scholarly journals Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

Computation ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 32
Author(s):  
Musa Demba ◽  
Poom Kumam ◽  
Wiboonsak Watthayu ◽  
Pawicha Phairatchatniyom
Keyword(s):  
Error Estimation ◽  
Variable Step Size ◽  
Local Error ◽  
Step Size ◽  
Local Error Estimation ◽  
Nyström Methods ◽  
Periodic Problems ◽  
Variable Step ◽  
Exponentially Fitted ◽  
Runge Kutta

In this work, a pair of embedded explicit exponentially-fitted Runge–Kutta–Nyström methods is formulated for solving special second-order ordinary differential equations (ODEs) with periodic solutions. A variable step-size technique is used for the derivation of the 5(3) embedded pair, which provides a cheap local error estimation. The numerical results obtained signify that the new adapted method is more efficient and accurate compared with the existing methods.

scholarly journals Variable Step Exponentially Fitted Explicit Sixth-Order Hybrid Method with Four Stages for Spring-Mass and Other Oscillatory Problems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 387
Author(s):  
Faieza Samat ◽  
Eddie Shahril Ismail
Keyword(s):  
Hybrid Method ◽  
Hybrid Methods ◽  
Variable Coefficients ◽  
Sixth Order ◽  
Variable Step Size ◽  
Local Error ◽  
Step Size ◽  
Oscillatory Solutions ◽  
Variable Step ◽  
Exponentially Fitted

For the numerical integration of differential equations with oscillatory solutions an exponentially fitted explicit sixth-order hybrid method with four stages is presented. This method is implemented using variable step-size while its derivation is accomplished by imposing each stage of the formula to integrate exactly { 1 , t , t 2 , … , t k , exp ( ± μ t ) } where the frequency μ is imaginary. The local error that is employed in the step-size selection procedure is approximated using an exponentially fitted explicit fourth-order hybrid method. Numerical comparisons of the new and existing hybrid methods for the spring-mass and other oscillatory problems are tabulated and discussed. The results show that the variable step exponentially fitted explicit sixth-order hybrid method outperforms the existing hybrid methods with variable coefficients for solving several problems with oscillatory solutions.

Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods

1992 ◽  
Vol 32 (1) ◽  
pp. 104-117 ◽  
Author(s):  
A. Bellen ◽  
Z. Jackiewicz ◽  
M. Zennaro
Keyword(s):  
Error Estimation ◽  
Local Error ◽  
Local Error Estimation ◽  
Runge Kutta

A Comparative Study of Digital Integration Methods

SIMULATION ◽  
1969 ◽  
Vol 12 (2) ◽  
pp. 87-94 ◽  
Author(s):  
Hinrich R. Martens
Keyword(s):  
Linear Systems ◽  
State Transition ◽  
General Purpose ◽  
The State ◽  
Variable Step Size ◽  
Step Size ◽  
Synchronous Interaction ◽  
Variable Step ◽  
Runge Kutta ◽  
Speed Accuracy

This paper is a brief report on an investigation into the performance of integration routines used in the simula tion of dynamic systems by general-purpose digital simu lation programs. The performance is compared with respect to overall speed, accuracy, and convenience. Several routines are considered, including rectangular, Runge-Kutta, Runge-Kutta-Blum, Runge-Kutta-Mer son, and Adams-Moulton methods. The last two may use either fixed- or variable-step-size methods. The Tus tin method and the State-Transition method are also included. By examining the performance of the routines through tests on representative systems, it is shown that for the simulation of linear systems the state transition method works best. For the general simulation of linear and non linear systems the variable-step-size Runge-Kutta-Mer son method proves to be most accurate and most efficient. The variable-step-size method may be designed to gen erate outputs at uniform intervals, thus greatly enhanc ing its value when synchronous interaction with other parts of a simulation is required.

scholarly journals A Variable Step-Size Exponentially Fitted Explicit Hybrid Method for Solving Oscillatory Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
F. Samat ◽  
F. Ismail ◽  
M. B. Suleiman
Keyword(s):  
Hybrid Method ◽  
Numerical Experiments ◽  
Variable Step Size ◽  
Step Size ◽  
Linear Combinations ◽  
Variable Step ◽  
Three Stages ◽  
Last Stage ◽  
Exponentially Fitted ◽  
The Stability

An exponentially fitted explicit hybrid method for solving oscillatory problems is obtained. This method has four stages. The first three stages of the method integrate exactly differential systems whose solutions can be expressed as linear combinations of{1,x,exp(μx),exp(−μx)},μ∈C, while the last stage of this method integrates exactly systems whose solutions are linear combinations of{1,x,x2,x3,x4,exp(μx),exp(−μx)}. This method is implemented in variable step-size code basing on an embedding approach. The stability analysis is given. Numerical experiments that have been carried out show the efficiency of our method.

scholarly journals IMPLEMENTATION OF EXPLICIT NORDSIECK METHODS WITH INHERENT QUADRATIC STABILITY

2013 ◽  
Vol 18 (2) ◽  
pp. 289-307 ◽  
Author(s):  
Michel Bras ◽  
Angelamaria Cardone ◽  
Raffaele D'Ambrosio
Keyword(s):  
Error Estimation ◽  
Numerical Experiments ◽  
General Linear ◽  
Local Error ◽  
Local Error Estimation ◽  
Stability Regions ◽  
Quadratic Stability ◽  
Variable Step ◽  
Nordsieck Methods ◽  
Step Algorithm

The present paper deals with the implementation in a variable-step algorithm of general linear methods in Nordsieck form with inherent quadratic stability and large stability regions constructed recently by Braś and Cardone. Various implementation issues such as rescale strategy, local error estimation, step-changing strategy and starting procedure are discussed. Some numerical experiments are reported, which show the performances of the methods and make comparisons with other existing methods.

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