Abstract and Applied Analysis

Volume 2012 (2012), Article ID 641236, 15 pages

http://dx.doi.org/10.1155/2012/641236

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

^{1}School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China^{2}Department of Mathematics, Chuzhou University, Chuzhou 239000, China

Received 9 May 2012; Revised 23 September 2012; Accepted 9 October 2012

Academic Editor: Malisa R. Zizovic

Copyright © 2012 Yonglei Fang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

In molecular dynamics, quantum physics, and chemistry, no other equation has been studied more profoundly than the Schrödinger equation [1–3]. The one-dimensional Schrödinger equation has the form
where is a real number denoting the *energy*, the function is the *effective potential* satisfying as . There have been a lot of numerical methods, such as exponentially fitted and phase fitted integrators based on the oscillatory property of the solution of the Schrödinger equation (1.1) (see, e.g., [4–13]). In [7], Simos and Aguiar constructed a modified Runge-Kutta method for the numerical integration of the Schrödinger equation by phase fitting based on the fifth-order RK method. Recently, Vyver improved this method and gave an embedded pair of modified RK methods by nullifying the dispersion (phase lag) of the fifth-order method and the fourth-order method [5].

In this paper, we derive a new kind of phase fitting RK embedded pair by nullifying the phase lag, the first derivative of phase lag of the fifth-order method, and the phase lag of the fourth-order method.

#### 2. Order Conditions and Phase Properties of Modified Runge-Kutta Methods

##### 2.1. Modified Runge-Kutta Methods

For the numerical integration of the initial-value problem of first-order differential equations we consider the -stage modified explicit Runge-Kutta method of the form which can be expressed in Butcher tableau as or equivalently by the triplet () with , , , . Here, following the approach of exponential fitting and/or phase fitting in [5, 7, 14], the frequency-depending parameters , , are introduced to adapt the traditional RK method to the oscillatory feature of the solution to the problem [15–27], for example, in this paper, to minimize the dispersion and/or dissipation (see next section).

The algebraic order conditions presented in [28] are not fit for the modified RK method (2.2). Writing where , , the third-to-fifth order conditions are listed as follows (see Vyver [5]).(i)Order 3 requires (ii) order 4 requires, in addition (iii)order 5 requires, in addition

##### 2.2. Dispersion and Dissipation of Modified Runge-Kutta Methods

Applying the modified RK method (2.2) to the test differential equation yields where with the identity matrix.

*Definition 2.1. *The quantities
are called the dispersion (or phase lag) and the dissipation (or amplification factor error) of the method (2.2), respectively. And the method is said to be dispersive of order and dissipative of order if
respectively.

Writing we have From the formula (2.10), and are polynomials in : which are completely determined by , , , and .

#### 3. Construction of the New Embedded Pair

In this section, we are concerned with the embedded modified pair which is simplify denoted by the Butcher tableau Choosing , the classical embedded RK5 pair derived by Dormand and Prince [29] is recovered, where the method is of order five and is of order four. It should be noted that the first method in this pair shares the FSAL property in the sense that it uses only six function evaluations at each step with seven stages. Simos and Aguiar [7] presented a modified phase fitted RK method by determining the one-parameter . Following this approach, Vyver [5] constructed a phase fitted embedded RK5 pair. Our main aim in this section is to derive a more efficient embedded RK pair.

Inspired by the ideas in [30–40], with a variant expression of dispersion (see Simos [41]), we compute the dispersion of the higher-order method and the dispersion of the lower-order method and the first derivative of dispersion of the higher-order method of the pair (3.1) as follows: in which Solving the systems of (3.2) we obtain where For small values of , say , the above formulae are subject to heavy cancelations and in that case the following Taylor series expansions must be used It is easy to check that the two schemes in the new pair (3.1) with -values (3.5) are of algebraic orders five and four, respectively. The Taylor series of the dissipations of the new fifth-order method and the fourth-order method are given by respectively.

#### 4. Stability Analysis

In this section, we are interested in the phase properties of the new methods. Lambert and Watson’s stability theory [42] was reconsidered by Coleman and Ixaru [43] for the periodicity of exponentially-fitted symmetric methods for . Vyver [44] formulated this theory to RK methods. Following Van de Vyver’s idea, we consider the test equation Applying the modified RK method (3.1) to test (4.1) yields the difference equation where with the identity matrix.

*Definition 4.1. *For the modified RK method (3.1) with stability function , the quantities
are called the phase lag (dispersion) and amplification factor error (dissipation), respectively. If
the method is said to be of *phase lag order * and *dissipation order *, respectively, where the and are called the *phase lag constant* and *dissipation constant*, respectively.

For the convenience of analyzing the phase lag and the dissipation, we denote the ratio . Then the phase lags and the dissipations of the higher-order method and the lower order method are respectively. Thus, the higher-order method has a phase lag of order six and a dissipation of order five and the low-order method is of phase lag order four and dissipation order five.

#### 5. Numerical Experiments

In this section, we will compare the numerical performance of the new pair with some existing well-known methods proposed in the scientific literature.

##### 5.1. Comparison with Fixed Step-Size Methods

The following fixed step-size methods are selected for comparison: (i)PHARK5S: the phase fitted fifth-order RK method derived by Simos [6]; (ii)MODPHARK5S: the modified phase fitted fifth-order RK method obtained by Simos and Aguiar in [7]; (iii)MODPHARK5V: the higher-order method of the modified phase fitted embedded RK5 pair obtained by Vyver presented in [5]; (iv)ARK5: an adapted fifth-order RK method given by Fang et al. in [45];(v)MODDPHARK5: the higher-order method of the new pair.

We consider the numerical integration of the Schrödinger equation (1.1) with the well-known Woods-Saxon potential in which , , , and . The domain of numerical integration is . It is appropriate to choose as follows [5, 46]: In the numerical experiments we consider the resonance problem (), the numerical results were compared with the analytical solution of the Woods-Saxon potential, rounded to six decimal places. In Figures 1, 2, 3, and 4, we plot the error of versus the integration step-size () for , respectively.

##### 5.2. Comparison with Variable Step-Size Methods

Next we select the following embedded RK5 pairs: (i)PHARK5S: the phase fitted embedded RK5 pair derived by Simos [6];(ii)MODPHARK5V: the modified phase fitted embedded RK5 pair obtained by Vyver presented in [5]; (iii)MODDPHARK5: the new embedded RK5 pair given in this paper.

We consider the numerical integration of the Schrödinger equation (1.1) with the well-known Lennard-Jones potential (see [5]) and we compute some phase shifts for this potential. In the numerical results, we compute the phase shifts correct to four decimal places for the energies and . We choose the fitting frequency . For the calculation of the phase shifts, we show the number of function evaluations as a function of in Figures 5-6.

Figures 1–6 show that our new methods are more efficient than the other methods we select for comparison.

#### 6. Conclusions and Discussions

A new kind of modified phase fitted explicit embedded RK pair for the numerical integration of the radial Schrödinger equation is presented in this paper. This new pair is based on the classical RK5 pair obtained by Dormand and Prince [29]. The phase fitted technique can be regarded as an improvement of the ideas from [5, 7, 30, 31]. The two schemes in this pair are of orders five and four, respectively. Numerical experiments show the effectiveness and competence of the new pair.

#### Acknowledgments

The authors are deeply grateful to the anonymous referees for their constructive comments and valuable suggestions. This research is partially supported by NSFC (no. 11101357), the foundation of Shandong Outstanding Young Scientists Award Project (no. BS2010SF031), the foundation of Scientific Research Project of Shangdong Universities (no. J11LG69), NSF of Shandong Province, China (no. ZR2011AL006), NSF of Universities of Anhui Province, China (no. KJ2010A248), and the Scientific Research Start-up Fund of Chuzhou University, China (no. 2010qd03).

#### 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. M. Blatt, “Practical points concerning the solution of the Schrödinger equation,”
*Journal of Computational Physics*, vol. 1, pp. 378–391, 1961. View at Google Scholar - 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. View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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). View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - 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). View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - 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). View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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). View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - 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. View at Google Scholar - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Google Scholar - T. E. Simos, “Runge-Kutta interpolants with minimal phase-lag,”
*Computers & Mathematics with Applications*, vol. 26, no. 8, pp. 43–49, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Scopus