P-Stable Higher Derivative Methods with Minimal Phase-Lag for Solving Second Order Differential Equations

Some new higher algebraic order symmetric various-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Basing on this formula, calculation of free parameters is performed to minimize the phase-lag. An explicit symmetric multistep method is presented. This method is of higher algebraic order and is fitted both exponentially and trigonometrically. Such methods are needed in various branches of natural science, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behavior. Many exponentially-fitted symmetric multistepmethods for the second-order differential equation are already developed. The stability properties of several existing methods are analyzed, and a new P -stable method is proposed, to establish the existence of methods to which our definition applies and to demonstrate its relevance to stiff oscillatory problems. The work is mainly concerned with two-stepmethods but extensions tomethods of larger step-number are also considered. To have an idea about its accuracy, we examine their phase properties. The efficiency of the proposed method is demonstrated by its application to well-known periodic orbital problems. The new methods showed better stability properties than the previous ones.


Introduction
In the recent years, there is much activity concerned with the numerical solution of the IVPs of the type y f x, y , y x 0 y 0 , y x 0 y 0 , 1.1 where f x, y is a function in which the first derivative does not appear explicitly.In addition the function f x, y also satisfies the Lipschitz condition of the first order with respect to y.
There are two main groups of methods for the numerical solution of the form given in 1.1 with periodic of an oscillatory solutions, see 1-3 .The first consists of methods in which

Stability and Periodicity
For problems with oscillatory solutions, linear stability analysis is based on the test equation of the form y −λ 2 y where λ is a real constant.This test equation is previously introduced by Lambert and Watson 7 as well as the interval of its periodicity, in order to investigate the periodic stability properties of numerical method for solving the initial value problem given in 1.1 .Stability means that the numerical solutions remain bounded as we move further away from the starting point, see Coleman 15 , Simos 16 , and Simos and Williams 17 .
In order to investigate the periodic stability properties of numerical methods for solving the initial-value problem 1.1 , in 7 they introduce the scalar test equation y −λ 2 y.

2.1
Based on the theory developed in 7, 18 , when a symmetric multistep method, given by k j 0 a j y n i h 2 k j 0 is applied to the scalar test equation 2.1 , a difference equation is obtained of the form: where χ λh, h is the step length, and y n is the computed approximation to y nh , n 0, 1, 2, . ... The general solution of the above difference equation can be written as where ξ j , j 1 1 k are the distinct roots of the polynomial: where ρ and σ are polynomials given by We note here that the roots of the polynomial 2.5 are perturbations of the roots of ρ 2.6 .We denote ξ 1 and ξ 2 the perturbations of the principal roots of ρ.
Based on 7 when a symmetric multistep method is applied to the scalar test equation 2.1 , a difference equation 2.3 is obtained.The characteristic equation associated with 2.3 is given by 2.5 .The roots of the characteristic polynomial 2.5 are denoted as ξ i , i 1 1 k.
According to Lambert and Watson 7 the following definitions are that given by: We have the following definitions.
As a modification of this method this paper aimed to develop a family of symmetric P -stable two step, four step and six step methods involving higher-order derivatives with minimal phase-lag error in the form: With the help of the associated formula, 2.9 This latter formula depends on an offpoint x n±a .The precise choice enables getting a P -stable formula with large interval of periodicity.

Construction of Two-Step P -Stable Higher-Order Derivative with Phase Fitted Schemes
Considering the two-step, P -stable formula that depends on one of f point involving higherorder derivative in the general form: where s is a constant; the basic idea behind our approach is to approximate y n±s by the expression involving the quantities y n , y n 1 , and y n−1 .We introduce P -stable methods of O h 4 , O h 6 , and O h 8 with minimal phase-lag errors using 2.10 with the associated formula 2.9 .This gives y n±s a 0 y n a 1 y n±1 a −1 y n∓1 h 2 b 0 y n b 1 y n±1 b −1 y n∓1 .

2.11
Applying the method given by 2.10 and 2.11 with i 1 to the scalar test problem 2.1 , we obtain the following difference equation: where h is the step length, and A χ and B χ are polynomials in χ.
The characteristic equation associated with 2.12 is obtained: The roots of 2.13 are complex and of modulus one, see 7, 19 , if R χ < 1.

2.14
Under this condition the roots of 2.13 can be written as ξ 1,2 e ±iθ χ where θ χ .The numerical solution of 2.13 is bounded if both roots are unequal and their magnitude less than one or equal to one.
The periodicity condition requires those roots to lie on the unit circle, that is, R χ 2 is then a rational approximation or cos χ 2 , 20-22 .A method is said to be P -stable if the interval of periodicity is infinite.
For a given method i.e., a given λ , one has to find a restriction which must be placed on the step length h to ensure that the condition |R χ 2 | < 1 is satisfied.
The following definition is taken from the work by 20, 23 : where θ χ is a real function of χ.For any method corresponding to the characteristic equation 2.13 , the phase-lag is defined by Hairer and Wanner in 5 as the leading term in the following expression:

2.16
If t O χ q 1 as χ → 0, the order of phase-lag is q.
Due to Definition 2.2, the method 2.10 is P -stable if its interval of periodicity is 0, ∞ .

Theorem 2.5 see 19 .
A method which has the characteristic equation 2.13 has an interval of periodicity 0, 2.17 When R nm ν 2 ; θ < 1, the roots of 2.13 are distinct and lie on the unit circle.
When R nm ν 2 ; θ > 1, the method is unstable since the corresponding difference equation has an unbounded solution, see 25, 26 .

2.18
Remark 2.8 see 27 .If the phase-lag order is q 2s, then By the last definition we have assumed that A χ > 0,

Two-Step P -Stable Involve Second Derivative with Minimal Phase-Lag Errors
Consider the symmetric family of second derivative, two-step methods of 2.10 and 2.11 as y n 1 − 2y n y n−1 h 2 2β 10 y n β 11 y n s y n−s .

2.22
Therefore, we introduce P -stable methods of higher order with minimal phase-lag errors using the formula: with the coefficients: where β 11 , a 0 , b 0 , b 1 , and b −1 are free parameters.Also, the local truncation error of the method is given by

2.25
We denote this family of methods 1 , setting χ λh, we obtain 2.13 , and we get

2.26
Then we get

2.27
As given in 2.14 , the method represented by 2.22 , 2.23 with 2.24 will be P -stable provided A χ ± B χ > 0 for all χ ∈ 0, ∞ ; consequently, it is easy to prove that

2.28
Following the phase-lag, which is denoted by P χ , as a leading coefficient in the expression of | θ χ − χ /χ|, then

2.29
The method has phase-lag error of

2.30
satisfying the P -stability conditions.we can obtain a symmetric P -stable second derivative, two-step method given by 2.22 with phase-lag error of order O h 4 , as the following cases.

Fourth-Order Scheme
As in Section 2.2.1 we introduced P -stable methods of higher order with minimal phase-lag errors using the formulae 2.22 and 2.23 with the coefficients: where β 11 , s, a 0 , and b 0 are free parameters.
Also, the local truncation error of the method is given by

2.32
For an M 4 a 0 , b 0 , b 1 , b −1 , setting χ λh, 2.13 is obtained with the coefficient given by, and we get
1 b 0 0, with phase-lag error of order:

Sixth-Order Scheme
The two-step P -stable methods of order six with minimal phase-lag errors using the formula

2.38
The P -stability conditions will be

2.39
The method has phase-lag error of O h 8 .

Construction of Four-Step P -Stable Higher-Order Derivative with Minimal Phase-Lag Errors
Consider the symmetric four-step methods in the form: the characteristic equation will be where

3.4
The P -stability conditions are

3.5
Following the phase-lag, denoted by P χ , is leading coefficient in the expression of The local truncation error of the method is given by 3.7

Second Derivative of Second-Order Scheme
We can establish some choices of the parameters to obtain symmetric four step P-stable methods involve second derivative with minimal phase-lag errors tabulated in Table 1.

Second Derivative of Fourth-Order Scheme
Let q 4, the coefficients of 3.1 and 3.2 will be

3.8
Also, the local truncation error of the method is given by LET The coefficients of the characteristic equation will be

3.10
The P -stability conditions are

3.11
We can obtain a symmetric P -stable second derivative, four-step methods with minimal phase-lag errors by suitable choices of the free parameters, see 8-10 .

Fourth Derivative of Sixth-Order Scheme
Consider the symmetric four-step methods involve fourth derivative in the form 3.1 with 3.2 to obtain the characteristic equation: with the following coefficients:

3.14
Now we have some cases for P -stable, symmetric four-step methods involve fourth derivative one is as follows. Let

Conclusions
In this paper a higher algebraic order exponentially fitted free-parameters method is developed.We have given explicitly the way for the construction of the method.Stability analysis of the new method is also presented.The numerical results, so far obtained in this paper, show the efficiency of the newly derived integrator of order five.We also observed that, for an exponentially fitted problems, our integrator do not use small step lengths, as may be required by many multistep methods before good accuracy is obtained.We exploit the freedom in the selection of the free parameters of one family with the purpose of obtaining specific class of the highest possible phase-lag order, which are also characterized by minimized principal truncation error coefficients.Finally, the new integrator derived in this paper is capable of handling stiff problems for which exponential fitting is applicable.

Definition 2 . 6 .
A region of stability is a region of the plane, throughout which |R χ | < 1.Any closed curve defined by |R χ | 1 is a stability boundary.
with phase-lag error of order: 10 y n β 11 y n s y n−s h 4 2β 20 y