Semi-implicit two-step hybrid method with FSAL property for solving second-order ordinary differential equations

Article history: Received 3 March 2017 Received in revised form 16 May 2017 Accepted 27 May 2017 Two semi-implicit two-step hybrid methods of order five and six designed using First Same as Last (FSAL) property are developed for solving secondorder ordinary differential equation. The stability analysis is determined by the interval of periodicity and the interval of absolute stability. The numerical results carried out show that the new method has smaller maximum error than existing method of similar type proposed in scientific literature, using constant step-size.


Introduction
* In this paper we are interested in the numerical solution of initial value problems (IVPs) associated with special second-order ordinary differential equation (ODE) of the forms (Eq. 1) " = ( , ), (0) = , ′ (0) = . (1) This problem does not incorporate the first derivative in ( , ) and the solution relates to oscillatory and periodic solutions. This type of problem commonly arises in the fields of applied sciences such as motion of planet in celestial mechanics, orbital problems, quantum mechanics and electronic. Since most differential equations of celestial mechanics take the form " = ( , ), it is not surprising that the first attempts at developing methods for Eq. 1 were made by astronomers (Hairer et al., 1993). In recent years, the special second-order ODEs have been extensively studied by many researchers for solving IVPs relates to oscillatory and periodic problems. There has been many research on multistep methods done for Eq. 1, particularly the two-step hybrid method (HM) (Tsitouras, 2003;Coleman, 2003;Franco, 2006;Fang and Wu, 2008;Samat et al., 2012;Ahmat et al., 2013;Jikantora et al., 2015;Franco et al., 2014;Franco and Randez, 2016;Kalogiratou et al., 2016). In a previous work, Coleman (2003), he has investigated the order condition of two-step hybrid method based on the theory of B-series. He discussed these order conditions for general class of two-step hybrid method for problem Eq. 1.
A new class of explicit two-step hybrid method (EHM) which requires less number of stages per step has been developed by Franco (2006). He has considered EHM of order four up to order six. The study by Ahmad et al. (2013) developed semiimplicit two-step hybrid method up to algebraic order five for solving oscillatory problems by taking dispersion relation and solving them together with algebraic conditions of the methods. Later, another study carried out by Jikantora (2015) also developed semi-implicit two-step hybrid method with fifth algebraic order with dispersion and dissipation of higher order.
In this paper, we constructed semi-implicit twostep hybrid method with algebraic order five and six with FSAL feature. The FSAL feature specifies that, the last row of coefficient matrix is same with the vector of output value coefficients. The interval of stability for new methods are also presented and followed by numerical experiments on second-order differential equation for oscillatory or periodic problems. An s-stage two-step hybrid method generally given by (Eqs. 2 and 3) (2) = 1, … , The method consists of coefficients which is called the generating matrix, the vector output and the vector abscissa which can be represented in Butcher tableau as shown in Table 1.

Preliminaries
The method of the form Eq. 2 and Eq. 3 can be defined as Eqs. 4, 5, and 6.
= 3, … , where, ℎ = +1 − is the step size while −1 and represent approximations for ( −1 , −1 ) and ( , ) respectively. The method only requires to evaluate − 1 function evaluation namely ( , ), ( + 3 ℎ, 3 ), …, ( + ℎ, ) in each step. Therefore, this method is considered as twostep hybrid method with − 1 stages per step. The tableau of semi-implicit two-step hybrid method with FSAL (SIHMF) feature is as in Table 2  The diagonal elements 33 , 44 … , in Table 2 are denoted by . The vector output corresponding to the output approximation is identical to the last row of . The s-stage implicit two-step hybrid method parameters are given by In this case, FSAL specifies the s-stage to be the same as the first-stage at the next step given by In order to investigate the phase property of twostep hybrid method for solving initial value problem Eq. 1 we consider the second order linear test equation as proposed by Franco (2006) If Eq. 2 and Eq. 3 are applied to the test problem Eq. 7, hence it can be written in the vector form as Eqs. 8 and 9 where = ( 1 , … , ) , = ( 1 , … , ) and = (1, … ,1) . Solving equation in Eq. 8 we obtain Eq. 10 where ( + 2 ) −1 = 1 − 2 + 4 2 = 1 − 2 + 4 2 − ⋯ + (−1) −2 2 −4 −2 .
Definition 1: The quantities ( ) and ( ) are called the dispersion error (or phase-error) and dissipation error, respectively (Eq. 12) 2√ ( 2 ) ) (12) According to Simos et al. (2003), the dispersion is the angle between the true and the approximate solution and the dissipation is the distance from a standard cyclic solution. The method is said to be dispersive of order and dissipative of order , if the stability of two-step hybrid method will be calculated using the interval of periodicity and interval of absolute stability which are determined by characteristic polynomial.

Definition 2:
The polynomial (Eq. 13) is called characteristic polynomial of Eq. 11. Twostep hybrid method has periodicity interval if coefficient of Eq. 13 satisfy the condition (Eq. 14) ( 2 ) ≡ 1, | ( 2 )| < 2, ∀ ∈ (0, ) The method satisfies the condition Eq. 14 are called zero dissipative ( ( ) = 0). When the methods have a finite order of dissipation, means the interval of periodicity is (0, ∝), the integration process is stable or remains bounded if the coefficient Eq. 13 satisfy the conditions (Eq. 15) | ( 2 )| < 1,  and | ( 2 )| < 1 + ( 2 ) (15) ∀ ∈ (0, ) The two-step hybrid method are derived by using order conditions which are the set of simultaneous equations which contain the coefficients. The solution of simultaneous equations gives the value of coefficients in terms of the free parameter associated to the local truncation error +1 . The coefficients are then substituted into the error constant +1 .
The minimized value of the free parameter is obtained by optimizing the error constant with respect to the free parameter. The th-order error constant is a quantity defined by where, is the number of trees of order + 2( ( ) = + 2) and +1 ( ) is local truncation error as defined in Coleman (2003). The order conditions up to algebraic order six given in Coleman (2003) are (Eqs. 17, 18, 19, 20, 21, 22) Order 2

Derivation of method SIHM with FSAL property
Presented in this section is the derivation of semiimplicit two-step hybrid method of order five designed using FSAL property.

Fifth order SIHM
In this case, FSAL specifies the four-stages to be the same as the first-stage at the next step given by The fourth-stage two-step hybrid method parameters are given by 4 = 1, 4 = , = 1,2,3,4 To derive fifth-order SIHMF method, we use the algebraic order conditions Eq. 17, Eq. 20 and Eq. 22. There are six equations and seven unknowns that have to be satisfied giving one free parameter which is chosen to be 3 . The system of equations are solved simultaneously to obtain the values of coefficients in terms of 3 which are given by the expression This method is denoted as SIHMF5 and can be expressed diagrammatically as in Table 3. The interval of periodicity is given by (0, √6).
To derive the new method, we use the algebraic order conditions Eq. 17 and Eq. 22. There are 10 equations and 12 unknowns that have to be satisfied giving two free parameter which is chosen to be 4 and . Solving all conditions simultaneously to obtain the values of coefficients in terms 4 and . By minimizing constant error Eq. 4, we obtain = − 1 40 and 4 = 163 100 and 6 = 4.4213 × 10 −2 . This method is denoted as SIHMF6 and can be expressed diagrammatically as in Table 4. The interval of absolute stability is given by (0, 2.18).

Results and discussion
In this section, we present five problems which have oscillatory solution. All the problems will be tested by the semi-implicit FSAL methods to evaluate the effectiveness of new method. The fifth order method, SIHMF5 is compared with semi-implicit hybrid method of order five with fourstages derived by Ahmad et al. (2013) and explicit two-step hybrid method of order four with threestages derived by Franco (2006). The sixth order method is compared with two other methods derived by Franco (2006).
The methods that have been used in comparisons are denoted by (i) SIHM4 (5): Fifth order semi-implicit method with four-stage derived by Ahmad et al. (2013). (ii) EHM4: Fourth order explicit hybrid method with four-stage derived by Franco (2006). (iii) EHM6: Sixth order explicit hybrid method with five-stage derived by Franco (2006). (iv) EHM5: Fifth order explicit hybrid method with four-stage derived by Franco (2006).
The criterion used in the numerical comparison is decimal logarithm of the maximum error versus step-sizes required by each method. Exact solution is = sin ( ). The numerical results are shown in Fig. 1 and Fig. 2.   Fig. 9 and Fig. 10 From Fig. 3, Fig. 5 and Fig. 7, we observed that the new SIHMF5 is performed better compared to SIHM4(5) and EHM4. However in Fig. 1 shows that SIHM4(5) is performed better than SIHMF5 and EHM4. While in Fig. 9 we observed that, all methods have almost equal performance. From Fig. 2, Fig. 4, Fig. 6, Fig. 8 and Fig. 10 we observed that the new SIHMF6 have almost equal performance with EHM6 and EHM5.

Conclusion
Two semi-implicit two-step hybrid method of order five and six designed using FSAL property for solving second-order IVPs with oscillatory solution are derived. The results of comparison based on maximum error evaluation at different step-sizes were used for comparison purpose as shown in Figs. 1-10. Our new method, SIHMF5 and SIHMF6 can be an alternative method for solving oscillatory problems and can be advantageous to the Science and Technology fields.