys. 151 2 1999 498-533
37. Anna-Karin Tornberg, Björn Enquist, Regularization techniques for numerical approximation of PDEs with singularities, UCLA CAM Report, 2 (37) (2002) J. Sci. Comp. 1-3, 527-552
38. Yen-Hsi Richard Tsai Yoshikazu Giga Stanley Osher A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations Math. Comp. 72 241 2003 159-181
39. E. Wigner On the quantum correction for thermodynamic equilibrium Phys. Rev. 40 1932 749-759
40. Ya.B. Zeldovich Gravitational instability: An approximate theory for large density perturbations Astron. Astrophys. 5 1970 84-89