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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 136961, 10 pages
Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems
1Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
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 Runge-Kutta (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.
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 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  or a multistep method for special second-order ODEs as in Yap et al. . Franco  proposed that (1) can be solved using a particular explicit hybrid algorithms or special multistep methods for second-order ODEs. Franco  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 .
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 . Van der Houwen and Sommeijer  proposed explicit RKN methods of order four, five, and six with reduced phase-lag of order , 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.  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.  also developed an OPRKN method based on the same explicit RKN method, in which he used the phase lag, amplification factor, and the first 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 semi-implicit 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 -stage two-step hybrid method for the numerical integration of the IVP(1) is of the form where , , and can be represented in Butcher notation by the table of coefficients as follows:
The methods of the form (2) are defined by where , , and the first two nodes are and . The method only requires to evaluate for each step after the starting procedure. This method is considered as a two-step hybrid method with s-1 stages per step and the semi-implicit hybrid with the diagonal elements being equal can be written in Butcher tableau as follows:
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 increases. As proposed by Franco , the phase analysis is investigated using the second-order homogeneous linear test model
Solving the difference system (12), the computed solution is given by where is the amplification factor, is the phase, and are real constants determined by and and the hybrid parameters. The exact solution of (8) is given by where is the number of term, and are real constants determined by initial conditions. Equations (14) and (15) led to the following definition.
Definition 1 (phase lag). Apply the hybrid method (2) to (8). Next we define the phase lag . If , and then the hybrid method is said to have phase-lag order . Additionally, the quantity is called amplification error. If , then the hybrid method is said to have dissipation order .
From Definition 1, it follows that
Let us denote and to be the following:
Based on the definition of phase lag, the dispersion relations are developed. For a zero dissipative method with three-stage (), the dispersion relation of order six ( is given by the following:
Order 6: and the dispersion relations up to order eight for are given by
The following quantity is used to determine that the dissipation constant of the formula for is(i)for (ii)for From (12), the stability polynomial of hybrid method can be written as The numerical solution defined by (12) should be periodic. The necessary conditions are and interval is known as the periodicity interval of the method. The method is called zero dissipative when , that is, if it satisfies conditions (16). Otherwise, as the method possesses a finite order of dissipation, the integration process is stable if the coefficients of polynomial in (23) satisfy the conditions and interval is known as the interval of absolute stability of the method.
3. Construction of the Methods
In this section, the fourth-and fifth-order SIHMs which require only three and four stages respectively are obtained. The derivations of the methods are based on the order conditions, dispersive and dissipative error, and minimization of the error constant of the method. The error constant is defined by where is the number of trees of order , for the th-order method and is the local truncation error defined in Coleman .
The Order conditions of hybrid method given in  are
Order 6 For the method, need to satisfy
3.1. SIHM with Three Stages
To derive the fourth-order SIHM method, we use the algebraic conditions up to order four (27)–(29), simplifying condition (32), zero dissipation conditions (), and dispersion relation of order six (, (19). The resulting system of equations consists of five nonlinear equations with seven unknown variables to be solved. Therefore, we have two degrees of freedom. The coefficients of the methods are determined in terms of the arbitrary parameters and which are given by the expressions
By minimizing the error constant from (26) we have and . This method is denoted as SIHM3 which is given below:
With this solution, the norms of the principal local truncation error coefficient for are given by where are the error equations for the fifth-order methods. This formula has dispersive order six and zero dissipation with a dispersion constant . The interval of periodicity of the method is (0, 2.96).
Meanwhile, to derive the three-stage fifth-order SIHM method, the algebraic conditions (27) to (30) and equation (32) with dispersion relation of order six (, equation (19), and zero dissipation conditions () are solved simultaneously. This involves seven equations and seven unknowns need to be solved; hence it has a unique solution. This method is denoted as SIHM3 whose coefficients are given below:
With this solution, the norm of the principal local truncation error coefficient for is given by where are the error equations for the sixth-order method. This formula has dispersive order six and zero dissipation with a dispersion constant . The interval of periodicity of the method is (0, 2.96).
3.2. SIHM Order Five with Four Stages
The SIHM method of order five is obtained by considering the order conditions up to order five which are (27) to (30) and (32) together with dispersion relations up to order eight, ((19) and (20)). Solving all the conditions simultaneously, and then the following family of solutions in terms of free parameters and is obtained:
By minimizing the error norm expression, we have and .
With this solution, the norm of the principal local truncation error coefficient for is given by
This fifth-order formula is dispersive order eight and dissipative order five with dispersion and dissipation constants are and respectively. This method is denoted as SIHM4, the coefficients are given below (see: The SIHM4 method) and the interval of absolute stability of the method is (0, 5.75).
Table 1 shows a comparison of the properties of the methods derived.
4. Problems Tested and Numerical Results
In this section, SIHM3 is compared with five other fourth-order methods: DIRKN three-stage fourth-order derived by Senu et al. , DIRKN three-stage fourth-order derived by Sommeijer , Classical Runge-Kutta fourth-order given in Dormand , explicit three-stage fourth-order hybrid method derived by Franco , and Classical RKN fourth order given in Hairer et al. . The fifth-order methods, SIHM3 and SIHM4 are compared with four other methods: DIRKN four-stage fourth-order derived by Senu et al. , Classical Runge-Kutta fifth order derived by Butcher , explicit four-stage fifth-order hybrid method derived by Franco , and Classical RKN fifth-order method given from Hairer et al. . All the problems were executed for except for Orbital problem . The test problems used are listed below.
Problem 1 (Chawla and Rao ). We have Exact solution is
Problem 2 (Attili et al. ). We have Exact solution is
Problem 3 (Lambert and Watson ). We have Exact solution is ,, and is chosen to be and parameters and are 20 and 0.1 respectively.
Problem 4 (an almost periodic orbit problem given in Stiefel and Bettis ). We have Exact solution is , .
Problem 5 (inhomogeneous system studied by Franco ). We have Exact solutions are
Problem 6 (Allen and Wing ). We have Exact solution is .
Problem 7 (inhomogeneous problem studied by Papadopoulos et al. ). We have where .
Exact solution is . Numerical result is for the case .
Problem 8 (orbital problem studied by van der Houwen and Sommeijer ). We have Exact solution is ,
The following notations are used in Figures 1–16.(i)SIHM3: a semi-implicit hybrid method of order four with dispersive order six and zero dissipation. (ii)SIHM3: a semi-implicit hybrid method of order five with dispersive order six and zero dissipation. (iii)SIHM4: a semi-implicit hybrid method of order five with dispersive order eight and dissipative order five. (iv)DIRKN(S1): a three-stage fourth-order dispersive order six method with “small” dissipation constant and principal local truncation errors derived by Senu et al. . (v)DIRKN(HS): a three-stage fourth-order derived by Sommeijer . (vi)DIRKN(S2): a four-stage fourth-order dispersive order eight method with “small” dissipation constant derived by Senu et al. . (vii)RKN4: a classical RKN method order four in . (viii)RK4: a classical RK method order four in . (ix)RKN5: a five-stage fifth-order RKN method derived by Butcher .(x)RK6: a seven-stage sixth-order RK method derived by Hairer et al. . (xi)EXHBRD4: a explicit three-stage fourth-order hybrid method derived by Franco . (xii)EXHBRD5: a explicit four-stage fifth-order hybrid method derived by Franco .
In order to evaluate the effectiveness of the semi-implicit hybrid methods, we solved several problems which have oscillatory solutions. To make a comparison of SIHM and other existing methods, one measure of the accuracy is examined using the absolute error which is defined by where is the exact solution and is the computed solution.
For comparison purposes, in analyzing the numerical results, methods of the same order will be compared. The results are given in Figures 1–16. We present the efficiency curves where the common logarithm of the maximum global error along the integration versus the CPU time is taken. From Figures 1, 2, 3, 4, 5, 6, 7, and 8, we observed that the new SIHM3 is the most efficient for integrating second-order differential equations possessing oscillatory solutions, followed by diagonally implicit DIRKN(S1), original explicit hybrid method EXHBRD4, and other methods like DIRKN(HS), RKN4, and RK4.
From Figures 9, 10, 11, 12, 13, 14, 15, and 16, for the fifth-order methods we observed that SIHM4 is the most efficient, followed by SIHM3 and EXHBRD5 and the rest of the methods. Even though the new methods are semi-implicit and fairly expensive in terms of time consumed, they are still more efficient compared to the explicit counterpart.
In this paper three-stage semi-implicit hybrid methods of order four and five are developed and denoted by SIHM3 and SIHM3, respectively, they have dispersion order six and zero dissipation. We also developed method of four-stage and fifth order denoted by SIHM4 which has dispersion order eight and dissipation order five. All the three methods developed are suitable for solving second-order IVPs which are oscillatory in nature. From the efficiency curves shown in Figures 1–16, we can conclude that all the methods are very efficient compared to the well-known existing methods of the same order in the scientific literature.
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