Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations

Applied Numerical Mathematics

Volumn 61, Issue 4, 2011, Pages 593-614

  Gao, Zhen ^a   Xie, Shusen ^a  

^a OCEAN UNIVERSITY OF CHINA   (China)

Author keywords

ADI compact difference scheme; Conservation law; Error estimate; Schr dinger equation

Indexed keywords

ALTERNATING DIRECTION IMPLICIT; COMPACT DIFFERENCE SCHEME; COMPACT FINITE DIFFERENCES; CONSERVATION LAW; CONVERGENCE RATES; DINGER EQUATION; ENGINEERING INTERESTS; ERROR ESTIMATE; FOURTH-ORDER; NUMERICAL EXPERIMENTS; NUMERICAL SOLUTION;

PHYSICAL PROPERTIES; TWO DIMENSIONAL;

CONVERGENCE OF NUMERICAL METHODS;

EID: 78651314088     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2010.12.004     Document Type: Article

References (49)

1. J. Argyris, and M. Haase An engineer's guide to solitons phenomena: application of the finite element method Comput. Methods Appl. Mech. Engrg. 61 1987 71 122
2. A. Arnold Numerically absorbing boundary conditions for quantum evolution equations VLSI Des. 6 1998 313 319
3. Q. Chang, E. Jia, and W. Sun Difference schemes for solving the generalized nonlinear Schrödinger equation J. Comput. Phys. 148 1999 397 415
4. C. Clavero, J.L. Gracia, and J.C. Jorge A uniformly convergent alternating direction HODIE finite difference scheme for 2D time-dependent convection-diffusion problems IMA J. Numer. Anal. 26 2006 155 172
5. C. Clavero, J.C. Jorge, F. Lisbona, and G.I. Shishkin A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems Appl. Numer. Math. 27 1998 211 231
6. C. Clavero, J.C. Jorge, F. Lisbona, and G.I. Shishkin An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problems IMA J. Numer. Anal. 20 2000 263 280
7. A.S. Davydov Solitons in Molecular Systems 1985 Reidel Dordrecht
8. M. Dehghan Alternating direction implicit methods for two-dimensional diffusion with a nonlocal boundary condition Int. J. Comput. Math. 72 1999 349 366 (Pubitemid 129585969)
9. M. Dehghan A new ADI technique for two-dimensional parabolic equation with an integral condition Comput. Math. Appl. 43 2002 1477 1488 (Pubitemid 34573351)
10. M. Dehghan Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices Math. Comput. Simulation 71 2006 16 30
11. M. Dehghan, and D. Mirzaei Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method Internat. J. Numer. Methods Engrg. 76 2008 501 520
12. M. Dehghan, and D. Mirzaei The meshless local Petrov-Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation Eng. Anal. Bound. Elem. 32 2008 747 756
13. M. Dehghan, and A. Shokri A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions Comput. Math. Appl. 54 2007 136 146
14. M. Dehghan, and A. Taleei A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients Comput. Phys. Comm. 181 2010 43 51
15. M. Dehghan, and A. Taleei Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method Numer. Methods Partial Differential Equations 26 2010 979 992
16. M. Delfour, M. Fortin, and G. Payre Finite difference solution of a nonlinear Schrödinger equation J. Comput. Phys. 44 1981 277 288
17. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris Solitons and Nonlinear Wave Equations 1982 Academic Press New York
18. J. Douglas Jr. On the numerical integration of b 2 u b x 2 + b 2 u b y 2 = b u b t by implicit methods J. Soc. Ind. Appl. Math. 3 1955 42 65
19. J. Douglas Jr., and D. Peaceman Numerical solution of two-dimensional heat flow problems AIChE J. 1 1955 505 512
20. J. Douglas Jr., and H. Rachford On the numerical solution of heat conduction problems in two and three space variables Trans. Amer. Math. Soc. 82 1960 421 439
21. L.R.T. Gardner, G.A. Gardner, S.I. Zaki, and Z. El Sahrawi B-spline finite element studies of the non-linear Schrödinger equation Comput. Methods Appl. Mech. Engrg. 108 1993 303 318
22. R.T. Glassey On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations J. Math. Phys. 18 1977 1794 1797
23. F.Y. Hajj Solution of the Schrodinger equation in two and three dimensions J. Phys. B: At. Mol. Phys. 18 1985 1 11
24. A. Hasegawa Optical Solitons in Fibers 1989 Springer-Verlag Berlin
25. B.M. Herbst, J.L. Morris, and A.R. Mitchell Numerical experience with the nonlinear Schrödinger equation J. Comput. Phys. 60 1985 282 305
26. W. Huang, C. Xu, S.T. Chu, and S.K. Chaudhuri The finite difference vector beam propagation method J. Lightwave Technol. 10 1992 295 304
27. L.Gr. Ixaru Operations on oscillatory functions Comput. Phys. Comm. 105 1997 1 9
28. J.C. Kalita, P. Chhabra, and S. Kumar A semi-discrete higher order compact scheme for the unsteady two-dimensional Schrödinger equation J. Comput. Appl. Math. 197 2006 141 149 (Pubitemid 44176245)
29. O. Karakashian, and C. Makridakis A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method Math. Comp. 67 1998 479 499 (Pubitemid 128381533)
30. S. Kim, and H. Lim High-order schemes for acoustic waveform simulation Appl. Numer. Math. 57 2007 402 414
31. M. Levy Parabolic Equation Methods for Electromagnetic Wave Propagation 2000 IEE
32. H. Lim, S. Kim, and J. Douglas Jr. Numerical method for viscous and non-viscous wave equations Appl. Numer. Math. 57 2007 194 212
33. A. Mohebbi, and M. Dehghan The use of compact boundary value method for the solution of two-dimensional Schrödinger equation J. Comput. Appl. Math. 225 2009 124 134
34. D. Pathria, and J.L. Morris Pseudo-spectral solution of nonlinear Schrödinger equations J. Comput. Phys. 87 1990 108 125
35. D. Peaceman, and H. Rachford The numerical solution of parabolic and elliptic equations J. Soc. Ind. Appl. Math. 3 1955 28 41
36. J.M. Sanz-Serna Methods for the numerical solution of the nonlinear Schrödinger equation Math. Comp. 43 1984 21 27
37. M. Subasi On the finite difference schemes for the numerical solution of two dimensional Schrödinger equation Numer. Methods Partial Differential Equations 18 2002 752 758
38. C. Sulem, and P.L. Sulem The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse 1999 Springer New York
39. P.L. Sulem, C. Sulem, and A. Patera Numerical simulation of singular solutions to the two-dimensional cubic Schrödinger equation Comm. Pure Appl. Math. 37 1984 755 778
40. T.R. Taha, and M.J. Ablowitz Analytical and numerical aspects of certain nonlinear evolution equations, II. Numerical nonlinear Schrödinger equation J. Comput. Phys. 55 1984 203 230
41. F.D. Tappert The parabolic approximation method J.B. Keller, J.S. Papadakis, Wave Propagation and Underwater Acoustics Lecture Notes in Physics vol. 70 1977 Springer Berlin 224 287
42. Y. Tourigny, and J.L. Morris An investigation into the effect of product approximation in the numerical solution of the cubic nonlinear Schrödinger equation J. Comput. Phys. 76 1988 103 130
43. E.H. Twizell, A.G. Bratsos, and J.C. Newby A finite-difference method for solving the cubic Schrödinger equation Math. Comput. Simulation 43 1997 67 75
44. L. Wu DuFort-Frankel-type methods for linear and nonlinear Schrödinger equations SIAM J. Numer. Anal. 33 1996 1526 1533
45. S. Xie, G. Li, and S. Yi Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schröinger equation Comput. Methods Appl. Mech. Engrg. 198 2009 1052 1060
46. Y. Xu, and C.-W. Shu Local discontinuous Galerkin methods for nonlinear Schrödinger equations J. Comput. Phys. 205 2005 72 97
47. N.N. Yanenko The Method of Fractional Steps 1971 Springer-Verlag New York
48. V.E. Zakharov, and V.S. Synakh The nature of self-focusing singularity Sov. Phys. JETP 41 1975 465 468
49. F. Zhang, V.M. Pérez-Grarciz, and L. Vázquez Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme Appl. Math. Comput. 71 1995 165 177