Abstract and Applied Analysis

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

Yonglei Fang, Qinghong Li, Qinghe Ming, and Kaimin Wang

Full-text: Open access

Abstract

A new embedded pair of explicit modified Runge-Kutta (RK) methods for the numerical integration of the radial Schrödinger equation is presented. The two RK methods in the pair have algebraic orders five and four, respectively. The two methods of the embedded pair are derived by nullifying the phase lag, the first derivative of the phase lag of the fifth-order method, and the phase lag of the fourth-order method. Nu merical experiments show the efficiency and robustness of our new methods compared with some well-known integrators in the literature.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 641236, 15 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475934

Digital Object Identifier
doi:10.1155/2012/641236

Mathematical Reviews number (MathSciNet)
MR2994964

Zentralblatt MATH identifier
1259.65117

Citation

Fang, Yonglei; Li, Qinghong; Ming, Qinghe; Wang, Kaimin. A New Optimized Runge-Kutta Pair for the Numerical Solution of the Radial Schrödinger Equation. Abstr. Appl. Anal. 2012 (2012), Article ID 641236, 15 pages. doi:10.1155/2012/641236. https://projecteuclid.org/euclid.aaa/1364475934


Export citation

References

  • 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.
  • J. M. Blatt, “Practical points concerning the solution of the Schrödinger equation,” Journal of Computational Physics, vol. 1, pp. 378–391, 1961.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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).
  • 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.
  • 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).
  • 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.
  • 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.
  • 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.
  • 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).
  • 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).
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations. I Nonstiff Problems, vol. 8, Springer, Berlin, Germany, 2nd edition, 1993.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • T. E. Simos, “Runge-Kutta interpolants with minimal phase-lag,” Computers & Mathematics with Applications, vol. 26, no. 8, pp. 43–49, 1993.
  • 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.
  • J. P. Coleman and L. Gr. Ixaru, “$P$-stability and exponential-fitting methods for ${y}^{\prime \prime }=f(x,y)$,” IMA Journal of Numerical Analysis, vol. 16, no. 2, pp. 179–199, 1996.
  • 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.
  • 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.
  • 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.