Two Optimized Runge-Kutta Methods for the Solution of the Schrödinger Equation

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MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. (8) 7-77 ISSN - Two Optimized Runge-Kutta Methods for the Solution of the Schrödinger Equation T.V. Triantafyllidis, Z.A. Anastassi and T.E. Simos Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese GR- Tripolis, GREECE (Received November, 7) Abstract In this paper we are constructing two explicit Runge-Kutta methods, one with tenth order of phase-lag and constant coefficients and one with infinite order of phase-lag and variable coefficients. The new methods have eight stages and sixth algebraic order and will be used for the numerical integration of the radial time-independent Schrödinger equation. The efficiency of the new constructed methods is compared to that of a wide range of known methods from the literature. The numerical results are shown through the graphs of the accuracy versus the function evaluations when applied to the Resonance problem. Introduction Much research has been done on the numerical integration of the radial Schrödinger equation: ( ) l(l +) y (x) = + V (x) E y(x) () x where l(l+) x and W (x) = l(l+) x is the centrifugal potential, V (x) isthepotential, E is the energy and therefore lim x W (x) =. +V (x)istheeffective potential. It is valid that lim x V (x) = Active Member of the European Academy of Sciences and Arts Corresponding author. Please use the following address for all correspondence: Dr. T.E. Simos, Menelaou Street, Amfithea - Paleon Faliron, GR-7 Athens, GREECE, Tel: 9 9, E-mail: tsimos@mail.ariadne-t.gr

- 7 - Many problems in chemistry, physics, physical chemistry, chemical physics, electronics etc., are expressed by equation (). In this paper we will study the case of E>. We divide [, ) into subintervals [a i,b i ] so that W (x) is a constant with value W i. After this the problem () can be expressed by the approximation y i =(W E) y ( i, whose ) solution is ( ) y i (x) =A i exp W Ex + B i exp W Ex, A i,b i R. The classical methods that have been constructed in the previous decades are not efficient enough when integrating the Schrödinger equation. Some well known classical methods compared in this paper can be found in [] and []. Many numerical methods have been constructed for the efficient solution of the Schrödinger equation. For example the explicit Runge-Kutta methods of Anastassi and Simos [9], [] and Vande Berghe et al. [8]. Some recent research work in numerical methods can be found in [], [], [], [], [], [], [7], [8], [9], [], [], [], [], [], [], [], [7] and [8]. () Basic theory. Explicit Runge-Kutta methods An s-stage explicit Runge-Kutta method used for the computation of the approximation of y n+ (x), when y n (x) is known, can be expressed by the following relations: y n+ = y n + b i k i ( ) i k i = hf x n + c i h, y n + h a ij k j,i=,...,s () j= The methods mentioned previously can also be presented using the Butcher table below: c a c. a. a. () c s a s a s... a s,s b b... b s b s The following equations must always hold: c i = a ij,i=,...,s () j= i=

- 7 - Definition [] A Runge-Kutta method has algebraic order p when the method s series expansion agrees with the Taylor series expansion in the p first terms: y (n) (x) =y n (n) (x), n =,,...,p. Equivalently a Runge-Kutta method must satisfy a number of equations, in order to have a certain algebraic order. These equations will be shown later in this paper.. Phase-Lag Analysis of Runge-Kutta Methods The phase-lag analysis of Runge-Kutta methods is based on the test equation y = i w y, wreal () Application of the Runge-Kutta method described in () to the scalar test equation () produces the numerical solution: y n+ = a n y n, a = A s (v )+ivb s (v ), (7) where v = wh and A s,b s are polynomials in v completely defined by Runge- Kutta parameters a i,j, b i and c i, as shown in (). Definition [] In the explicit s-stage Runge-Kutta method, presented in (), the quantity t(v) =v arg[a (v)] is called the phase-lag or dispersion error. Ift(v) =O(v q+ ) then the method is said to be of phase-lag order q. Although dispersion (or phase-lag) was introduced for cyclic orbit, Runge- Kutta methods with high phase-lag order are more efficient in many other problem types than methods with lower phase-lag order and higher algebraic order with the same number of stages. They are even more effective in problems with oscillating solutions. Construction of the New Methods We consider a 8-Stage explicit Runge-Kutta method as shown in (8): c a c a a c a a a c a a a a c a a a a a c 7 a 7 a 7 a 7 a 7 a 7 a 7 c 8 a 8 a 8 a 8 a 8 a 8 a 8 a 87 b b b b b b b 7 b 8 (8)

- 7 - There are unknowns totally. The necessary equations that must hold so that the method has th algebraic order are the following 7: st Alg. Order ( equation) th Algebraic Order (7) b i = b i c i = i= i= nd Alg. Order ( equations) b i c i a ij c j = i,j= b i c i = b i c i a ij c j = 8 i= i,j= rd Alg. Order ( equations) b i c i a ij c j a ik c k = i,j,k= b i c i = b i c i a ij a jk c k = i= i,j,k= b i a ij c j = b i c i a ij c j = i,j= i,j= th Alg. Order (8 equations) b i c i a ij c j a jk c k = 8 i,j,k= b i c i = b i a ij c j a ik c k = i= i,j,k= b i c i a ij c j = b 8 i c i a ij a jk a kl c l = i,j= i,j,k,l= b i a ij c j = b i a ij c j a ik a kl c l = 7 i,j= i,j,k,l= b i a ij a jk c k = b i c i a ij a jk c k = 7 i,j,k= i,j,k= th Algebraic Order (7) b i a ij c j = b i c i = i= i,j= i,j,k= b i a ij c j a jk c k = (9)

- 77 - b i c i a ij c j = i,j= b i c i a ij c j = i,j= b i c i a ij a jk c k = b i a ij c j = i,j= b i a ij c j a jk c k = b i c i a ij a jk c k = b i a ij a jk a kl c l = b i a ij c j a ik c k = i,j,k= i,j,k= i,j,k= i,j,k,l= i,j,k= i,j,k= i,j,k,l= i,j,k= i,j,k= i,j,k,l= i,j,k,l= i,j,k,l,m= b i a ij c j a jk c k = 9 b i a ij c j a jk a kl c l = 8 b i a ij a jk c k a jl c l = b i a ij a jk c k = b i a ij a jk c k a kl c l = b i a ij a jk a kl c l = b i a ij a jk a kl a lm c m = 7 () We select some coefficients giving them constant values in order to simplify the calculations afterwards: a =,b 8 =,b =,b 7 =, c =, c =, c =, c =, c =,c 7 = and c 8 =. After satisfying consistency conditions () and th algebraic order conditions (9), the coefficients depend now on a 8 and a 7. At this point we have two options: To create a method with constant coefficients, which has finite order of phase-lag, or to create a method with variable coefficients, which has infinite order of phase-lag. In order to create a method with constant coefficients we expand the phase-lag (after satisfying the above equations) using Taylor series. ( 7 phl = 8 a 8, 8 7 a 7, + 7 9 a 8, + 8 ( 7 9 a 8, 9 8 a 8, + + ( + 7 88 a 8, 9 a 8, + 7 a 8,a 7, 7 ) 97 a 8,a 7, ) 7 a 8,a 7, v 7 v 9 ) v +... By solving the coefficients of the lowest powers of the series for a 8 and a 7, we achieve a higher order of phase-lag. So for a 7, = 7 + 89 7 98 + 7 and a 8, = 8 + 7 8

- 78 - phase-lag becomes phl = 988 v v v +... which means that the new method has th order of phase-lag. The coefficients of the new method are given below: a =, a = 7, a = 7, a + 7 =/ 7 + 7 a = / + 7 7+ 7, a = 7 + 7 7 + 7 7 + 7 a = 7+ 7, a = 8 89 + 7 7 + 7 a = 8 + 7 7+ 7, a = a = 97 + 89 7 ( )( ) 7+ 7 + 7 a = 97 + 8 7 ( 7+ 7 )( + 7 ) a = 7 + 8 7 ( )( ), a = 7+ 7 + 7 8, a = 8 a 7 = a 7 = 8 7878 + 8 7 7+, a 7 = + 7 7 7 + 7 7 + 89 7 7+, a 7 = 7 7 7 8 a 7 = 777 + 7 7, a7 = 777 8 7 8 99 + 8 7 a 8 = 8 7+ 7 + 7 a 8 = 7+ 7, a 8 = 997 + 77 7 7+ 7 a 8 = 7 + 7, a8 = 7879 8 777 7 7 777 a 8 = 8 + 7, a87 = 8 As regarded to the new method with variable coefficients we want to produce the method that is corresponding to the previous method. This means that the constant coefficients of the previous method must be equal to the limit of the respective coefficients for v. In order to achieve this we set

- 79 - a 7, = 7 + 89 7 98 + 7 that is the same as before and then we satisfy phl = obtaining the value of a 8, as shown below together with the other coefficients: a =, a = 7, a = 7 a = 9 7 + 89 7 8 7+ + 7 7 a 8, a = 7 + 89 7 8 7+ 7 a 8, a = + a 8, 7 + 89 7 8 7+ 7 a = 9 + 7 + 89 7 8 7+ 9 7 a 8, a = 7 + 89 7 8 7+ + 7 a 8, a = 9/ a 8, + 78 7 + 89 7 8 7+ 7 a = ( a = 8 a 8, + + 79 a 8, 8a 8, 8 7 + 89 7 7 + a 8, 88 7 8 7 + 89 7 7+ + a 8, ( ) 7 8 ) ( a = 8 a 8, + a 8, a 8, 8 7 + 89 7 7+ a 8, 89 7 8 7 + 89 7 ) 7+ 7 ( a = 7 a 8, a 8, a 8, 8 7 + 89 7 7+ + a 8, + 89 7 8 7 + 89 7 7+ + a 8, ( ) 7 8 )

- 7 - a 7 = 7 a 8, + 88 88 a 7 = a 8, + 89 a 7 = 8 a = 8, a = 8 ( 8 ) 7+89 7 8 7+ 7 8 7 + 89 7 7+ 7 a 7 = 7+89 7 7+ 7, a 7 = 9 a 8 = 7 8 8 7 + 89 7 8 7 + 7 a 8 = 8 + 8 a 8 = 89 + a 8, 8 a 8 = 8 a 8 = + v ( + tan (v) v) 7 a 8, +, + a 8,, a 7 = + 9 a 8, a 8, 7 + 89 7 7+ a 8, 7 7 + 89 7 8 7+ 7 8 a 8,, a 8 = 99 7 a 8, ( 7 v tan (v) 7 + +7 v tan (v) 8 v 77 v 7 + ( 99 v (tan (v)) + 9788 v (tan (v)) 7 787 v 9 tan (v) 7 87978 v 9 tan (v) +79 v 8 + 8 v 8 7 + +87 (tan (v)) v, 7977 (tan (v)) v + +89978 (tan (v)) v 88 (tan (v)) v 8 +87 v 7 tan (v) 878 v tan (v) +88799888 tan (v) v 877 tan (v) v ) / ) +97 v 79 v + 877 v Numerical Results a 87 =. The resonance problem The efficiency of the two new constructed methods will be measured through the integration of problem () with l = at the interval [, ] using the well known Woods-Saxon potential

- 7 - V (x) = u +q + u q ( + q), q = exp ( x x u =, a =., x = 7 and u = u a and with boundary condition y() =. a ), where () The potential V (x) decays more quickly than l (l+), so for large x (asymptotic x region) the Schrödinger equation () becomes ( ) l(l +) y (x) = E y(x) () x The last equation has two linearly independent solutions kxj l (kx) and kxn l (kx), where j l and n l are the spherical Bessel and Neumann functions. When x the solution takes the asymptotic form y(x) Akxj l (kx) Bkxn l (kx) () D[sin(kx πl/) + tan(δ l ) cos (kx πl/)], where δ l is called scattering phase shift and it is given by the following expression: tan (δ l )= y(x i) S(x i+ ) y(x i+ ) S(x i ) y(x i+ ) C(x i ) y(x i ) C(x i+ ), () where S(x) =kxj l (kx), C(x) =kxn l (kx) and x i <x i+ and both belong to the asymptotic region. Given the energy we approximate the phase shift, the accurate value of which is π/ for the above problem.. The Methods We will compare the new constructed method to a wide range of already known methods. These are explicit Runge-Kutta methods with algebraic order up to six, some of which are optimized for solving problems with oscillating solutions or Schrödinger equation specifically. Formulae Butcher (th order), Fehlberg I (th), Fehlberg II (th), Fehlberg th, Fehlberg th, Kutta-Nyström (th), England I (th), England II (th) and Gill (th) from []. Formulae. (--),. (--),. (--),.7 (--),.8 (--8),.9 (--),. (--),. (--),. (--) and.7 (--) from []. In parentheses there are the number of stages, the algebraic order and the order of phase-lag respectively. The well known classical Runge-Kutta method with stages and th algebraic order and the method of Zonneveld with stages and th algebraic order.

- 7 - The trigonometrically fitted method of Vande Berghe et al. with stages and th algebraic order from [8]. The two trigonometrically fitted methods of Anastassi and Simos based on the classical method of England with stages, th algebraic order and exponential orders one and two [9]. The two trigonometrically fitted methods of Anastassi and Simos based on the classical method of Kutta-Nyström with stages, th algebraic order and exponential orders one and two []. The two trigonometrically fitted methods based on the classical method of Fehlberg I with stages, th algebraic order and exponential orders one and two. The trigonometrically fitted method based on the classical method of Butcher with 8 stages, th algebraic order and first exponential order. The two new formulae 8--Inf and 8-- constructed in this paper, where A-B-C means that the method has A stages, algebraic order B and phaselag order C.. Comparison We will use four values for the energy: 989.79,.987,. and.8887. As for the frequency w we will use the suggestion of Ixaru and Rizea [7]: { E x [,.] w = () E x [., ] We present the accuracy of the tested methods expressed by the log (error at the end point) when comparing the phase shift to the actual value π/ versus the log (total function evaluations). The function evaluations per step are equal to the number of stages of the method multiplied by two that is the dimension of the vector of the functions y(x) and z(x) of the resonance problem. In Figures - we use E = 989.79, in Figures - we use E =.987, in Figures 7-9 we use. and in Figures - we use.8887. The methods are divided into three groups depending on their efficiency with the first group including the most efficient ones. As we see in the figures, the new method with constant coefficients is the most accurate among all the methods and for all values of energy. It also has the advantage of being independent from the frequency of the problem and the step-length. The second method developed in this paper that has variable coefficients is more accurate than all the other known methods when searching for high accuracy and this can be more clearly seen for the low values of energy (. and.8887).

- 7 - Efficiency (E=989.79) 9 8 7 New 8 New 8 Inf Kutta Nystrom th (nd) Fehlberg I th (nd) England II th (nd) Fehlberg II th Fehlberg I th (st) Kutta Nystrom th (st) England II th (st) Zonneveld th......7.8.9. Log (Function Evaluations) Figure : Efficiency for the first group of methods using E = 989.79 Efficiency (E=989.79) 8 8 Fehlberg th Kutta Nystrom th Butcher th England II th Gill th.......7.8.9. Log (Functions Evaluations) Figure : Efficiency for the second group of methods using E = 989.79

- 7 - Vanden Berghe th (st) Butcher th (st) Fehlberg th Classical th England I th Efficiency (E=989.79).......7.8.9. Log (Functions Evaluations) Figure : Efficiency for the third group of methods using E = 989.79 8 Efficiency (E=.987) 7 New 8 New 8 Inf England II th (nd) Kutta Nystrom th (nd) Fehlberg I th (nd) Fehlberg II th Fehlberg I th (st) Kutta Nystrom th (st) Zonneveld th England II th (st).......7.8.9. Log (Functions Evaluations) Figure : Efficiency for the first group of methods using E =.987

- 7 - Efficiency (E=.987) 8 8 Fehlberg th Kutta Nystrom th Butcher th England II th Gill th......7.8.9. Log (Functions Evaluations) Figure : Efficiency for the second group of methods using E =.987 Efficiency (E=.987) Vanden Berghe th (st) Fehlberg th Butcher th (st) Classical th England I th......7.8.9. Log (Functions Evaluations) Figure : Efficiency for the third group of methods using E =.987

- 7 - Efficiency (E=.) 8 7 New 8 New 8 Inf Kutta Nystrom th (nd) Fehlberg I th (nd) England II th (nd) Fehlberg II th Fehlberg I th (st) Zonneveld th Kutta Nystrom th (st) England II th (st)........7.8.9. Log (Functions Evaluations) Figure 7: Efficiency for the first group of methods using E =. Efficiency (E=.) 7 8 8 Fehlberg th Kutta Nystrom th Butcher th England II th Gill th........7.8.9. Log (Functions Evaluations) Figure 8: Efficiency for the second group of methods using E =.

- 77 - Efficiency (E=.) Butcher th (st) Classical th England I th Fehlberg th Vanden Berghe th........7.8.9. Log (Functions Evakuations) Figure 9: Efficiency for the third group of methods using E =. Efficiency (E=.8887) 7 New 8 New 8 Inf Fehlberg II th Fehlberg I th (nd) Fehlberg I th (st) Kutta Nystrom th (nd) Zonneveld th England II th (nd) Kutta Nystrom th (st) England II th (st)........7.8.9. Log (Functions Evaluations) Figure : Efficiency for the first group of methods using E =.8887

- 78 - Efficiency (E=.8887) 7 8 Fehlberg th 8 Kutta Nystrom th Butcher th England II th Gill th......7.8.9. Log (Functions Evaluations) Figure : Efficiency for the second group of methods using E =.8887 Efficiency (E=.8887) 7 Butcher th (st) Fehlberg th V.Berghe th (st) England I th Classical th......7.8.9. Log (Functions Evakuations) Figure : Efficiency for the third group of methods using E =.8887

- 79 - Among the rest of the methods the ones that were competitive were the exponentially-fitted versions of Kutta-Nyström, Fehlberg I and England II. Especially the second exponential order methods were very efficient followed by the first exponential order methods of the families. The high efficiency of all optimized methods is more clear for high values of energy. Method of Zonneveld and high phase-lag order methods from [], like --, --8 etc. were next in order of efficiency followed by the classical methods found in [], i.e. Fehlberg th, Kutta-Nyström th etc., the optimized method of Vande Berghe, followed by the low algebraic and phase-lag order methods. Conclusions Two new explicit Runge-Kutta methods with eight stages, sixth algebraic order and order of phase-lag infinite and tenth are produced in this paper. After applying the new methods and a great variety of known methods we conclude that the newly developed methods are highly efficient when integrating the Resonance problem. This reveals the importance of phase-lag when solving ODEs with oscillating solutions, especially in the case of non-zero phase-lag, that is when we want to produce a method with constant coefficient. References [] Engeln-Mllges G. and Uhlig F., Numerical Algorithms with Fortran, 99, Springer-Verlag, Berlin Heidelberg, -88 [] Franco J.M., Palacios M., 99, J. Comput. Appl. Math.,, [] Hairer E., Nørsett S.P., Wanner G., Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag, Berlin Heidelberg, 99 [] Simos T.E., Chemical Modelling - Applications and Theory Vol., Specialist Periodical Reports,, The Royal Society of Chemistry, Cambridge, - [] Stiefel E. and Bettis, D.G., 99, Numer. Math.,, [] P.J. Van der Houwen and B.P Sommeijer, Explicit Runge-Kutta Nyström methods with reduced phase-errors for computation oscillating solutions, 987, SIAM J. Numer. Anal.,, 9-7 [7] L.Gr. Ixaru, M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, Comp. Phys. Comm. 9, -7 (98) [8] Vanden Berghe G., De Meyer H., Van Daele M. and Van Hecke T., Exponentially-fitted explicit RungeKutta methods, Comp. Phys. Comm., 7- (999)

- 77 - [9] Anastassi Z.A. and Simos T.E., Trigonometrically Fitted Runge-Kutta Methods for the Numerical Solution of the Schrödinger Equation, Journal of Mathematical Chemistry, 8-9 () [] Anastassi Z.A. and Simos T.E., Trigonometrically Fitted Fifth Order Runge-Kutta Methods for the Numerical Solution of the Schrödinger Equation, Mathematical and Computer Modelling, (7-8), 877-88 () [] T.E. Simos, P-stable Four-Step Exponentially-Fitted Method for the Numerical Integration of the Schrödinger Equation, CoLe,,, 7- () [] T.E. Simos, Closed Newton-Cotes Trigonometrically-Fitted Formulae for Numerical Integration of the Schrödinger Equation, CoLe,,, -7 (7) [] G. Psihoyios, A Block Implicit Advanced Step-point (BIAS) Algorithm for Stiff Differential Systems, CoLe, -,, -8 () [] L. Aceto, R. Pandolfi, D. Trigiante, Stability Analysis of Linear Multistep Methods via Polynomial Type Variation, JNAIAM,, -, -9 (7) [] S. D. Capper, J. R. Cash and D. R. Moore, Lobatto-Obrechkoff Formulae for nd Order Two-Point Boundary Value Problems, JNAIAM,,, - () [] S. D. Capper and D. R. Moore, On High Order MIRK Schemes and Hermite-Birkhoff Interpolants, JNAIAM,,, 7-7 () [7] J. R. Cash, N. Sumarti, T. J. Abdulla and I. Vieira, The Derivation of Interpolants for Nonlinear Two-Point Boundary Value Problems, JNAIAM,,, 9-8 () [8] J. R. Cash and S. Girdlestone, Variable Step Runge-Kutta-Nyström Methods for the Numerical Solution of Reversible Systems, JNAIAM,,, 9-8 () [9] J. R. Cash and F. Mazzia, Hybrid Mesh Selection Algorithms Based on Conditioning for Two-Point Boundary Value Problems, JNAIAM,,, 8-9 () [] F. Iavernaro, F. Mazzia and D. Trigiante, Stability and Conditioning in Numerical Analysis, JNAIAM,,, 9- () [] F. Iavernaro and D. Trigiante, Discrete Conservative Vector Fields Induced by the Trapezoidal Method, JNAIAM,,, - () [] F. Mazzia, A. Sestini and D. Trigiante, BS Linear Multistep Methods on Non-uniform Meshes, JNAIAM,,, - () [] G. Vanden Berghe and M. Van Daele, Exponentially-fitted Stormer/Verlet methods, JNAIAM,,, - ()

- 77 - [] T.E. Simos and P.S. Williams, A new Runge-Kutta-Nyström method with phase-lag of order infinity for the numerical solution of the Schrödinger equation, MATCH Commun. Math. Comput. Chem.,, -7 () [] T.E. Simos, Multiderivative methods for the numerical solution of the Schrödinger equation, MATCH Commun. Math. Comput. Chem.,, 7- () [] G. Psihoyios and T.E. Simos, A family of fifth algebraic order trigonometrically fitted P-C schemes for the numerical solution of the radial Schrödinger equation, MATCH Commun. Math. Comput. Chem.,, - () [7] D.P. Sakas and T.E. Simos, Trigonometrically-fitted multiderivative methods for the numerical solution of the radial Schrödinger equation, MATCH Commun. Math. Comput. Chem.,, 99- () [8] T.E. Simos and I. Gutman, Papers presented on the International Conference on Computational Methods in Sciences and Engineering (Castoria, Greece, September -, ), MATCH Commun. Math. Comput. Chem., (): A-A ()

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