New Phase Fitted and Amplification Fitted Numerov-Type Methods for Periodic IVPs with Two Frequencies

and Applied Analysis 3 Applying the explicit Numerov-type method 2.1 to 2.3 , we obtain the following relation yn 1 − S ( H2 ) yn P ( H2 ) yn−1 0, H λh 2.4


Introduction
In this paper we are interested in the numerical integration of the initial value problem IVP of ordinary differential equations in the form y x f x, y , y x 0 y 0 , y x 0 y 0 , x ∈ x 0 , x end , 1.1 whose solutions exhibit a pronounced oscillatory character. Such problems are frequently encountered in a variety of scientific fields and engineering applications. Since the analytical solutions of these equations are usually not available, the numerical solutions become very important. To our knowledge, most existing methods of fitted type trigonometrically/exponentially fitted, phase fitted and amplification fitted, etc. are adapted to one frequency see 1-3 . We call them one-frequency methods. For example, by phase fitting, Van de Vyver 4 constructed an adapted explicit Numerov-type method for oscillatory problems with one frequency. Other related work can be found in 5-20 . However, for oscillatory problems with many frequencies in the solution, when we apply a one-frequency method, it is difficult to choose the most suitable fitting frequency. Contrast to the onefrequency methods, papers on integrators that are adapted to more than one frequency are few. In 21 , Wang obtained a trigonometrically fitted Numerov-type method for periodic IVPs with two frequencies and provided some numerical examples to show that his method associated with two frequencies is more accurate and more efficient than the trigonometrically fitted Numerov methods with one frequency. More recently, Fang et al. 22 gave a trigonometrically fitted explicit Numerov-type method for the numerical integration of oscillatory problems with two frequencies. Inspired by Van de Vyver's idea in 4 , we investigate, in this paper, explicit Numerov-type methods for the numerical integration of periodic IVPs with two frequencies. The paper is organized as follows. In Section 2, the preliminaries of stability and phase properties of explicit Numerov-type methods are recalled. In Section 3, we construct two new phase fitted and amplification fitted explicit Numerov-type methods for periodic initial value problems with two frequencies. In Section 4, the stability and phase properties of the newly derived methods are analyzed. In Section 5, we present some numerical results. Section 6 is devoted to the conclusions.

Numerov-Type Methods
In this section, we recall some elementary notions about Numerov-type methods. The explicit Numerov-type methods considered here have the form with c 1 −1, c 2 0, and can be expressed in the Butcher tableau as c s a s1 a s2 · · · a ss−1 0 Since the coefficient matrix A satisfies A s−1 0, we have where e is the s-dimensional vector of units, and the vectors c, b, and matrix A are defined by the Butcher tableau of the scheme 2.

Construction of the New Methods
In this section, we construct explicit Numerov-type methods which are both phase fitted and amplification fitted. The construction is based on the following Butcher tableau of coefficients , 2791/3450, 307328/3056775, 125/636732}, the classical Numerov-type method is recovered in 26 . In the following, we derive the frequencydependent weights b i , i 1, . . . , 4 of a one-frequency method and a two-frequency method by phase fitting and amplification fitting.

A One-Frequency Numerov-Type Method
Applying the method 3.1 to the test equation we obtain the following expressions for S and P where M −19000 952935u 2 11707u 4 . It is easy to verify the above coefficients satisfy the algebraic fifth-order conditions for the Numerov-type methods with one frequency see 23 :

3.6
This one-frequency phase fitted and amplification fitted fifth-order method is denoted as PFAFI.

A Two-Frequency Numerov-Type Method
For the method 3.1 , we only need the weights b i i 1, . . . , 4 . In order to solve the oscillatory problems with two frequencies, we consider another test equation With the same idea stated above, we find the following relation Solving 3.4 and 3.8 gives

3.9
In It is also easy to verify the algebraic fifth order conditions for the Numerov-type methods with two frequencies see 23 for reference

3.10
The two-frequency fifth-order Numerov-type method with the weights in 3.9 is denoted as PFAFII.
Abstract and Applied Analysis 7

Stability and Phase Analysis
In this section, we analyze the stability and phase properties of the two new methods derived in Section 3. The theory of Lambert and Watson was reconsidered by Coleman and Ixaru 27 .

Stability and Phase Analysis of PFAFI
We apply a four-stage one-frequency Numerov-type method to the test equation 2.3 and obtain the functions P and S in 2.4 as follows

4.1
The stability and the phase properties depend on the bivariate functions S H 2 , u and P H 2 , u . In this situation, the interval of absolute stability is replaced by a two-dimensional region. then the method is said to be dissipative of order q and dissipative of order p, respectively.
For convenience of analyzing the phase-lag and the dissipation, we denote the ratio r u/H. Then the phase-lag and the dissipation of the new method PFAFI are given by

4.4
Thus the method PFAFI is dispersive of order six and dissipative of order five. is a stability boundary of the method.
In Figure 1, we plot the region of the absolute stability of the new method PFAFI derived in Section 3.1 a similar discussion can be found in 27, 28 .
It should be noted that when the exact value of the frequency of the problem is known i.e., the fitted frequency ω equals the test frequency λ , then u H and we have that S H 2 , H 2 cos H and P H 2 , H 1. Consequently, when the main frequency is known the interval of periodicity is 0, ∞ except for H 2π, 4π, . . .. So the new method PFAFI is almost P -stable.

Stability and Phase Analysis of Phase Fitted Numerov-Type Method with Two Frequencies
We apply a four-stage two-frequency Numerov-type method to the test equation 2.3 and obtain the functions P and S in 2.4 as follows

4.7
Abstract and Applied Analysis

4.10
Thus the new method PFAFII is dispersive of order six and dissipative of order five. Following the main idea of Coleman and Ixaru in 27 , the absolute stability of the two-frequency method PFAFII can be described by a three-dimensional region.
is called the region of absolute stability of a Numerov-type method with the S H 2 , u, v and P H 2 , u, v given by 4.7 , and any closed curve defined by is a stability boundary of the method.
In Figure 2, we plot the region of absolute stability for the method PFAFII, and Figure 3 shows the projection of the three-dimensional region onto the u-H plane. A similar discussion can be found in 22, 29 . It should be noted that, for the new method PFAFII and for a problem with one frequency, when the exact value of the frequency of the problem is known, that is, ω 1 ω 2 λ, then u v H and we have that S H 2 , H, H 2 cos H and P H 2 , H, H 1. Consequently, the interval of periodicity is 0, ∞ with the points H 2π, 4π, . . . deleted. That is, the new method PFAFII is also almost P -stable.

Numerical Experiments
In this section, we compare the numerical performance of the new methods with some wellknown methods from the literature. We select three periodic IVPs with two frequencies. The methods we choose for comparison are listed as follows.
i EFRKN4F: the trigonometrically fitted RKN method of order 4 with the FSAL property given in 2 .
ii VPHA: the phase fitted and amplification fitted Numerov-type method with one frequency given in 4 .
iii TFNTONE: the trigonometrically fitted Numerov-type method obtained by Fang and Wu in 3 . In our test we choose ω 1 10 for VPHA, TFNTONE, EFRKN4F and PFAFI, and ω 1 10, ω 2 1 for PFAFII. The problem is integrated on the interval 0, 1000 and the numerical results are presented in Table 1.

Problem 2.
Consider the famous Kramarz system see 30 The eigenvalues of the matrix of coefficients of the equations for y and z are −1 and −2500.
For this stiff equation, we choose ω 1 1 for VPHA, TFNTONE, EFRKN4F and PFAFI, and ω 1 50, ω 2 1 for PFAFII. The problem is integrated on the interval 0, 1000 and the numerical results are presented in Table 2.

5.5
whose analytic solution is given by y t cos t sin t − cos t − sin t cos ωt sin ωt cos ωt sin ωt .

5.8
In our test we choose ω λ for VPHA, TFNTONE, EFRKN4F and PFAFI, and ω 1 λ, ω 2 1 for PFAFII. The problem is integrated on the integral interval 0, 1000 and the numerical results are stated in Table 5.

Conclusions
In this paper, we investigate Numerov-type methods adapted to oscillatory problems with two frequencies obtained by phase fitting and amplification fitting. The numerical stability and the phase properties of the new method are analyzed. Unlike the case of one-frequency methods, the absolute stability region of a two-frequency method is a region in the threedimensional space. The results of numerical experiments on four oscillatory test problems with two frequencies, including non-stiff and stiff problems, show that the new methods are superior to some well-known high quality codes proposed in the recent scientific literature.