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.
Similar content being viewed by others
References
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)
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)
Simos T.E.: Chemical Modelling—Applications and Theory Vol. 1, Specialist Periodical Reports, pp. 32–140. The Royal Society of Chemistry, Cambridge (2000)
Engeln-Mullges G., Uhlig F.: Numerical Algorithms with Fortran, pp. 423–488. Springer-Verlag, Berlin Heidelberg (1996)
Franco J.M.: Runge-Kutta methods adapted to the numerical integration of oscillatory problems. Appl. Numer. Math. 50(3–4), 427–443 (2004)
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)
Aceto L., Sestini A.: Numerical aspects of the coefficient computation for LMMs. J. Numer. Anal., Ind. Appl. Math. 3, 181–191 (2008)
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)
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)
Enachescu C.: Approximation capabilities of neural networks. J. Numer. Anal., Ind. Appl. Math. 3, 221–230 (2008)
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)
Ogata H.: Fundamental solution method for periodic plane elasticity. J. Numer. Anal., Ind. Appl. Math. 3, 249–267 (2008)
An P.T.: Some computational aspects of helly-type theorems. J. Numer. Anal., Ind. Appl. Math. 3, 269–274 (2008)
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)
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)
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
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
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)
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)
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)
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)
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)
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)
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)
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)
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)
Simos T.E.: Exponentially-fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Simos T.E.: A numerov-type method for the numerical-solution of the radial Schrödinger-equation. Appl. Numer. Math. 7(2), 201–206 (1991)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
T. E. Simos is an active member of the European academy of sciences and arts, active member of the European academy of sciences.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-009-9571-z