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Symmetric Eighth Algebraic Order Methods with Minimal Phase-Lag for the Numerical Solution of the Schrödinger Equation

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Abstract

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.

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References

  1. L.Gr. Ixaru and M.Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978).

    Google Scholar 

  2. L.Gr. Ixaru and M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, Comput. Phys. Commun. 19 (1980) 23–27.

    Google Scholar 

  3. L.D. Landau and F.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965).

    Google Scholar 

  4. G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, Toronto, 1950).

    Google Scholar 

  5. J.R. Cash and A.D. Raptis, A high order method for the numerical solution of the one-dimensional Schrödinger equation, Comput. Phys. Commun. 33 (1984) 299–304.

    Google Scholar 

  6. J.R. Cash, A.D. Raptis and T.E. Simos, A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation, J. Comput. Phys. 91 (1990) 413–423.

    Google Scholar 

  7. L.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl. 18 (1976) 189–202.

    Google Scholar 

  8. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).

    Google Scholar 

  9. R.M. Thomas, Phase properties of high order, almost P-stable formulae, BIT 24 (1984) 225–238.

    Google Scholar 

  10. T.E. Simos, Numerical solution of ordinary differential equations with periodical solution, Doctoral Dissertation, National Technical University of Athens (1990).

  11. T.E. Simos, A new Numerov-type method for computing eigenvalues and resonances of the radial Schrödinger equation, Internat. J. Modern Phys. C 7 (1996) 33–41.

    Google Scholar 

  12. T.E. Simos, An eighth order method with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem, Internat. J. Modern Phys. C 7 (1996) 825–835.

    Google Scholar 

  13. T.E. Simos, An eighth order exponentially-fitted method for the numerical solution of the Schrödinger equation, Internat. J. Modern Phys. C 9 (1998) 271–288.

    Google Scholar 

  14. T.E. Simos, High algebraic order methods with minimal phase-lag for accurate solution of the Schrödinger equation, Internat. J. Modern Phys. C 9 (1998) 1055–1071.

    Google Scholar 

  15. T.E. Simos and P.S.Williams, Exponentially-fitted Runge-Kutta third algebraic order methods for the numerical solution of the Schrödinger equation and related problems, Internat. J. Modern Phys. C 10 (1999) 830–851.

    Google Scholar 

  16. T.E. Simos and P.S. Williams, On finite difference methods for the solution of the Schrödinger equation, Comput. & Chemistry 23 (1999) 513–554.

    Google Scholar 

  17. T.E. Simos, Atomic structure computations in chemical modeling, in: Applications and Theory UMIST, Vol. 1, ed. A. Hinchliffe (The Royal Society of Chemistry, 2000) pp. 38–142.

  18. T.E. Simos, Numerical methods for 1D, 2D and 3D differential equations arising in chemical problems in chemical modeling, in: Applications and Theory, UMIST, Vol. 2, ed. A. Hinchliffe (The Royal Society of Chemistry) (to appear).

  19. M.M. Chawla and P.S. Rao, A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems, J. Comput. Appl. Math. 11 (1984) 277–281.

    Google Scholar 

  20. E. Stiefel and D.G. Bettis, Stabilization of Cowell's method, Numer. Math. 13 (1969) 154–175.

    Google Scholar 

  21. G.D. Quinlan and S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, Astronom. J. 100 (1990) 1694–1700.

    Google Scholar 

  22. A. Konguetsof and T.E. Simos, On the construction of exponentially-fitted methods for the numerical solution of the Schrödinger equation, J. Comput. Methods Sci. Engrg. 1 (2001) 143–162.

    Google Scholar 

  23. T.E. Simos, Eighth order methods with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem, J. Math. Chem. 21 (1997) 359–372.

    Google Scholar 

  24. T.E. Simos, Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations, J. Math. Chem. 24 (1998) 23–37.

    Google Scholar 

  25. T.E. Simos, A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 25 (1999) 65–84.

    Google Scholar 

  26. G. Avdelas and T.E. Simos, Embedded eighth order methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 26 (1999) 327–341.

    Google Scholar 

  27. J. Vigo-Aguiar and T.E. Simos, A family of P-stable eighth algebraic order methods with exponential fitting facilities, J. Math. Chem. 29 (2001) 177–189.

    Google Scholar 

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Simos, T., Vigo-Aguiar, J. Symmetric Eighth Algebraic Order Methods with Minimal Phase-Lag for the Numerical Solution of the Schrödinger Equation. Journal of Mathematical Chemistry 31, 135–144 (2002). https://doi.org/10.1023/A:1016259830419

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