scholarly journals Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations y′′(x)=f(x,y,y′)

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
T. S. Mohamed ◽  
N. Senu ◽  
Z. B. Ibrahim ◽  
N. M. A. Nik Long
Keyword(s):  
Special Class ◽  
Numerical Experiments ◽  
Stability Property ◽  
Second Derivative ◽  
New Methods ◽  
Nyström Methods ◽  
Three Stages ◽  
Runge Kutta ◽  
The Stability ◽  
Third Derivative

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y′′(x)=f(x,y,y′) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.

scholarly journals A Phase-Fitted and Amplification-Fitted Explicit Runge–Kutta–Nyström Pair for Oscillating Systems

2021 ◽  
Vol 26 (3) ◽  
pp. 59
Author(s):  
Musa Ahmed Demba ◽  
Higinio Ramos ◽  
Poom Kumam ◽  
Wiboonsak Watthayu
Keyword(s):  
Stability Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Truncation Error ◽  
Local Truncation Error ◽  
Nyström Methods ◽  
Oscillating Systems ◽  
The Common ◽  
Runge Kutta ◽  
The Stability

An optimized embedded 5(3) pair of explicit Runge–Kutta–Nyström methods with four stages using phase-fitted and amplification-fitted techniques is developed in this paper. The new adapted pair can exactly integrate (except round-off errors) the common test: y″=−w2y. The local truncation error of the new method is derived, and we show that the order of convergence is maintained. The stability analysis is addressed, and we demonstrate that the developed method is absolutely stable, and thus appropriate for solving stiff problems. The numerical experiments show a better performance of the new embedded pair in comparison with other existing RKN pairs of similar characteristics.

scholarly journals On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 924 ◽  
Author(s):  
Khai Chien Lee ◽  
Norazak Senu ◽  
Ali Ahmadian ◽  
Siti Nur Iqmal Ibrahim
Keyword(s):  
Thin Film Flow ◽  
Third Order ◽  
New Methods ◽  
Algebraic Order ◽  
Fourth Derivative ◽  
Three Stages ◽  
Runge Kutta ◽  
Two Stages ◽  
Order Four ◽  
Third Derivative

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals On order 5 symplectic explicit Runge-Kutta Nyström methods

2000 ◽  
Vol 4 (2) ◽  
pp. 143-150 ◽  
Author(s):  
Lin-Yi Chou ◽  
P. W. Sharp
Keyword(s):  
Extra Cost ◽  
Optimal Method ◽  
New Methods ◽  
Nyström Methods ◽  
Stage Order ◽  
Principal Error ◽  
Numerical Performance ◽  
Free Parameters ◽  
Runge Kutta ◽  
Error Coefficients

Order five symplectic explicit Runge-Kutta Nyström methods of five stages are known to exist. However, these methods do not have free parameters with which to minimise the principal error coefficients. By adding one derivative evaluation per step, to give either a six-stage non-FSAL family or a seven-stage FSAL family of methods, two free parameters become available for the minimisation. This raises the possibility of improving the efficiency of order five methods despite the extra cost of taking a step.We perform a minimisation of the two families to obtain an optimal method and then compare its numerical performance with published methods of orders four to seven. These comparisons along with those based on the principal error coefficients show the new method is significantly more efficient than the five-stage, order five methods. The numerical comparisons also suggest the new methods can be more efficient than published methods of other orders.

scholarly journals Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
N. A. Ahmad ◽  
N. Senu ◽  
F. Ismail
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
First Order ◽  
Runge Kutta Method ◽  
Algebraic Order ◽  
Runge Kutta ◽  
The Stability ◽  
Order Four

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.

scholarly journals New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
K. W. Moo ◽  
N. Senu ◽  
F. Ismail ◽  
M. Suleiman
Keyword(s):  
Fourth Order ◽  
Variable Coefficients ◽  
Phase Lag ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Nyström Methods ◽  
Runge Kutta ◽  
Fifth Order ◽  
New Phase

Two new Runge-Kutta-Nyström (RKN) methods are constructed for solving second-order differential equations with oscillatory solutions. These two new methods are constructed based on two existing RKN methods. Firstly, a three-stage fourth-order Garcia’s RKN method. Another method is Hairer’s RKN method of four-stage fifth-order. Both new derived methods have two variable coefficients with phase-lag of order infinity and zero amplification error (zero dissipative). Numerical tests are performed and the results show that the new methods are more accurate than the other methods in the literature.

scholarly journals Sin-Cos-Taylor-Like method for solving stiff ordinary diffrential equations

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Rokiah @ Rozita Ahmad ◽  
Nazeeruddin Yaacob
Keyword(s):  
Fourth Order ◽  
Stability Property ◽  
Stiff Problems ◽  
Explicit Method ◽  
Implicit Methods ◽  
Stiff Ordinary Differential Equations ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability ◽  
The Cost

This paper discusses the derivation of an explicit Sin-Cos-Taylor-Like method for solving stiff ordinary differential equations, which is a formulation of the combination of a polynomial and the exponential function. This new method requires extra work to evaluate a number of differentiations of the function involved. However, the result shows smaller errors when compared to the results from the explicit classical fourth-order Runge-Kutta (RK4) and the Adam-Bashforth-Moulton (ABM) methods. Implicit methods could work well for stiff problems but have certain drawbacks especially when discussing about the cost. Although extra work is required, this explicit method has its own advantages. Besides providing excellent results, the cost of computation using this explicit method is much cheaper than the implicit methods. We also considered the stability property for this method since the stability property of the classical explicit fourth order Runge-Kutta method is not adequate for the solution of stiff problems. As a result, we find that this explicit method is of order-6, which has been developed, and proved to be both A-stable and L-stable.

scholarly journals A New Family of Phase-Fitted and Amplification-Fitted Runge-Kutta Type Methods for Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Zhaoxia Chen ◽  
Xiong You ◽  
Xin Shu ◽  
Mei Zhang
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
Order Conditions ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
New Methods ◽  
New Family ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Phase Fitting

In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta (RK) methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase-fitted and amplification-fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.

A Class of Methods with Optimal Stability Properties for the Numerical Solution of IVPs: Construction and Implementation

2017 ◽  
Vol 14 (01) ◽  
pp. 1750007
Author(s):  
Masoumeh Hosseini Nasab ◽  
Gholamreza Hojjati ◽  
Ali Abdi
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Point Of View ◽  
Variable Step Size ◽  
Second Derivative ◽  
Step Size ◽  
Stability Properties ◽  
Variable Step ◽  
The Stability ◽  
Size Mode

Considering the methods with future points technique from second derivative general linear methods (SGLMs) point of view, makes it possible to improve their stability properties. In this paper, we extend the stability regions of a modified version of E2BD formulas to optimal one and show its effectiveness by numerical verifications. Also, implementation issues, with numerical experiments, of these methods are investigated in a variable step-size mode.

scholarly journals A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-17 ◽  
Author(s):  
F. F. Ngwane ◽  
S. N. Jator
Keyword(s):  
Second Derivative ◽  
New Methods ◽  
Block Form ◽  
Trigonometric Basis Functions ◽  
Derivative Formulas ◽  
Numerical Integrators ◽  
Oscillatory Initial Value Problems ◽  
The Stability ◽  
Predictor Corrector ◽  
Second Derivative Methods

A family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed. The methods which can be applied in predictor-corrector form are implemented in block form as simultaneous numerical integrators over nonoverlapping intervals. Numerical results obtained using the proposed block form reveal that the new methods are efficient and highly competitive with existing methods in the literature.

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