eu NEW EFFICIENT PHASE-FITTED AND AMPLIFICATION-FITTED RUNGE-KUTTA METHOD FOR OSCILLATORY PROBLEMS

A new Runge-Kutta (RK) method is constructed to solve first-order differential equations with oscillatory solutions. This new method is based on the Runge-Kutta method of order four with seven-stage. Numerical tests are performed, and the results of the new method is compared with the existing methods. The numerical results show that the new method is more efficient. AMS Subject Classification: 65L05, 65L06


Introduction
In this study, we focus on the initial value problems (IVPs) of the form: y ′ (x) = f (x, y), y(x 0 ) = y 0 (1) where the solutions of the ordinary differential equations (1) are oscillating.This type of problem can be found in many fields of applied science such as physical chemistry, astronomy, quantum mechanics, mechanics, and electronics.Some authors have suggested adapting traditional integration for the oscillatory character of the solution to problem (1).Bettis [1] constructs three-and four-stage methods that solve the equation y ′ = iwy without truncation error.
Franco [2] improves the update of Runge-Kutta Nyström methods adapted to the numerical integration of perturbed oscillators.Anastassi and Simos [3] construct phase-and amplification-fitted explicit RK methods for the numerical solution of orbital problems.Many researchers focus on the effective numerical integration of specific categories of oscillatory problems, such as Vigo-Aguiar and Simos [4] constructed exponentially and trigonometrically-fitted methods for orbital problems.This current work focus on phase-fitted and amplification-fitted RK integrators whose coefficients depend on the product of fitting frequency w and step size h.Section 2 introduces the notion of phase-fitted and amplification-fitted RK-type methods and derivation of the conditions.In Section 3, new method of order four is constructed.We compare the results with existing methods in Section 4.

Phase-Fitted and Amplification-Fitted Conditions
An s-stage Runge-Kutta method: where c i , a ij , i, j = 1, ..., s are real numbers, h is step size, and b i (z), i = 1, ..., s are even functions of z = wh.Scheme 2 can be represented by the Butcher tableau below: Hairer et al [5], explained the order conditions for the RK method (2) which can be derived by just considering the autonomous equation y ′ = f (y).
Van der Houwen and Sommeijer [6] and Paternoster [7] suggested apart from the algebraic order, the analysis of phase-lag and dissipation is important.We consider the following linear scalar equation: The exact solution of this equation with the initial value y(x 0 ) = y 0 satisfies where R(z) = exp(z).This means that after a period of time h, the exact solution experiences a phase advance z = hw and the amplification remains constant.
When applying the RK method (2) to (3) yield where e = (1, ..., 1) The numerical solution attain a phase advance arg R(z) and the amplification factor|R(z)|.R(z) is called the stability function of the method (2).Denote the real and imaginary part of R(z) by U (z) and V (z) respectively.Then, for small h we have For small h, arg Van der Houwen and Sommeijer [6] stated, the quantities which are called the phase-lag (or dispersion) and the error of amplification factor (or dissipation) of the method, respectively.If then the method is called dispersive of order q and dissipative of order p, respectively.If the method is called phase-fitted (or zero-dispersive) and amplificationfitted (or zero-dissipative), respectively.It is interesting to consider the phase properties of the update of the scheme (2).Suppose that the internal stages have been exact for the linear equation (3), that is, Y i = exp(c i z) y 0 , then the update gives Denote the real and imaginary part of R(z) by U (z) and V (z), respectively.Then, for small h.
Theorem 1.The method (2) is phase-fitted and amplification-fitted if and only if

Construction of New Method
In this section, we consider a seven-stage RK method as given in Butcher [2] with the following Butcher tableau: Table 1 Butcher Tableau for 7-stage fourth order RK method For this method, the phase-fitted and amplification-fitted conditions ( 13) become From equations ( 14), ( 15) and following sufficient conditions in [8], we obtain: Solving ( 14), ( 15) and ( 16), we obtain: As z → 0, we obtain the following Taylor expansions: The result is compared, then we analysed which problem will give the small error.This new method is denoted as PHAFRK4D.
Using equation ( 6   we next obtained the stability region of the new method from the above three stability polynomials by equating each to the Euler formula and then solve for h using maple package. i.e R( ĥ) = e Iθ = cos(θ) + I sin(θ).
The stability region for the new method is shown in Figure 1.

Error Analysis
In this section, we will compute the local truncation error analysis (LTE) of the new method is based on the Taylor series expansion of the differences y n+1 and y( ) From equation (23), it is clear that the order of the new method is four because all the terms of h lower than h 5 are vanished.

Numerical Results
In this section, we will apply the new method to solve differential equations (1).The following explicit RK methods are selected for the numerical comparison.
• PHAFRK4D: the seven-stage fourth-order phase-fitted and amplificationfitted RK method given in Section 3 of this paper.

Discussion and Conclusion
In this study, we have presented a new phase-fitted and amplification-fitted (PHAFRK4D) method that can be used to solve first-order ordinary differential equations with the solutions are oscillating.The numerical results are plotted in Figures 2, 3 ), we obtained the stability polynomial are three different stages of the solutions.First, we take the value of b 1 , b 2 , b 3 , b 4 , b 5 , b 6 and b 7 up to h 6 from their series solution. b

Figure 2 :
Figure 2: Efficiency curves of all the methods for problem 1 with h=0.00625, 0.0125, 0.025 and 0.05 for b=10000

Figure 3 :Figure 4 :Figure 5 :Figure 6 :
Figure 3: Efficiency curves of all the methods for problem 2 with h=0.00625, 0.0125, 0.025 and 0.05 for b=10000 Secondly, we take the values b 1 , b 2 , b 3 , b 4 , b 5 , b 6 and b 7 up to h 8 from their series solution.Lastly, we take the values b 1 , b 2 , b 3 , b 4 , b 5 , b 6 and b 7 up to h 10 from their series solution,