Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 136961, 10 pages http://dx.doi.org/10.1155/2013/136961 Research Article Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems S. Z. Ahmad,1 F. Ismail,2 N. Senu,2 and M. Suleiman2 1 2 Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia Correspondence should be addressed to S. Z. Ahmad; sufia Received 11 January 2013; Accepted 4 March 2013 Academic Editor: Juan Carlos Cortés López Copyright © 2013 S. Z. Ahmad 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. We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and RungeKutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented. 1. Introduction Second-order ordinary differential equations (ODEs) which are oscillatory in nature often arise in many scientific areas of engineering and applied sciences such as celestial mechanics, molecular dynamics, and quantum mechanics. Consider the numerical solution of the initial value problem (IVP) for second-order ODEs in the form 𝑦󸀠󸀠 = 𝑓 (𝑡, 𝑦) , 𝑦 (𝑡0 ) = 𝑦0 , 𝑦󸀠 (𝑡0 ) = 𝑦0󸀠 , (1) in which the first derivative does not appear explicitly. Apparently, some of the most common methods used for solving second-order ODEs numerically are Runge-Kutta Nyström (RKN) and Runge-Kutta methods for Runge-Kutta method the IVPs need to be reduced to a system of first-order ODEs twice the dimension. The IVP can be solved using a particular explicit hybrid algorithms which were developed by Franco [1] or a multistep method for special second-order ODEs as in Yap et al. [2]. Franco [3] proposed that (1) can be solved using a particular explicit hybrid algorithms or special multistep methods for second-order ODEs. Franco [3] constructed explicit two-step hybrid methods of order four up to order six for solving second-order IVPs by considering the local truncation error and order conditions developed by Coleman [4]. Most of the IVPs represented by (1) have solutions which are oscillatory in nature, making it difficult to get the accurate numerical results. To address the problem several authors [5–7] focused their research on developing methods with reduced phase lag and dissipation, where phase-lag or dispersion error is the difference of the angle for the computed solution and the exact solution and dissipation is the distance of the computed solution from the standard cyclic solution. The analysis of phase-lag or dispersion errors was first introduced by Brusa and Nigro [8]. Van der Houwen and Sommeijer [9] proposed explicit RKN methods of order four, five, and six with reduced phase-lag of order 𝑞 = 6, 8, 10, respectively. Senu et al. [7, 10] developed diagonally implicit RKN method with dispersion of higher order for solving oscillatory problems. In a related work Kosti et al. [11] constructed an optimized RKN method (OPRKN) based on the existing explicit four-stage fifth-order RKN method for the integration of oscillatory IVPs. In his derivation he used the phase-lag, amplification factor and the first derivative of the amplification factor by equating them to zero. Later, Kosti et al. [12] also developed an OPRKN method based on the same explicit RKN method, in which he used the phase lag, amplification factor, and the first 2 Abstract and Applied Analysis derivative of the phase-lag properties instead of using only the first derivative of amplification factor in his previous work. In this paper, we constructed three-stage fourth-order and three-stage fifth-order methods with dispersion order six and zero dissipation and also four-stage fifth-order method with dispersion order eight and dissipation order five. It is done by taking the dispersion relations for the semiimplicit hybrid methods and solving them together with the algebraic conditions of the methods. The intervals of stability of the methods are also presented. Finally, numerical tests on second-order differential equation for oscillatory problems are given. 2. Analysis Phase Lag of the Methods An 𝑠-stage two-step hybrid method for the numerical integration of the IVP(1) is of the form 𝑌𝑖 = (1 + 𝑐𝑖 ) 𝑦𝑛 − 𝑐𝑖 𝑦𝑛−1 step and the semi-implicit hybrid with the diagonal elements being equal can be written in Butcher tableau as follows: −1 0 𝑐3 𝑎3,1 𝑎3,2 𝛾 .. .. .. . d 𝛾 . . 𝑐𝑠 𝑎𝑠,1 𝑎𝑠,2 ⋅ ⋅ ⋅ 𝑎𝑠,𝑠−1 𝛾 𝑏1 𝑏2 ⋅ ⋅ ⋅ 𝑏𝑠−1 𝑏𝑠 . Phase analysis can be divided into two parts. First is the inhomogeneous part, in which the phase error is constant in time and second is the homogeneous one, in which the phase errors are accumulated as 𝑛 increases. As proposed by Franco [3], the phase analysis is investigated using the second-order homogeneous linear test model 𝑦󸀠󸀠 (𝑡) = −𝜆2 𝑦 (𝑡) 𝑠 𝑖 = 1, . . . , 𝑠, 2 (2) 𝑦𝑛+1 = 2𝑦𝑛 − 𝑦𝑛−1 + ℎ ∑𝑏𝑖 𝑓 (𝑡𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖 ) , 𝑖=1 Y = (e + c) 𝑦𝑛 − c𝑦𝑛−1 − 𝐻2 AY, (9) 𝑦𝑛+1 = 2𝑦𝑛 − 𝑦𝑛−1 − 𝐻2 bT Y, (10) −1 where 𝑏𝑖 , 𝑐𝑖 , and 𝑎𝑖𝑗 can be represented in Butcher notation by the table of coefficients as follows: 𝑐1 𝑎1,1 .. .. . . = 𝑇 𝑏 𝑐𝑠 𝑎𝑠,1 𝑏1 ⋅ ⋅ ⋅ 𝑎1,𝑠 . d .. 𝑐 𝐴 ⋅ ⋅ ⋅ 𝑎𝑠,𝑠 ⋅ ⋅ ⋅ 𝑏𝑠 . (3) (12) where −1 𝑃 (𝐻2 ) = 1 − 𝐻2 bT (I + 𝐻2 A) c. (4) (5) 𝑗=1 𝑖 = 3, . . . , 𝑠, (6) 𝑖=3 where 𝑓𝑛−1 = 𝑓(𝑡𝑛−1 , 𝑦𝑛−1 ), 𝑓𝑛 = 𝑓(𝑡𝑛 , 𝑦𝑛 ), ℎ = Δ𝑡 = 𝑡𝑛+1 − 𝑡𝑛 and the first two nodes are 𝑐1 = −1 and 𝑐2 = 0. The method only requires to evaluate 𝑓(𝑡𝑛 , 𝑦𝑛 ), 𝑓(𝑡𝑛 + 𝑐3 ℎ, 𝑌3 ), . . . , 𝑓(𝑡𝑛 + 𝑐𝑠 ℎ, 𝑌𝑠 ) for each step after the starting procedure. This method is considered as a two-step hybrid method with 𝑠−1 stages per (13) Solving the difference system (12), the computed solution is given by 󵄨 󵄨𝑛 (14) 𝑦𝑛 = 2 |𝑐| 󵄨󵄨󵄨𝜌󵄨󵄨󵄨 cos (𝜔 + 𝑛𝜙) , where 𝜌 is the amplification factor, 𝜙 is the phase, 𝜔 and 𝑐 are real constants determined by 𝑦0 and 𝑦0󸀠 and the hybrid parameters. The exact solution of (8) is given by 𝑦 (𝑡𝑛 ) = 2 |𝜎| cos (𝜒 + 𝑛𝐻) , 𝑦𝑛+1 = 2𝑦𝑛 − 𝑦𝑛−1 + ℎ2 [𝑏1 𝑓𝑛−1 + 𝑏2 𝑓𝑛 + ∑𝑏𝑖 𝑓 (𝑡𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖 )] , Substituting (11) into (10), the following recursion relation is obtained: −1 𝑖 𝑠 (11) 𝑆 (𝐻2 ) = 2 − 𝐻2 bT (I + 𝐻2 A) (e + c) , 𝑌𝑖 = (1 + 𝑐𝑖 ) 𝑦𝑛 − 𝑐𝑖 𝑦𝑛−1 + ℎ2 ∑ 𝑎𝑖𝑗 𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑌𝑗 (...truncated)