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

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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.

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References

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Engeln-Mullges G., Uhlig F.: Numerical Algorithms with Fortran, pp. 423–488. Springer-Verlag, Berlin Heidelberg (1996)

    Google Scholar 

  5. Franco J.M.: Runge-Kutta methods adapted to the numerical integration of oscillatory problems. Appl. Numer. Math. 50(3–4), 427–443 (2004)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. 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. Enachescu C.: Approximation capabilities of neural networks. J. Numer. Anal., Ind. Appl. Math. 3, 221–230 (2008)

    Google Scholar 

  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. Ogata H.: Fundamental solution method for periodic plane elasticity. J. Numer. Anal., Ind. Appl. Math. 3, 249–267 (2008)

    Google Scholar 

  13. An P.T.: Some computational aspects of helly-type theorems. J. Numer. Anal., Ind. Appl. Math. 3, 269–274 (2008)

    Google Scholar 

  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. 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)

    Article  Google Scholar 

  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. 38–142

  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. 170–270

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  27. Simos T.E.: Exponentially-fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

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Authors and Affiliations

  1. Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, 22 100, Tripolis, Greece

    A. A. Kosti, Z. A. Anastassi & T. E. Simos

  2. 10 Konitsis Street, Amfithea-Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

Authors
  1. A. A. Kosti
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  2. Z. A. Anastassi
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  3. T. E. Simos
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Correspondence to T. E. Simos.

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T. E. Simos is an active member of the European academy of sciences and arts, active member of the European academy of sciences.

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Kosti, A.A., Anastassi, Z.A. & Simos, T.E. An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems. J Math Chem 47, 315–330 (2010). https://doi.org/10.1007/s10910-009-9571-z

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  • DOI: https://doi.org/10.1007/s10910-009-9571-z

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