Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials

Journal of Computational Physics

Volumn 229, Issue 13, 2010, Pages 4869-4883

  Jin, Shi ^a   Wu, Hao ^b   Yang, Xu ^a,c   Huang, Zhongyi ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b TSINGHUA UNIVERSITY   (China)

^c PRINCETON UNIVERSITY   (United States)

Author keywords

Bloch decomposition; Gaussian beam method; Liouville equation; Periodic potential; Schr dinger equation

Indexed keywords

GAUSSIAN BEAMS; GAUSSIAN DISTRIBUTION; NUMERICAL METHODS;

BEAM METHODS; BLOCH DECOMPOSITION; DIRECT SIMULATION; EULERIAN; GAUSSIAN BEAM METHOD; LIOUVILLE'S EQUATION; MESH SIZE; PERIODIC POTENTIALS; SCHRÖDINGE EQUATION; SOLID-STATE PHYSICS;

MESH GENERATION;

EID: 77952425757     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.01.025     Document Type: Article

References (40)

1. Ashcroft N.W., Mermin N.D. Solid State Physics 1976, Saunders, New York.
2. Bal G., Fannjiang A., Papanicolaou G., Ryzhik L. Radiative transport in a periodic structure. J. Statist. Phys. 1999, 95(1-2):479-494.
3. Bao W., Jin S., Markowich P.A. On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 2002, 175(2):487-524.
4. Bensoussan A., Lions J.L., Papanicolaou G. Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications 1978, vol. 5. North-Holland Publishing Co., Amsterdam, New York.
5. Carles R., Markowich P.A., Sparber C. Semiclassical asymptotics for weakly nonlinear Bloch waves. J. Stat. Phys. 2004, 117(1-2):343-375.
6. Cheng L.T., Liu H.L., Osher S. Computational high frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations. Commun. Math. Sci. 2003, 1:593-621.
7. Dimassi M., Guillot J.C., Ralston J. Gaussian beam construction for adiabatic pertubations. Math. Phys. Anal. Geom. 2006, 9(3):187-201.
8. C. Garcia-Cervera, J.F. Lu, Y. Xuan, W. E, A linear scaling subspace iteration algorithm with optimally localized non-orthogonal wave functions for Kohn-Sham density functional theory, preprint.
9. Gosse L. Using K-branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys. 2002, 180(1):155-182.
10. Gosse L. Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice II. Impurities, confinement and Bloch oscillations. J. Comput. Phys. 2004, 201:344-375.
11. Gosse L. The numerical spectrum of a one-dimensional Schrödinger operator with two competing periodic potentials. Commun. Math. Sci. 2007, 5:485-493.
12. Gosse L., Markowich P.A. Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice-I. Homogeneous problems. J. Comput. Phys. 2004, 197:387-417.
13. Gosse L., Mauser N. Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice III. From ab initio models to WKB for Schrödinger-Poisson. J. Comput. Phys. 2006, 211:326-346.
14. Hagedorn G.A. Semiclassical quantum mechanics. I. The h→0 limit for coherent states. Commun. Math. Phys. 1980, 71:77-93.
15. Heller E.J. Cellular dynamics: a new semiclassical approach to time-dependent quantum mechanics. J. Chem. Phys. 1991, 94:2723-2729.
16. Heller E.J. Guided Gaussian wave packets. Acc. Chem. Res. 2006, 39:127-134.
17. Hill N.R. Gaussian beam migration. Geophysics 1990, 55(11):1416-1428.
18. Huang Z., Jin S., Markowich P.A., Sparber C. A Bloch decomposition-based split-step pseudospectral method for quantum dynamics with periodic potentials. SIAM J. Sci. Comput. 2007, 29(2):515-538.
19. Huang Z., Jin S., Markowich P.A., Sparber C. Numerical simulation of the nonlinear Schrodinger equation with multi-dimensional periodic potentials. SIAM Multiscale Model. Simul. 2008, 7:539-564.
20. Huang Z., Jin S., Markowich P.A., Sparber C. On the Bloch decomposition-based spectral method for wave propagation in periodic media. Wave Motion 2009, 46:15-28.
21. Jin S., Liu H., Osher S., Tsai R. Computing multivalued physical observables the semiclassical limit of the Schrödinger equations. J. Compun. Phys. 2005, 205:222-241.
22. Jin S., Osher S. A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations. Commun. Math. Sci. 2003, 1(3):575-591.
23. Jin S., Wu H., Yang X. Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations. Commun. Math. Sci. 2008, 6(4):995-1020.
24. Jin S., Yang X. Computation of the semiclassical limit of the Schrödinger equation with phase shift by a level set method. J. Sci. Comput. 2008, 35(2-3):144-169.
25. Kohn W. Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 1959, 115:809-821.
26. Leung S., Qian J., Burridge R. Eulerian Gaussian beams for high frequency wave propagation. Geophysics 2007, 72:61-76.
27. Liu H., Wang Z. A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations. J. Comput. Phys. 2009, 228(9):3326-3344.
28. Min C. Simplicial isosurfacing in arbitrary dimension and codimension. J. Comput. Phys. 2003, 190(1):295-310.
29. M. Motamed, O. Runborg, Taylor expansion errors in Gaussian beam summation, preprint.
30. M. Motamed, O. Runborg, A wave front-based Gaussian beam method for computing high frequency waves, preprint.
31. Osher S., Cheng L.-T., Kang M., Shim H., Tsai Y.-H. Geometric optics in a phase-space-based level set and Eulerian framework. J. Comput. Phys. 2002, 179(2):622-648.
32. Panati G., Spohn H., Teufel S. Motions of electrons in adiabatically perturbed periodic structures. Analysis, modeling and simulation of multiscale problems 2006, 595-617. Springer, Berlin.
33. Peng D., Merriman B., Osher S., Zhao H., Kang M. A PDE based fast local level set method. J. Comput. Phys. 1999, 155:410-438.
34. Popov M.M. A new method of computation of wave fields using Gaussian beams. Wave Motion 1982, 4:85-97.
35. Ralston J. Gaussian beams and the propagation of singularities. Studies in PDEs, MAA Stud. Math. 1982, 23:206-248.
36. Sundaram G., Niu Qian Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects. Phys. Rev. B 1999, 59:14915-14925.
37. Tanushev N.M. Superpositions and higher order Gaussian beams. Commun. Math. Sci. 2008, 6(2):449-475.
38. Tanushev N.M., Qian J.L., Ralston J. Mountain waves and Gaussian beams. SIAM Multiscale Model. Simul. 2007, 6:688-709.
39. X. Wen, High order numerical methods to two dimensional delta function integrals in level set methods, preprint.
40. Wilcox C.H. Theory of Bloch waves. J. Anal. Math. 1978, 33:146-167.