Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2016, Article ID 8181927, 20 pages
http://dx.doi.org/10.1155/2016/8181927
Research Article

An Efficient Numerical Method for the Solution of the Schrödinger Equation

1School of Information Engineering, Chang’an University, Xi’an 710064, China
2Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
3Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00 Tripolis, Greece

Received 27 May 2016; Accepted 5 July 2016

Academic Editor: Maria L. Gandarias

Copyright © 2016 Licheng Zhang and Theodore E. Simos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. C. Allison, “The numerical solution of coupled differential equations arising from the Schrödinger equation,” Journal of Computational Physics, vol. 6, no. 3, pp. 378–391, 1970. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. J. D. Lambert and I. A. Watson, “Symmetric multistep methods for periodic initial values problems,” Journal of the Institute of Mathematics and Its Applications, vol. 18, no. 2, pp. 189–202, 1976. View at Google Scholar
  3. T. E. Simos and P. S. Williams, “A finite–difference method for the numerical solution of the Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 79, no. 2, pp. 189–205, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. R. M. Thomas, “Phase properties of high order, almost P-stable formulae,” BIT Numerical Mathematics, vol. 24, no. 2, pp. 225–238, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. A. D. Raptis and T. E. Simos, “A four-step phase-fitted method for the numerical integration of second order initial value problems,” BIT Numerical Mathematics, vol. 31, no. 1, pp. 160–168, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Z. A. Anastassi and T. E. Simos, “A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems,” Journal of Computational and Applied Mathematics, vol. 236, no. 16, pp. 3880–3889, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. Z. A. Anastassi and T. E. Simos, “An optimized Runge-Kutta method for the solution of orbital problems,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 1–9, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press, Oxford, UK, 2011.
  9. M. M. Chawla and P. S. Rao, “A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: explicit method,” Journal of Computational and Applied Mathematics, vol. 15, no. 3, pp. 329–337, 1986. View at Publisher · View at Google Scholar
  10. M. M. Chawla and P. S. Rao, “An explicit sixth–order method with phase–lag of order eight for y′′=ft,y,” Journal of Computational and Applied Mathematics, vol. 17, no. 3, pp. 363–368, 1987. View at Google Scholar
  11. C. J. Cramer, Essentials of Computational Chemistry, John Wiley & Sons, Chichester, UK, 2004.
  12. J. M. Franco and M. Palacios, “High-order P-stable multistep methods,” Journal of Computational and Applied Mathematics, vol. 30, no. 1, pp. 1–10, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. L. G. Ixaru and M. Rizea, “Comparison of some four-step methods for the numerical solution of the Schrödinger equation,” Computer Physics Communications, vol. 38, no. 3, pp. 329–337, 1985. View at Publisher · View at Google Scholar · View at Scopus
  14. L. G. Ixaru and M. Micu, Topics in Theoretical Physics, Central Institute of Physics, Bucharest, Romania, 1978.
  15. L. G. Ixaru and M. Rizea, “A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies,” Computer Physics Communications, vol. 19, no. 1, pp. 23–27, 1980. View at Publisher · View at Google Scholar · View at Scopus
  16. F. Jensen, Introduction to Computational Chemistry, John Wiley & Sons, Chichester, UK, 2007.
  17. Z. Kalogiratou, T. Monovasilis, and T. E. Simos, “New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation,” Computers & Mathematics with Applications, vol. 60, no. 6, pp. 1639–1647, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems,” Computers& Mathematics with Applications, vol. 61, no. 11, pp. 3381–3390, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. J. D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, New York, NY, USA, 1991.
  20. A. R. Leach, Molecular Modelling—Principles and Applications, Pearson, Essex, UK, 2001.
  21. T. Lyche, “Chebyshevian multistep methods for ordinary differential equations,” Numerische Mathematik, vol. 19, no. 1, pp. 65–75, 1972. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. T. Monovasilis, Z. Kalogiratou, and T. E. Simos, “A family of trigonometrically fitted partitioned Runge-Kutta symplectic methods,” Applied Mathematics and Computation, vol. 209, no. 1, pp. 91–96, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. T. Monovasilis, Z. Kalogiratou, and T. E. Simos, “Exponentially fitted symplectic Runge–Kutta–Nyström methods,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 81–85, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. G. A. Panopoulos and T. E. Simos, “An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown,” Journal of Computational and Applied Mathematics, vol. 290, pp. 1–15, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. G. A. Panopoulos and T. E. Simos, “An optimized symmetric 8-step semi-embedded predictor-corrector method for IVPs with oscillating solutions,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 73–80, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. G. A. Panopoulos and T. E. Simos, “A new optimized symmetric Embedded Predictor-Corrector Method (EPCM) for initial-value problems with oscillatory solutions,” Applied Mathematics and Information Sciences, vol. 8, no. 2, pp. 703–713, 2014. View at Publisher · View at Google Scholar · View at Scopus
  27. D. F. Papadopoulos and T. E. Simos, “A modified Runge-Kutta-Nyström method by using phase lag properties for the numerical solution of orbital problems,” Applied Mathematics & Information Sciences, vol. 7, no. 2, pp. 433–437, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. D. F. Papadopoulos and T. E. Simos, “The use of phase lag and amplification error derivatives for the construction of a modified Runge-Kutta-Nyström method,” Abstract and Applied Analysis, vol. 2013, Article ID 910624, 11 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  29. G. D. Quinlan and S. Tremaine, “Symmetric multistep methods for the numerical integration of planetary orbits,” Astronomical Journal, vol. 100, no. 5, pp. 1694–1700, 1990. View at Publisher · View at Google Scholar · View at Scopus
  30. A. D. Raptis and A. C. Allison, “Exponential-fitting methods for the numerical solution of the Schrödinger equation,” Physics Communications, vol. 14, no. 1-2, pp. 1–5, 1978. View at Publisher · View at Google Scholar
  31. D. P. Sakas and T. E. Simos, “Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 161–172, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. T. E. Simos, “Dissipative trigonometrically-fitted methods for linear second-order IVP s with oscillating solution,” Applied Mathematics Letters, vol. 17, no. 5, pp. 601–607, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. T. E. Simos, “Closed Newton–Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616–1621, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. T. E. Simos, “Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1331–1352, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. T. E. Simos, “New stable closed Newton-Cotes trigonometrically fitted formulae for long-time integration,” Abstract and Applied Analysis, vol. 2012, Article ID 182536, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  36. T. E. Simos, “Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag,” Journal of Applied Mathematics, vol. 2012, Article ID 420387, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  37. T. E. Simos, “High order closed Newton–Cotes trigonometrically–fitted formulae for the numerical solution of the Schrödinger equation,” Applied Mathematics and Computation, vol. 209, no. 1, pp. 137–151, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. H. Ramos, Z. Kalogiratou, Th. Monovasilis, and T. E. Simos, “An optimized two-step hybrid block method for solving general second order initial-value problems,” Numerical Algorithms, vol. 72, no. 4, pp. 1089–1102, 2016. View at Publisher · View at Google Scholar · View at Scopus
  39. T. Monovasilis, Z. Kalogiratou, and T. E. Simos, “Construction of exponentially fitted symplectic Runge-Kutta-Nyström methods from partitioned Runge-Kutta methods,” Mediterranean Journal of Mathematics, vol. 13, no. 4, pp. 2271–2285, 2016. View at Publisher · View at Google Scholar · View at Scopus
  40. T. E. Simos, “On the explicit four-step methods with vanished phase-lag and its first derivative,” Applied Mathematics α Information Sciences, vol. 8, no. 2, pp. 447–458, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. T. E. Simos, “Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems,” Computational Materials Science, vol. 18, no. 3-4, pp. 315–332, 2000. View at Publisher · View at Google Scholar · View at Scopus
  42. S. Stavroyiannis and T. E. Simos, “Optimization as a function of the phase–lag order of nonlinear explicit two–step P-stable method for linear periodic IVPs,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2467–2474, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. E. Stiefel and D. G. Bettis, “Stabilization of Cowell's method,” Numerische Mathematik, vol. 13, pp. 154–175, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  44. K. Tselios and T. E. Simos, “Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 173–181, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. C. Tsitouras, I. T. Famelis, and T. E. Simos, “On modified Runge-Kutta trees and methods,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 2101–2111, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. R. Vujasin, M. Senčanski, J. Radić-Perić, and M. Perić, “A comparison of various variational approaches for solving the one-dimensional vibrational Schrödinger equation,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 63, no. 2, pp. 363–378, 2010. View at Google Scholar
  47. A. D. Raptis and J. R. Cash, “A variable step method for the numerical integration of the one–dimensional Schrödinger equation,” Computer Physics Communications, vol. 36, no. 2, pp. 113–119, 1985. View at Publisher · View at Google Scholar · View at Scopus
  48. J. R. Dormand, M. E. A. El-Mikkawy, and P. J. Prince, “Families of Runge-Kutta-Nyström formulae,” IMA Journal of Numerical Analysis, vol. 7, no. 2, pp. 235–250, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  49. J. R. Dormand and P. J. Prince, “A family of embedded Runge-Kutta formulae,” Journal of Computational and Applied Mathematics, vol. 6, no. 1, pp. 19–26, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  50. K. Mu and T. E. Simos, “A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 53, no. 5, pp. 1239–1256, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  51. H. Ning and T. E. Simos, “A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 53, no. 6, pp. 1295–1312, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  52. M. Liang and T. E. Simos, “A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 54, no. 5, pp. 1187–1211, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  53. X. Xi and T. E. Simos, “A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems,” Journal of Mathematical Chemistry, vol. 54, no. 7, pp. 1417–1439, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  54. Z. Zhou and T. E. Simos, “A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation,” Journal of Mathematical Chemistry, vol. 54, no. 2, pp. 442–465, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  55. T. E. Simos, “Multistage symmetric two-step P-stable method with vanished phase-lag and its first, second and third derivatives,” Applied and Computational Mathematics, vol. 14, no. 3, pp. 296–315, 2015. View at Google Scholar · View at MathSciNet
  56. R. B. Bernstein, A. Dalgarno, H. Massey, and I. C. Percival, “Thermal scattering of atoms by homonuclear diatomic molecules,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 274, no. 1359, pp. 427–442, 1963. View at Publisher · View at Google Scholar
  57. R. B. Bernstein, “Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams,” The Journal of Chemical Physics, vol. 33, no. 3, pp. 795–804, 1960. View at Publisher · View at Google Scholar · View at Scopus