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A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

Consider the numerical solution of the IVPs for first-order Ordinary Differential Equations (ODEs) in the form of

Several well-known authors in their papers have developed phase-fitted and amplification-fitted RK methods. Simos and Vigo-Aguiar [

With the evolution of RK methods, Papadopoulos et al. [

In the last few years, Senu et al. [

Chan and Tsai [

Awoyemi and Idowu [

A seventh-order three-step hybrid linear multistep method (HLMM) with three nonstep points is proposed by Jator [

Up until now, there are no research findings related to phase-fitting in TDRK methods. Researchers have not yet explored the advantages or disadvantages of applying phase-fitted techniques to TDRK methods. Hence, in this paper, a new sixth-order four-stage phase-fitted and amplification-fitted TDRK methods is constructed. In Section

Consider the scalar ODEs (

An explicit TDRK method for the numerical integration of IVPs (

The explicit TDRK methods are presented with the coefficients in (

Explicit methods with minimal number of function evaluations can be developed by considering the methods in the form

The above method is called special explicit TDRK methods. The unique part of this method is that it involves only one evaluation of

The TDRK parameters

The order conditions for special explicit TDRK methods are given in Table

Order conditions for special explicit TDRK methods.

Order | Conditions | ||||
---|---|---|---|---|---|

1 | | ||||

2 | | ||||

3 | | ||||

4 | | ||||

5 | | | |||

6 | | | | ||

7 | | | | | |

Consider the following linear scalar equation

Applying the test equation (

The quantities

The method is phase-fitted and amplification-fitted if and only if

When (

Denote

A TDRK method is phase-fitted and amplification-fitted if and only if Theorem

For this study, two-derivative sixth-algebraic-order method presented by Chan and Tsai [

Considering the stability function (

Substitute matrices (

Solving (

As

The following expansions are obtained by direct calculation:

Hence, the coefficients given in (

Hence, it is a sixth-order method. The original method is obtained by Chan and Tsai [

Since we have verified that this new method is order six, hence it is called PFAFTDRK4(

PFAFTDRK4(

In this section, the linear stability of the method developed is analyzed. Consider the test equation (

A TDRK method is said to be absolutely stable if

The stability polynomial of the PFAFTDRK4(

The comparison of the stability region of the PFAFTDRK4(

Stability region of PFAFTDRK4(

The stability interval of this method with the coefficients of

From the stability interval, we can actually find the biggest value of

A method is stable if the maximum global error is small and converges to its exact solution. Otherwise, the method is unstable if it has a bigger maximum global error which means it is actually diverging from its exact solution. We will show the relationship between

Stability test for PFAFTDRK4(

| | Global error |
---|---|---|

4.55 | 2.3772841250 | |

4.35 | 1.5452837990 | |

4.15 | 0.9838244901 | |

1.00 | 0.3678739259 | |

0.15 | 0.8607079765 | |

0.01 | 0.9900498337 | |

In this section, the performance of the proposed method PFAFTDRK4(

Exact solution is

The following notations are used in Figures

Energy conservation. The logarithm error of energy (MAXERR) at each integration point when solving the harmonic oscillator (Problem

The error at each integration point when solving the harmonic oscillator (Problem

The global error at each integration point when solving the inhomogeneous problem (Problem

The global error at each integration point when solving the “almost” periodic problem (Problem

The global error at each integration point when solving the two-body problem (Problem

The global error at each integration point when solving the Duffing problem (Problem

The global error at each integration point when solving the Prothero-Robinson problem (Problem

The efficiency curve for the harmonic oscillator (Problem

The efficiency curve for the inhomogeneous problem (Problem

The efficiency curve for the “almost” periodic problem (Problem

The efficiency curve for the two-body problem (Problem

The efficiency curve for the Duffing problem (Problem

The efficiency curve for the Prothero-Robinson problem (Problem

We represent the performance of these numerical results graphically in Figures

The results show the typical properties of the new phase-fitted and amplification-fitted TDRK method, PFAFTDRK4(

Next, the global error and the efficiency of the method are plotted over a long period of integration. Figures

In this research, a new phase-fitted and amplification-fitted higher order TDRK method is developed. Based on the numerical results obtained, it can be concluded that the new PFAFTDRK4(

The authors declare that there is no conflict of interests regarding the publication of this paper.