A New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution of Oscillatory Problems

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1 Applied Mathematical Sciences, Vol. 1, 16, no. 44, HIKARI Ltd, A New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution of Oscillatory Problems F. A. Fawzi Department of Mathematics, Universiti Putra Malaysia, 434 UPM Serdang Selangor, Malaysia & Department of Mathematics Faculty of Computer Science and Mathematics Tikrit University, Iraq N. Senu, F. Ismail and Z. A. Majid Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia, 434 UPM Serdang Selangor, Malaysia Copyright c 16 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid. This article is 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 In this work, a new way for constructing an efficiently modified Runge-Kutta (RK) method to solve first-order ordinary differential equations with oscillatory solutions is provided. The proposed method solves the first-order ODEs by first converting the second order ODEs to an equivalent first-order ODEs. The method of the embedded has algebraic orders five and four. The numerical results of the new method have been compared with those of existing methods and showed that the new method is more efficient. Mathematics Subject Classification: 65L5, 65L

2 158 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid Keywords: Embedded Runge Kutta Methods; Variable Step-size; Oscillatory Problems 1 Introduction The last few decades, a lot of research has been done in the area of numerical solution of first-order initial value problems of the form: where y (x) = f(x, y), y(x ) = y, y (x ) = y, x [a, b] (1) y(x) = [y 1 (x), y (x),..., y s (x)] T f(x, y) = [f 1 (x, y), f (x, y),..., f s (x, y)] T y is a given vector of initial condition and their solution is oscillating. This type of problem appears often in scientific areas such as astronomy, quantum mechanics, mechanics, and electronics. A lot of researches have been conducted for the construction of numerical methods that are both accurate and computationally efficient to solve the oscillatory problems. Moreover, they are a lot of numerical integration of adapting kind which is based on significant properties like exponentially-fitted, trigonometrically-fitted and phase-fitted [1-4]. Van de Vyver [5] presented an embedded phase-fitted modified RK method for the numerical solution of the Schrödinger equation. Fang et al. [6] constructed new embedded pair explicit RK methods with (FSAL) properties. Recently, Fang et al. [7] presented a new phase-fitted modified RK pair for the numerical solution of the radial Schrödinger equation. Simos and Vigo Aguiar [8] derived a modified RK method with phase-lag for the numerical solution of the Schrödinger equation. Also, they were concerned with phase-fitted in [9] based on the fifth - order method of Dormand and Prince. We used the phase-fitted technique to determine g. In [8, 9] the constant step size mode for phase-fitted modified Runge-Kutta methods have been used. Our purpose is to extend this technique for the development of variable step size algorithm. The aim of this paper is to develop a new efficient embedded phase-fitted modified Runge-Kutta method. In Section the preliminaries on the phase properties of the explicit modified RK method is discussed. In Section 3 we construct the new pair of explicit modified Runge-Kutta method and the stability and error analysis are also provided. The efficiency of the new method, when compared with the existing method, is given in Section 4.

3 A new embedded phase-fitted modified Runge-Kutta method 159 Phase Properties of Modified Runge-Kutta Method An explicit m-stage modified Runge-Kutta method formula is given by y n+1 = y n + h m b i f(x n + c i h, Y i ), i=1 i 1 Y i = g i y n + h a ij f(x n + c i h, Y j ), i = 1,..., m. () j=1 where a ij, c i, b i, i = 1,..., m are constant, h is the step size and the parameters g i, i = 1,..., m are even function of v = hw. The method is determined by means of Butcher tableau (see Table 1) [1] Table 1: The modified Runge-Kutta method c g a 1 c 3 g 3 a 31 a c s g s a s1 a s... a s,s 1 b 1 b... b s 1 b s The method is stated to be explicit when i j where a ij = and implicit otherwise. In RK method the embedded pair q(p) is based on the RK method (c, A, b) of order q and another RK method (c, A, b ) of order p < q. An embedded pair is characterized by Butcher tableau c g A An embedded pair of explicit modified Runge-Kutta method is used in a variable step size algorithm because they provide a cheap error estimate. b T b T

4 16 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid From embedded method we obtain an estimated error. EST n+1 = y n+1 y n+1 (3) For the numerical integration of the equation (1) we used stepsize control procedure which is given by [11]: if EST n+1 < T ol 1, h n+1 = h n, if T ol 1 EST n+1 < T ol, h n+1 = h n, if EST n+1 T ol, h n+1 = hn and repeat the step. where T ol is the requested local error. It should be noted that the q th-order approximation y n is used as the initial value for the (n+1) th step, that mean the embedded pair is applied in local extrapolation mode or higher order mode. To develop the new method we utilize the test equation with analytical solution y = iwy, w ɛ R. (4) y(x n ) = y e iwx. (5) Based on reasons described by Houwen and Sommeijer [1], we shall confine our considerations to the homogeneous phase-lag. Applying the method () to the test equation (4) yields y n+1 = R(iH)y n, H = wh. (6) Definition.1 An explicit m-stage modified RK, (presented in Table 1) the quantities: φ(h) = H arg[r(ih)], α(h) = 1 R(iH) are called the dispersion or phase error or phase-lag and the amplification error respectively. If φ(h) = O(H q+1 ), and α(h) = O(H r+1 ) then the method is said to be phase-lag order q and dissipative order r. For explicit modified Runge-Kutta method R(iH) = A m (H ) + ihb m (H ), (7) where A m and B m are polynomials in H. From equation (7) it follows that [ φ(h) = H tan H B ] m(h ), (8) A m (H ) α(h) = 1 A m(h ) + H B m(h ). (9)

5 A new embedded phase-fitted modified Runge-Kutta method Analysis of Stability An m-stage modified Runge-Kutta method () is applied to equation (4), we obtain y n+1 = y n + HBY, (1) where and From (11), we have Y = y n G + HAY (11) Y = [Y 1, Y,..., Y s ], G = [g 1, g,..., g s ] B = [b 1 b... b s ] T, H = hw (I HA)Y = y n G, Y = (I HA)y n G, Y n+1 = y n + HB(I HA) 1 y n G (1) substituting equation (1) into equation (1), we obtain: where y n+1 = R(H)y n, is the stability function of the method. R(H) = 1 + HB(1 HA) 1 G (13) Definition. A modified Runge-kutta method is said to be absolutely stable if R(H) < 1 3 Construction of the New Method We consider an embedded pair of explicit Runge-Kutta which denoted by Butcher tableau which is given in [1]:

6 16 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid 1 5 Table : Butcher Tableau RK method In this study, our focus is to have modified RK method with the phase-lag of order infinity. Therefore, the coefficients of this method is depending on the product H = wh. To get zero-phase error (φ(h) = ) for the fifth order method then the relation (9) become [ ] Bm (H ) tan(h) = H (14) A m (H ) = H [( g ) H H + 1 ] H H4 1 (15) H Solving equation (15) for g, we obtain g = 1 15H 4 tan(h) + 3 tan(h)h6 1 tan(h)h tan(h)h +5H 5 4H 3 + 4H (16) As H, g have the following Taylor expansions: g = H H H H H H (17)

7 A new embedded phase-fitted modified Runge-Kutta method 163 The same case for the fourth-order method tan(h) = ( H4 + ( 11g ) H + 19g ) H6 + ( 7 g ) H4 + ( 41 g ) 683 H. (18) g 3 = 9 tan(h) + 79 tan(h)h tan(h)h tan(h)h +... H(31H 3 tan(h) 41 tan(h)h + H 95) (19) As H, g 3 have the following Taylor expansions: g 3 = H H H H H H H () This new method is denoted as PHRK5(4). The stability function of the higher order of the new method is given by R(H) = 1 + H + 1 H H H H H H H H H H (1) It is not possible to find the stability region for infinity series of coefficients. Here, three different finite degrees of coefficients are considered, which are order four, six and eight. we obtained the stability interval to the new method of three different finite degrees of coefficients is ( 3.5, ) for order four, (.8, ) for order six and (.5, ) for order eight. If we continue to cut up to order ten the stability interval is (.4, ) that mean when we increase the order of coefficients the stability interval will be smaller. For the low order of the new method, we follow the same procedure then the stability function is given by R(H) = 1 + H + 1 H H H H H H H H H1

8 164 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid H H H H () The stability region of the new method is obtained by equating the stability function R(H) with e Iθ and then solve for H using maple package. i.e R(H) = e iθ = cos(θ) + i sin(θ). The stability region for the high order method is shown in Figure 1. Figure 1: The stability region for the fifth order of the PHRK5(4) method for different order of the coefficients

9 A new embedded phase-fitted modified Runge-Kutta method 165 The stability region for the low order method is shown in Figure. Figure : The stability region for the fourth order of the PHRK5(4) method for different order of the coefficients The local truncation error analysis (LTE) of the new method based on the Taylor series expansion of the differences y n+1 and y(x n = h) will be considered The LTE of the high order is given by LT E = y n+1 y(x n + h) (3) LT E = h 6 [ 1 1 (f xf y f xyy y + f x f xy f yy y f y f xyy y ) 1 8 (f yf xy f yy y () f y f xyy y () ) (f xf y f yyy y () f x f xy f xy f xxy y f yy f xyy y () +f x f y f xxy f xy f xyy y () f yy f xyy y (3) f x f yyy () + f xf y f yy ) 1 7 (f yyf y f xx y +f xy f xxx f y f xxx f 3 y f xx f 4 y f xx f 4 y f x + f 5 y y + f xy f yyy y (3) + f y f xy f xx +f yy f xxx y + f yy f yyy y (4) ) 1 36 (f y f yy y f 3 y f xy y ) + f yy f xxx y + f yy f yyy y (4) )

10 166 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid 1 36 (f y f yy y fy 3 f xy y ) f yfxyy 1 18 (f xfy f xy fy f yyy y (3) )] + O(h 7 ) (4) From equation (4), it is clear that the order of the new method is five because all the terms of h lower than h 6 are vanished. and the LTE for the low order: LT E = h 5 11 [ 7 (f xxxx+f yyyy y (4) ) f yf xxy y (f xf xyy y +f xy f yy y () ) 11 9 f xf y f xy (f y f xy y + f yy f xx y + f xxxy + y + f xyyy y (3) + f xx f xy +f yyy (3) ) f yf xyy y () (f xf yyy y () + f xxyy y () + f x f xxy ) f y f yy y () f xf y f yy y f xyy f xf yy 11 3 (f yf xxx + fy f xx ) (f y 3 f x + fy 4 y )] + O(h 6 ) (5) From equation (5), it is clear that the order of the new method is four because all the terms of h lower than h 5 are vanished. 4 Numerical Experiments In this section, we will apply the new method to solve different problems. The following explicit MRK method are selected for the numerical comparison. 4.1 Comparison with fixed step-size methods RK5: The phase-fitted fifth-order derived in this paper. PLRK4: The modified RK method derived by Simos and Vigo-Aguiar [8]. RK5B: Fifth-order six-stage RK method derived by Sakas and Simos [13]. RK4M: Fourth-order five-stage RK method given in Butcher [1]. RK4Z: Fourth-order five-stage RK method given in Hairer et. al [14].

11 A new embedded phase-fitted modified Runge-Kutta method 167 Problem 1: ( Homogeneous) [15] y 1 = y, y 1 (x) = 1, y = 64 y 1, y (x) = Theoretical solution : y 1 (x) = 1 sin(8x) + cos(8x) 4 y (x) = cos(8x) 8 sin(8x) Problem : ( Inhomogeneous) [1] y 1 = y, y 1 () = 1 y = v y 1 + (v 1) sin(x), y () = v + 1 Estimated frequency: v = 1 Theoretical solution : y 1 (x) = cos(vx) + sin(vx) + sin(x) y (x) = v sin(vx) + v cos(vx) + cos(x) Problem 3: ( Periodic orbit system) [3] y 1 = y, y 1 (x) = 1, y (x) = y = y cos(x) y 3 = y 4, y 3 (x) =, y 4 (x) =.9995 y 4 = y sin(x) Theoretical solution: y 1 (x) = cos(x) +.5x sin(x) y (x) = sin(x).5x cos(x) y 3 (x) = sin(x) +.5x cos(x) y 4 (x) = cos(x) +.5x sin(x)

12 168 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid Problem 4: ( Nonlinear system) [17] y 1 = y 3, y 1 () = 1 y 3 = y 1 ( ) 3, y 3 () = y 1 + y y = y 4, y () = y 4 = y ( ) 3, y 4 () = 1 y 1 + y Theoretical solution : y 1 (x) = cos(x) y (x) = sin(x) y 3 (x) = sin(x) y 4 (x) = cos(x) Problem 5: ( Inhomogeneous) [18] y 1 = y, y 1 () = 1 y = y 1 + x, y () = Theoretical solution: y 1 = sin(x) + cos(x) + x y = cos(x) sin(x) + 1

13 A new embedded phase-fitted modified Runge-Kutta method log1(maxerr) 1 3 RK5 RK5B 4 PLRK4 RK4M RK4Z log 1 (STEP-SIZE) Figure 3: Comparison for RK5, RK5B, PLRK4, RK4M and RK4Z problem 1 with b=1 1 log1(maxerr) 1 RK5 3 RK5B PLRK4 RK4M RK4Z log 1 (STEP-SIZE) Figure 4: Comparison for RK5, RK5B, PLRK4, RK4M and RK4Z problem with b=1

14 17 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid 1 RK5 RK5B PLRK4 RK4M RK4Z log1(maxerr) log 1 (STEP-SIZE) Figure 5: Comparison for RK5, RK5B, PLRK4, RK4M and RK4Z problem 3 with b=1 1 log1(maxerr) RK5 RK5B 6 PLRK4 RK4M RK4Z log 1 (STEP-SIZE) Figure 6: Comparison for RK5, RK5B, PLRK4, RK4M and RK4Z problem 4 with b=1

15 A new embedded phase-fitted modified Runge-Kutta method 171 log1(maxerr) RK5 RK5B PLRK4 RK4M RK4Z log 1 (STEP-SIZE) Figure 7: Comparison for RK5, RK5B, PLRK4, RK4M and RK4Z problem 5 with b=1

16 17 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid 4. Comparison with variable step-size methods The methods we choose for comparison are as follows: PHMRKD5(4): The new pair of phase-fitted modified RK method presented in this paper. MODPHARK5: The phase-fitted modified RK pair given by Yonglei et al. [7]. MODRK54: The modified RK pair given by Van de Vyver [5]. RK5(4)F: The RK pair of Fehlbreg given in Butcher [1]. RK5(4)D: The RK pair of Dormand given by Hairer et al. [14]. Problem 6: ( Inhomogeneous system) y 1 = y 3, y 1 () = 1.1 y = y 4, y () = 1 Theoretical solution : y 3 = 16y e 1x + 1e 1x, y 3 () = 1 y 4 = 16y + 16e 1x + 1e 1x, y 4 () = 9.6 Source : Moo. et al. [17]. y 1 =.1 cos(4x) + e 1x y =.1 sin(4x) + e 1x y 3 =.4 sin(4x) 1e 1x y 4 =.4 cos(4x) 1e 1x Problem 7: ( Nonlinear system) Theoretical solution : y 1 = y 3, y 1 () = 1 y = y 4, y () = y 3 = 4x y y 1, y 3 () = y 1 + y y 4 = 4x y Source : Yonglei et al. [19]. y 1, y 4 () = y 1 + y3 y 1 (x) = cos(x ) y (x) = sin(x ) y 3 (x) = x sin(x ) y 4 (x) = x cos(x )

17 A new embedded phase-fitted modified Runge-Kutta method 173 log1(maxerr) PHMRKD5(4) RK5(4)D RK5(4)F MODRK54 MODPHARK5(4) log 1 (Function Evaluations) Figure 8: Accuracy curves for embedded 5(4) of MRK methods for Problem1 1 log1(maxerr) PHMRKD5(4) RK5(4)D RK5(4)F MODRK54 MODPHARK5(4) log 1 (Function Evaluations) Figure 9: Accuracy curves for embedded 5(4) of MRK methods for Problem

18 174 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid log1(maxerr) PHMRKD5(4) RK5(4)D RK5(4)F MODRK54 MODPHARK5(4) log 1 (Function Evaluations) Figure 1: Accuracy curves for embedded 5(4) of MRK methods for Problem4 log1(maxerr) PHMRKD5(4) 1 RK5(4)D RK5(4)F MODRK54 MODPHARK5(4) log 1 (Function Evaluations) Figure 11: Accuracy curves for embedded 5(4) of MRK methods for Problem5

19 A new embedded phase-fitted modified Runge-Kutta method log1(maxerr) PHMRKD5(4) RK5(4)D RK5(4)F MODRK54 MODPHARK5(4) log 1 (Function Evaluations) 4 Figure 1: Accuracy curves for embedded 5(4) of MRK methods for Problem6 1 log1(maxerr) PHMRKD5(4) RK5(4)D RK5(4)F MODRK54 MODPHARK5(4) log 1 (Function Evaluations) Figure 13: Accuracy curves for embedded 5(4) of MRK methods for Problem7

20 176 F. A. Fawzi, N. Senu, F. Ismail and Z. A. Majid For comparison purpose, in analyzing the numerical results, methods with the same order will be compared. The results are plotted in Figures 3, 4, 5, 6 and 7 we present efficiency curves where the common logarithm of the maximum global error throughout the integration versus computational cost measured by h step-size. From the Figures 8-13, we present efficiency curves where the common logarithm of the maximum global error throughout the integration versus computational cost measured by number of function evaluation and we observed that the new PHMRKD5(4) method is more efficient for integration first-order differential equations possessing an oscillatory solution compared with other methods which are RK5(4)D, RK5(4)F, MODRK54 and MOD- PHARK5(4). 5 Conclusion In this paper, we derived a new embedded phase-fitted RK method for solving first-order ordinary differential equations with oscillatory problems. The methods of the embedded have algebraic orders five and four. The phase-fitting technique we used is an extension of the idea from [8]. At first, we introduced for fixed step-size then numerical findings indicate that the proposed method is more efficient than the existing method. Lastly, we used variable step-size become clear that the new embedded is more robustness than the existing method for solving oscillatory problems. References [1] T. E. Simos, Exponentially fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation and related problems, Comput. Mater. Sci., 18 (), [] T. E. Simos, An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions, Comput. Phys. Commun., 115 (1998), [3] G. Vanden Berghe, H. De Meyer, M. Van Daele, T. Van Hecke, Exponentially fitted explicit Runge-Kutta methods, Comput. Phys. Commun., 13 (1999), [4] Y. Fang, X. Wu, A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions, Appl.

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