New Third Order Runge Kutta Based on Contraharmonic Mean for Stiff Problems

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1 Applied Mathematical Sciences, Vol., 2009, no. 8, New Third Order Runge Kutta Based on Contraharmonic Mean for Stiff Problems Osama Yusuf Ababneh 1 and Rokiah Rozita School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Abstract In this paper we introduce an explicit one step method that can be used for solving stiff problems. This method can be viewed as a modification of the explicit third order runge-kutta method using the contraharmonic mean (C o M) that allows rducing the stiffiness in some sense. The stability of the method is analyzed and numerical results shown to verify the conclusions.numerical examples indicate that this method is superior compared to some existing methods including the third and fourth order contraharmonic mean methods,abm method, classical third order Runge-Kutta, and Wazwaz method. Mathematics Subject Classifications: 51N20, 62J05, 70F99 Keywords: ODE solver; Runge-Kutta method; contraharmonic mean; stiff problems 1 Introduction It has been suggested that The problem of stiffness is very difficult to be solved by explicit methods but recently many explicit methods were introduced and developed to solve the stiff problems like 1 ]. In this paper we introduced a Runge-Kutta like explicit method can be used to solve stiff problems and give a good accuracy butter than wide of an explicit methods. Theorm 1.1 ]: If an p stage explicit Runge-Kutta contraharmonic mean(c 0 M) method has order N where N then p N. 1 ababneh@math.com

2 66 O. Y. Ababneh and R. Rozita As a 2 stage formula of second order the contraharmonic mean( C 0 M ) method is k 2 y n+1 = y n + h 1 + k2 2 ] k 1 + k 2 where k 1 = f(x n,y n ), k 2 = f(x n + h, y n + hk 1 ). A rd -order method for -stages of the ( C 0 M ) method are given in the form where, y n+1 = y n + h 2 k k 2 2 k 1 + k 2 + k2 2 + k2 k 2 + k k 1 = f(x n,y n ), k 2 = f(x n + h 2,y n + h 2 k 1), k = f(x n + h 2,y n + h 2 k 2). The fourth order contraharmonic mean ( C 0 M ) method can be expressed in the form y n+1 = y n + h k 2 ] 1 +k2 2 k 1 +k 2 + k2 2 +k2 k 2 +k + k2 +k2 k +k where k 1 = f(x n,y n ), k 2 = f(x n + h 2,y n + h 2 k 1), k = f(x n + h 2,y n hk hk 2). k = f(x n + h, y n + 1 hk 1 hk k ). 2 Modified C 0 M Weights Runge-Kutta Method (MCHW-RK) It is possible to establish a three-stage Runge-Kutta formula based on the contraharmonic mean using the mean in the main formula which can be presented as follows: ] y n+1 = y n + h w 1 k k2 2 k 1 + k 2 + w 2 k k2 k 2 + k ] (1)

3 Third order Runge Kutta based on contraharmonic mean 67 where k 1 = f(x n,y n )=f, k 2 = f(x n + ha 1,y n + ha 1 k 1 ), k = f(x n + h(a 2 + a ),y n + h(a 2 k 1 + a k 2 ). w 1 and w 2 are the weights chosen in such a way so that a 1 and a 2 are parameters to be determined and k2 i +k2 i+1 k i +k i+1 is defined as the contraharmonic mean. Notice that for simplicity of the algebra f have been considered as a function of y only, without loss of generality. This will considerably reduce the Taylor series expansions of k i i =1, 2,, to the following k 1 = f (2) k 2 = f + ha 1 ff y f 2 a 2 1 h2 f yy f h a 1 f yyy +... () k = f + h(a 2 + a )ff y + h 2 (a 1 a ff 2 y (a 2 + a ) 2 f 2 f yy )+ () h ( 1 2 a2 1 a f 2 f y f yy + a 1 a (a 2 + a )f 2 f y f yy (a 2 + a ) f f yyy )+... Traditionally, the equations (2)-() would be substituted to obtain an expression of y n+1 in terms of the function together with the parameters a i, i =2, and its derivatives. Since the algebra involved is the division of two series, k 2 i + k 2 i+1 k i + k i+1,i= 1(1) (5) direct substitution cannot be done. These problems are alleviated by multiplying the terms across with the common denominator(k 1 + k 2 )(k 2 + k )and can be written as y n+1 = y n + upper (6) lower with and upper = h(w 1 (k k2 2 )(k 2 + k )+w 2 (k k2 )(k 1 + k 2 )) lower =(k 1 + k 2 )(k 2 + k )

4 68 O. Y. Ababneh and R. Rozita Taylor series expansion of y(x n+1 ) may be written as Taylor = y n + hf h2 ff y h (ff 2 y + f 2 f yy )+ (7) h (f f yyy +f 2 f y f yy + ff y )+.. Since the error of the method can be measured using the expression thus, we get Error = y(x n+1 ) y n+1 Error = Taylor upper lower which could be written as Error lower = Taylor lower upper (8) Comparing the coefficients of the same terms in(8) up to the term h,the following equations of conditions were obtained: f 2 h : w 1 w 2 + = 0 (9) h 2 f f y : 6a 1 w 1 6a 1 w 2 2a 2 w 1 a 2 w 2 (10) 2a w 1 a w 2 +2+a 1 +2a 2 +2a =0 h f f 2 y : a 2 1w 1 a 2 1w 2 2a 1 a 2 w 1 2a 1 a 2 w 2 2a 2 2w 2 a 1 a w 1 (11) 6w 2 a 1 a 6w 2 a 1 a w 2 a 2 a h f f yy : a 2 1 w 1 a 2 1 w 2 a 2 2 w 1 2a 2 2 w 2 2a 2 a w 1 a 2 a w 2 (12) a 2 w 1 2w 2 a (2 + 6a2 1 +a a 2 a +a 2 )=0 Solving equations (9)-(12) using MATHEMATICA 5] we obtained a set of parameters and weights a shown below. w 2 =,w 1 = 1,a = 21 ( + 2)

5 Third order Runge Kutta based on contraharmonic mean 69 a 2 =0,a 1 = 1 7 ( 2) The third order contraharmonic mean.rk formula MCHW-RK can be represented by k 1 = f(x n,y n ), k 2 = f(x n + h 2,y n + h 2 k 1), k = f(x n + h 21 ( + 2),y n + h 21 ( + 2)k 2 ). y n+1 = y n + h 1 k1 2 + k2 2 + k2 2 + k 2 ] k 1 + k 2 k 2 + k (1) (1) Stability Analysis To check on the stability, the equations in(1) and(1) are substituted into the simple test equation proposed by Dahlquist6] is used y = λy and it yields k 1 = f(x n,y n )=λy n (15) k 2 = f(x n + h 1 7 ( 2),y n + h 1 7 ( 2)hλy n ) (16) = λ(y n ( 2)hλy n )=λy n ( ( 2)hλ) k = f(x n + h 21 ( + 2),y n + h 21 ( + 2)λy n ( ( 2)hλ)) (17) = λ(y n + h 21 ( + 2)λy n ( ( 2)hλ)) = λy n (1 + h 21 ( + 2)λ( ( 2)hλ)) Substituting (15), (16) and (17) in (1), and letting z = hλ, we obtain the simplified equation y n+1 = y n + h ( λ 2 y 2 n 1 λ 2 yn( z 2 7 2z z2 8 2z2 9 ) λy n(1+( ( 2)z) 2z z z z 2 +( )z ) z λy n(( ( 2)z)+(1+ 21 (+ 2)z( ( 2)z))) z )

6 70 O. Y. Ababneh and R. Rozita y n+1 = y n + h 1 λy n ( z λy n( z 2 7 2z z2 8 2z2 9 ) (1+( ( 2)z) 2z z (( ( 2)z)+(1+ + 2z 2 +( )z z (+ 2)z( ( 2)z))) z ) y n+1 = y n + zy n ( z ( z 2 7 2z z2 8 2z2 9 ) (1+( ( 2)z) 2z z (( ( 2)z)+(1+ which yield the stability polynomial: 1 (2+ y n+1 = y n (1 + z 8 7 z z 2 +( )z z (+ 2)z( ( 2)z))) 2z z2 8 2z2 9 ) (1+( ( 2)z) ( z z z (( ( 2)z)+(1+ or in more simplified form, where R(z) =1+z 1 ( z z ) 2z 2 +( )z ) z y n+1 = y n R(z) ( z 2 7 2z z2 8 2z2 9 ) (1+( ( 2)z)) 2z z (( ( 2)z)+(1+ 21 (+ 2)z( ( 2)z))) z ) 2z 2 +( )z z (+ 2)z( ( 2)z))) z ) The stability region for the above formula is illustrated in the graphs below: Stiff problem ] ).1 Definition8]: if a numerical method is forced to use, in an interval of integration, a stepsize is forced to be excessively small relative to dominant time-scale of the solution to get a smooth approximation of the exact solution in that interval,then the problem is said to be stiff in that interval. According to Definition.1, in order to get a smooth approximation of the solution need to use very small stepsize for stiff problems. In practice use large stepsize to reduce computational costs.

7 Third order Runge Kutta based on contraharmonic mean 71 5 Numerical Experiments The MCHW RK is tested on two examples of ordinary differential equations which are stiff problems to check on the accuracy of this method,we will comparing the new method by some existing methods with defrrent h and n,including the third and fourth order contraharmonic mean (C 0 M) methods],adam Bashforth Moulton method method(abm), classical third order Runge-Kutta,Nystrom method and Waswas method7], where the third order classical Runge-Kutta method uses the formula where k 1 = f(x n,y n ), k 2 = f(x n + h 2,y n + h 2 k 1), k = f(x n + h, y n hk 1 +2hk 2 ). y n+1 = y n + h 6 k 1 +k 2 + k ] Example 1: Consider the stiff ordinary differential equation with the exact solution y (t) = 100y(t)+e 2t ; y(0) = 0 y(t) = e 100t ( 1+e 98t ) was considered over the range 0 t 1 using a stepsize h =, and h = Discussion and Conclusion The research done in this paper shows the possibility of constructing method out of the various forms of explicit three-stage third order contraharmonic mean Runge-Kutta formula to solve stiff problem. With the purpose of verifying the accuracy, the example used is solved using the MCHW-RK method and some existing methods.figures 1, 2 and demonstrate the stability for the new method, the results show excellent accuracy of the MCHW RK method using tow step sizes,h= and h=0.02 which are better than all the methods results unless classical RK method,nevertheless classical RK method shows a better results when h= compared to h=0.02 which shows that this method requires more iteration to obtain a better accuracy and same thing for the third order CH-RK,but the new method gives abetter accuracy than calssical RK

8 72 O. Y. Ababneh and R. Rozita and CH-RK when h=0.02 Table 1 shows the numerical solutions and the absolute errors of the example for the methods when h= whilst Table 2 when h=0.02. The results of using the MCHW RK method and the other methods for solving this stiff problem on the interval 0,1] are presented in Figures,5,6,7,8,9 and 10. Figures 6 and 7 show that the classical RK and RK methods definitely seems to have difficulty of the aproximation when h=0.02 but without difficulties with h=.however, the MCHW RK method performs perfectly well, even for h= and h=0.02 as shown in Figure (a) and (b).in Figures 7 10 the computed solutions of this problem with h=0.02 using the third and fourth order contraharmonic mean (C 0 M) methods(ch-rk and CH-RK),Adam Bashforth Moulton method(abm) and Waswas methed are performed poorly. from this discussion it is clearly confirmed that the MCHW RK method is appropriate for stiff problems. References 1] P. Novati, An Explicit One-Step Method for Stiff Problems,Springer Journal, Volume 71, Number 2 / October, ] Xin-Yuan Wu,A Sixth-Order A-Stable Explicit One-Step Method for Stiff Systems, Computers Math. Applic. Volume 5, No.9,pp. 59-6, ] R. R. Ahmad and N.Yaacob,Third-order composite Runge-Kutta method for stiff problems, International Journal of Computer Mathematics Vol. 82, No. 10, October 2005, ] A.R.Yaakub and D.J.Evans, A fourt order Runge Kutta RK(,) method with error control,international Journal of Computer Mathematics Vol. 71, October 1999, 811 5] Wolfram S Mathematica : A System For Doing Mathematics By Computer. 2nd Ed. Addison-Wesley Publishing Company. 6] G.Dahlquist and A.Bjorck.,Numerical Methods,Prentice Hall Inc., ] Wazwaz A. M., A Modified Third Order Runge-Kutta method. Appl. Math Letter. Vol () ] Shirley Jun Ying Huang. implementation of Genral Linear Methods for Stiff Ordinary Differential Equations. PHD thesis.the university of Auckland p 28, 2005 Received: July, 2008

9 Third order Runge Kutta based on contraharmonic mean 7 Table 1: the absolute error of the Explicit MCHW-RK method, h= on the equations y = -100y(t)+exp(-2t) compared to the RK,RK,C 0 M,C 0 M, ABM and WAZWAZ methods Y EXACT MCHW RK RK C 0 M C 0 M ABM WAZWAZ E E-08.10E-08.1E E-08.0E-05.18E E E-07.67E E-08.2E-07.8E E E E E-08.96E E-07.16E E E E E-08.88E E E E E E E-08.00E E E E E E E-08.27E E-07.00E E E E E E E E E E E-08.00E E E E E E-08.E E E E E E E-08.55E E E E E E-05 Table 2: the absolute error of the Explicit MCHW-RK method, h =0.02 on the equations y = -100y(t)+(-2t) compared to the RK,RK,C 0 M,C 0 M, ABM and WAZWAZ methods Y EXACT MCHW RK RK C 0 M C 0 M ABM WAZWAZ E-07.07E-05.98E E E E E E E E E E E E E E E E E+0 1.9E-02.07E E-0 9.9E E E E E-02 6.E E E E-07.8E+00.8E E E E E E-07.96E+00.96E+00.76E E E E E E E E E E-08.2E-07.22E E E E+02.69E E-08.5E-07.5E E E+00.8E E E E E E E E E+00

10 7 O. Y. Ababneh and R. Rozita Figure 1: The stability region of the MCHW-RK method in 2D Q z Im z Re z Figure 2: The stability region of the MCHW-RK method in D.

11 Third order Runge Kutta based on contraharmonic mean 75 Figure : Analytical solution of Example 1 Figure : The MCHW-RK method used to solve the stiff problem for(a) h = and (b) h=0.02. Figure 5: The RK method used to solve the stiff problem for(a) h = and (b) h=0.02. Figure 6: The RK method used to solve the stiff problem for (a) h = and (b) h=0.02.

12 76 O. Y. Ababneh and R. Rozita Figure 7: The (C o M) method used to solve the stiff problem for(a)h = and(b) h= Figure 8: he (C o M) method used to solve the stiff problem for(a)h = and(b)h=0.02. Figure 9: The ABM method used to solve the stiff problem for(a) h=and (b)h= Figure 10: The WAZWAZ method used to solve the stiff problem for(a) h =and(b)h=0.02.

Fourth Order RK-Method

Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)