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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 641236, 15 pages
http://dx.doi.org/10.1155/2012/641236
Research Article

A New Optimized Runge-Kutta Pair for the Numerical Solution of the Radial Schrödinger Equation

1School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China
2Department of Mathematics, Chuzhou University, Chuzhou 239000, China

Received 9 May 2012; Revised 23 September 2012; Accepted 9 October 2012

Academic Editor: Malisa R. Zizovic

Copyright © 2012 Yonglei Fang et al. 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, pp. 378–391, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J. M. Blatt, “Practical points concerning the solution of the Schrödinger equation,” Journal of Computational Physics, vol. 1, pp. 378–391, 1961.
  3. J. W. Cooley, “An improved eigenvalue corrector formula for solving the Schrödinger equation for central fields,” Mathematics of Computation, vol. 15, pp. 363–374, 1961. View at Zentralblatt MATH
  4. T. E. Simos, “A family of fifth algebraic order trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation,” Computational Materials Science, vol. 34, no. 4, pp. 342–354, 2005. View at Publisher · View at Google Scholar · View at Scopus
  5. H. Van De Vyver, “An embedded phase-fitted modified Runge-Kutta method for the numerical integration of the radial Schrödinger equation,” Physics Letters A, vol. 352, no. 4-5, pp. 278–285, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. T. E. Simos, “An embedded Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schrödinger equation,” International Journal of Modern Physics C, vol. 11, no. 6, pp. 1115–1133, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. T. E. Simos and J. V. Aguiar, “A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 30, no. 1, pp. 121–131, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. 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 Zentralblatt MATH
  9. 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 Zentralblatt MATH
  10. 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 Zentralblatt MATH
  11. Z. Kalogiratou, Th. Monovasilis, and T. E. Simos, “Symplectic integrators for the numerical solution of the Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 83–92, 2003, Selected papers from the Conference on Computational and Mathematical Methods for Science and Engineering (Alicante, 2002). View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. T. E. Simos and P. S. Williams, “On finite difference methods for the solution of the Schrödinger equation,” Computers and Chemistry, vol. 23, no. 6, pp. 513–554, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. A. Konguetsof and T. E. Simos, “A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 93–106, 2003, Selected papers from the Conference on Computational and Mathematical Methods for Science and Engineering (Alicante, 2002). View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. G. Vanden Berghe, H. De Meyer, M. Van Daele, and T. Van Hecke, “Exponentially-fitted explicit Runge-Kutta methods,” Computer Physics Communications, vol. 123, no. 1–3, pp. 7–15, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. A. D. Raptis and T. E. Simos, “A four-step phase-fitted method for the numerical integration of second order initial value problems,” BIT, vol. 31, no. 1, pp. 160–168, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Z. A. Anastassi and T. E. Simos, “Numerical multistep methods for the efficient solution of quantum mechanics and related problems,” Physics Reports, vol. 482-483, pp. 1–240, 2009. View at Publisher · View at Google Scholar
  17. Z. Kalogiratou and T. E. Simos, “Newton-Cotes formulae for long-time integration,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 75–82, 2003, Selected papers from the Conference on Computational and Mathematical Methods for Science and Engineering (Alicante, 2002). View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. G. Psihoyios and T. E. Simos, “Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 135–144, 2003, Selected papers from the Conference on Computational and Mathematical Methods for Science and Engineering (Alicante, 2002). View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. T. E. Simos, I. T. Famelis, and C. Tsitouras, “Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions,” Numerical Algorithms, vol. 34, no. 1, pp. 27–40, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. T. E. Simos, “Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution,” Applied Mathematics Letters, vol. 17, no. 5, pp. 601–607, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. 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 Zentralblatt MATH
  22. G. Psihoyios and T. E. Simos, “A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 137–147, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. 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
  24. 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 Zentralblatt MATH
  25. 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
  26. 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 Zentralblatt MATH
  27. 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
  28. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations. I Nonstiff Problems, vol. 8, Springer, Berlin, Germany, 2nd edition, 1993.
  29. 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
  30. 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 Zentralblatt MATH
  31. A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “An optimized explicit Runge-Kutta-Nyström method for the numerical solution of orbital and related periodical initial value problems,” Computer Physics Communications, vol. 183, no. 3, pp. 470–479, 2012. View at Publisher · View at Google Scholar
  32. I. Alolyan and T. E. Simos, “High algebraic order methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 48, no. 4, pp. 925–958, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. I. Alolyan and T. E. Simos, “A new hybrid 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. 50, no. 7, pp. 1861–1881, 2011. View at Publisher · View at Google Scholar · View at Scopus
  34. I. Alolyan and T. E. Simos, “A family of ten-step methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 49, no. 9, pp. 1843–1888, 2011. View at Publisher · View at Google Scholar
  35. I. Alolyan and T. E. Simos, “A family of eight-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 49, no. 3, pp. 711–764, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. I. Alolyan and T. E. Simos, “Multistep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 48, no. 4, pp. 1092–1143, 2010. View at Publisher · View at Google Scholar
  37. I. Alolyan and T. E. Simos, “On eight-step methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation,” Communications in Mathematical and in Computer Chemistry, vol. 66, no. 2, pp. 473–546, 2011.
  38. T. E. Simos, “A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 49, no. 10, pp. 2486–2518, 2011.
  39. I. Alolyan and T. E. Simos, “A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation,” Computers & Mathematics with Applications, vol. 62, no. 10, pp. 3756–3774, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. I. Alolyan and T. E. Simos, “A new high order two-step method with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation,” Journal of Mathematical Chemistry, vol. 50, no. 9, pp. 2351–2373, 2012.
  41. T. E. Simos, “Runge-Kutta interpolants with minimal phase-lag,” Computers & Mathematics with Applications, vol. 26, no. 8, pp. 43–49, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. J. D. Lambert and I. A. Watson, “Symmetric multistep methods for periodic initial value problems,” Journal of the Institute of Mathematics and its Applications, vol. 18, no. 2, pp. 189–202, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. J. P. Coleman and L. Gr. Ixaru, “P-stability and exponential-fitting methods for y=f(x,y),” IMA Journal of Numerical Analysis, vol. 16, no. 2, pp. 179–199, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. H. Van de Vyver, “Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems,” Computer Physics Communications, vol. 173, no. 3, pp. 115–130, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. Y. Fang, Y. Song, and X. Wu, “New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems,” Physics Letters. A, vol. 372, no. 44, pp. 6551–6559, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  46. 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