Two new RungeKuttaNyström (RKN) methods are constructed for solving secondorder differential equations with oscillatory solutions. These two new methods are constructed based on two existing RKN methods. Firstly, a threestage fourthorder Garcia’s RKN method. Another method is Hairer’s RKN method of fourstage fifthorder. Both new derived methods have two variable coefficients with phaselag 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.
Throughout this paper, we are dealing with the initial value problems (IVP) related to secondorder ODEs in the form:
Over the last three decades, there are a number of numerical methods that have been derived by several authors based on different approaches such as minimal phaselag, phasefitted, and exponentialfitted for solving secondorder oscillatory IVPs. See van der Houwen and Sommeijer [
In this paper, we will combine the idea of phaselag of order infinity and zero amplification error together. First, we will construct a threestage phasefitted and amplificationfitted RKN method which is based on Garcia's method of algebraic order four [
The general
In the investigation of phaselag error of the method, we will use the test equation in the following:
It is given that the exact solution of (
Apply RKN method (
Apply RKN method (
Let us denote that
To obtain phaselag of order infinity the relation
When at a point
To achieve phaselag of order infinity and zero amplification error, the relations below must hold
In this section, we will present the construction of two new RKN methods. The first method is based on a threestage RKN method with algebraic order four (see [
In this section, we want to derive an RKN method with phaselag of order infinity and zero amplification error which is based on Garcia's RKN method of threestage and algebraic order four as follows
First of all, we have to compute the polynomials
For small value of
In this section, we want to derive a new RKN method which is based on Hairer's RKN method of fourstage and algebraic order five as follows:
By using the same strategy, we can obtain a new fifthorder RKN method with phaselag of order infinity and zero amplification error. In this case, we set
Thus, we have obtained a new method which has two variable coefficients,
In this section, we will compute the local truncation error (LTE) of the new methods and verify the algebraic order of the methods.
Firstly, we compute the Taylor expansions of the exact solution
The LTE and
Summary of the properties of the methods.
Method 




DPC  DSC  S.I/P.I 

SRKN3V 





—  (0, 7.57) 
PFERKN4P 


—  —  —  —  — 
OPTRKN4P 



—  —  —  — 
RKN4G 






(0, 8.77) 
PFAFRKN4 



—  —  —  (0, 8.94) 
 
OPTRKN5K1 


— 

—  —  — 
OPTRKN5K2 


— 

—  —  — 
RKN5H 






— 
PFAFRKN5 


—  —  —  —  (0, 8.01) 
Note:
DPC is dispersion constant.
DSC is dissipation constant.
P.I is periodicity interval.
S.I is stability interval.
In this section, we will apply the new methods to some secondorder differential equation problems. The following explicit RKN methods are selected for the numerical comparisons.
Fourthorder:
PFAFRKN4: the new derived fourthorder RKN method;
RKN4G: the threestage fourthorder RKN method derived by García et al. [
SRKN3V: the thirdorder symplectic RKN method with minimal phaselag derived by van de Vyver [
PFERKN4P: the fourthorder RKN method with phaselag of order infinity derived by Papadopoulos et al. [
OPTRKN4P: the fourthorder optimized RKN method derived by Papadopoulos and Simos [
Fifthorder:
PFAFRKN5: the new derived fifthorder RKN method;
RKN5H: the fourstage fifthorder RKN method derived by Hairer et al. [
OPTRKN5K1: the fifthorder optimized RKN method with zero first derivative of phaselag derived by Kosti et al. [
OPTRKN5K2: the fifthorder optimized RKN method with zero first derivative of amplification factor derived by Kosti et al. [
Those methods are categorized into two categories according to algebraic order of each method for comparison purposes. The accuracy criteria taken is calculating the
Exact solution:
Estimated frequency:
Source: Kosti et al. [
Exact solution:
Estimated frequency:
This case is using
Exact solution:
Estimated frequency:
Source: van der Houwen and Sommeijer [
Exact solution:
Estimated frequency:
Source: Allen, Jr. and Wing [
Exact solution:
Estimated frequency:
Source: Lambert and Watson [
The numerical results are plotted in Figures
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods for Problem
Efficiency curves for all methods of order five for Problem
Efficiency curves for all methods of order five for Problem
Efficiency curves for all methods of order five for Problem
Efficiency curves for all methods of order five for Problem
From Figures
In this paper, we have derived two new phasefitted and amplificationfitted RKN methods for solving secondorder IVPs which are oscillatory in nature. First method is based on Garcia's fourth algebraic order RKN method and the second method is based on Hairer's fourstage fifth algebraic order RKN method. Numerical results show that both methods are more accurate and efficient for solving secondorder differential equations with oscillating solutions.