Advertisement

An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems

  • A. A. Kosti
  • Z. A. Anastassi
  • T. E. Simos
Original Paper

Abstract

In this paper we present an optimized explicit Runge-Kutta method, which is based on a method of Fehlberg with six stages and fifth algebraic order and has improved characteristics of the phase-lag error. We measure the efficiency of the new method in comparison to other numerical methods, through the integration of the Schrödinger equation and three other initial value problems.

Keywords

Numerical solution Initial value problems (IVPs) Explicit methods Runge-Kutta methods Schrödinger equation 

References

  1. 1.
    Ixaru L.Gr., Rizea M.: A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comp. Phys. Comm. 19, 23–27 (1980)CrossRefGoogle Scholar
  2. 2.
    Simos T.E., Tsitouras Ch.: A P-stable eighth order method for the numerical integration of periodic initial value problems. J. Comput. Phys. 130, 123–128 (1997)CrossRefGoogle Scholar
  3. 3.
    Simos T.E.: Chemical Modelling—Applications and Theory Vol. 1, Specialist Periodical Reports, pp. 32–140. The Royal Society of Chemistry, Cambridge (2000)Google Scholar
  4. 4.
    Engeln-Mullges G., Uhlig F.: Numerical Algorithms with Fortran, pp. 423–488. Springer-Verlag, Berlin Heidelberg (1996)Google Scholar
  5. 5.
    Franco J.M.: Runge-Kutta methods adapted to the numerical integration of oscillatory problems. Appl. Numer. Math. 50(3–4), 427–443 (2004)CrossRefGoogle Scholar
  6. 6.
    Van de Vyver H.: Comparison of some special optimized fourth-order Runge-Kutta methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 166, 109–122 (2005)CrossRefGoogle Scholar
  7. 7.
    Aceto L., Sestini A.: Numerical aspects of the coefficient computation for LMMs. J. Numer. Anal., Ind. Appl. Math. 3, 181–191 (2008)Google Scholar
  8. 8.
    Chatterjee S., Isaiay M., Bonay F., Badinoy G., Venturino E.: Modelling environmental influences on wanderer spiders in the Langhe region (Piemonte-NW Italy). J. Numer. Anal., Ind. Appl. Math. 3, 193–209 (2008)Google Scholar
  9. 9.
    Dagnino C., Demichelis V., Lamberti P.: A nodal spline collocation method for the solution of cauchy singular integral equations. J. Numer. Anal., Ind. Appl. Math. 3, 211–220 (2008)Google Scholar
  10. 10.
    Enachescu C.: Approximation capabilities of neural networks. J. Numer. Anal., Ind. Appl. Math. 3, 221–230 (2008)Google Scholar
  11. 11.
    Frederich O., Wassen E., Thiele F.: Prediction of the flow around a short wall-mounted finite cylinder using LES and DES. J. Numer. Anal., Ind. Appl. Math. 3, 231–247 (2008)Google Scholar
  12. 12.
    Ogata H.: Fundamental solution method for periodic plane elasticity. J. Numer. Anal., Ind. Appl. Math. 3, 249–267 (2008)Google Scholar
  13. 13.
    An P.T.: Some computational aspects of helly-type theorems. J. Numer. Anal., Ind. Appl. Math. 3, 269–274 (2008)Google Scholar
  14. 14.
    Verhoeven A., Tasic B., Beelen T.G.J., ter Maten E.J.W., Mattheij R.M.M.: BDF compound-fast multirate transient analysis with adaptive stepsize control. J. Numer. Anal., Ind. Appl. Math. 3, 275–297 (2008)Google Scholar
  15. 15.
    Anastassi Z.A., Simos T.E.: Special optimized Runge-Kutta methods for IVPs with oscillating solutions. Int. J. Mod. Phys. C 15, 1–15 (2004)CrossRefGoogle Scholar
  16. 16.
    T.E. Simos, in Atomic Structure Computations in Chemical Modelling: Applications and Theory, ed. by A. Hinchliffe. UMIST, (The Royal Society of Chemistry, Cambridge, 2000), pp. 38–142Google Scholar
  17. 17.
    T.E. Simos, Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems. In Chemical Modelling: Application and Theory, Vol. 2, (The Royal Society of Chemistry, Cambridge, 2002), pp. 170–270Google Scholar
  18. 18.
    Anastassi Z.A., Simos T.E.: A family of exponentially-fitted Runge-Kutta methods with exponential order up to three for the numerical solution of the Schrödinger equation. J. Math. Chem. 41(1), 79–100 (2007)CrossRefGoogle Scholar
  19. 19.
    Monovasilis T., Kalogiratou Z., Simos T.E.: Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 40(3), 257–267 (2006)CrossRefGoogle Scholar
  20. 20.
    Psihoyios G., Simos T.E.: The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order predictor-corrector methods. J. Math. Chem. 40(3), 269–293 (2006)CrossRefGoogle Scholar
  21. 21.
    Simos T.E.: A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem. 40(3), 305–318 (2006)CrossRefGoogle Scholar
  22. 22.
    Monovasilis T., Kalogiratou Z., Simos T.E.: Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 37(3), 263–270 (2005)CrossRefGoogle Scholar
  23. 23.
    Kalogiratou Z., Monovasilis T., Simos T.E.: Numerical solution of the two-dimensional time independent Schrödinger equation with numerov-type methods. J. Math. Chem. 37(3), 271–279 (2005)CrossRefGoogle Scholar
  24. 24.
    Anastassi Z.A., Simos T.E.: Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 281–293 (2005)CrossRefGoogle Scholar
  25. 25.
    Psihoyios G., Simos T.E.: Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 37(3), 295–316 (2005)CrossRefGoogle Scholar
  26. 26.
    Sakas D.P., Simos T.E.: A family of multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 317–331 (2005)CrossRefGoogle Scholar
  27. 27.
    Simos T.E.: Exponentially-fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)CrossRefGoogle Scholar
  28. 28.
    Tselios K., Simos T.E.: Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation. J. Math. Chem. 35(1), 55–63 (2004)CrossRefGoogle Scholar
  29. 29.
    Simos T.E.: A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems. J. Math. Chem. 34(1–2), 39–58 (2003)CrossRefGoogle Scholar
  30. 30.
    Tselios K., Simos T.E.: Symplectic methods for the numerical solution of the radial Shrödinger equation. J. Math. Chem. 34(1–2), 83–94 (2003)CrossRefGoogle Scholar
  31. 31.
    Vigo-Aguiar J., Simos T.E.: Family of twelve steps exponential fitting symmetric multistep methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 32(3), 257–270 (2002)CrossRefGoogle Scholar
  32. 32.
    Avdelas G., Kefalidis E., Simos T.E.: New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 31(4), 371–404 (2002)CrossRefGoogle Scholar
  33. 33.
    Simos T.E., Vigo-Aguiar J.: Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation. J. Math. Chem. 31(2), 135–144 (2002)CrossRefGoogle Scholar
  34. 34.
    Kalogiratou Z., Simos T.E.: Construction of trigonometrically and exponentially fitted Runge-Kutta-Nystrom methods for the numerical solution of the Schrödinger equation and related problems a method of 8th algebraic order. J. Math. Chem. 31(2), 211–232 (2002)CrossRefGoogle Scholar
  35. 35.
    Simos T.E., Vigo-Aguiar J.: A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. J. Math. Chem. 30(1), 121–131 (2001)CrossRefGoogle Scholar
  36. 36.
    Avdelas G., Konguetsof A., Simos T.E.: A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 1. Development of the basic method. J. Math. Chem. 29(4), 281–291 (2001)CrossRefGoogle Scholar
  37. 37.
    Avdelas G., Konguetsof A., Simos T.E.: A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 2. Development of the generator; optimization of the generator and numerical results. J. Math. Chem. 29(4), 293–305 (2001)CrossRefGoogle Scholar
  38. 38.
    Vigo-Aguiar J., Simos T.E.: A family of P-stable eighth algebraic order methods with exponential fitting facilities. J. Math. Chem. 29(3), 177–189 (2001)CrossRefGoogle Scholar
  39. 39.
    Simos T.E.: A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation. J. Math. Chem. 27(4), 343–356 (2000)CrossRefGoogle Scholar
  40. 40.
    Avdelas G., Simos T.E.: Embedded eighth order methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 26(4), 327–341 (1999)CrossRefGoogle Scholar
  41. 41.
    Simos T.E.: A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 25(1), 65–84 (1999)CrossRefGoogle Scholar
  42. 42.
    Simos T.E.: Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations. J. Math. Chem. 24(1–3), 23–37 (1998)CrossRefGoogle Scholar
  43. 43.
    Simos T.E.: Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem. J. Math. Chem. 21(4), 359–372 (1997)CrossRefGoogle Scholar
  44. 44.
    Simos T.E.: Predictor corrector phase-fitted methods for y”=f(x,y) and an application to the Schrödinger-equation. Int. J. Quantum Chem. 53(5), 473–483 (1995)CrossRefGoogle Scholar
  45. 45.
    Simos T.E.: A new numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation. Int. J. Mod. Phys. C-Phys. Comput. 7(1), 33–41 (1996)CrossRefGoogle Scholar
  46. 46.
    Simos T.E., Mousadis G.: Some new numerov-type methods with minimal phase-lag for the numerical-integration of the Radial Schrödinger-equation. Mol. Phys. 83(6), 1145–1153 (1994)CrossRefGoogle Scholar
  47. 47.
    Simos T.E.: A numerov-type method for the numerical-solution of the radial Schrödinger-equation. Appl. Numer. Math. 7(2), 201–206 (1991)CrossRefGoogle Scholar
  48. 48.
    Avdelas G., Simos T.E.: Dissipative high phase-lag order numerov-type methods for the numerical solution of the Schrödinger equation. Phys. Rev. E 62(1), 1375–1381 (2000)CrossRefGoogle Scholar
  49. 49.
    Simos T.E., Williams P.S.: Bessel and Neumann-fitted methods for the numerical solution of the radial Schrödinger equation. Comput. Chem. 21(3), 175–179 (1997)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and TechnologyUniversity of PeloponneseTripolisGreece
  2. 2.AthensGreece

Personalised recommendations