scholarly journals Nine-stage multi-derivative Runge-Kutta method of order 12

2009 ◽  
Vol 86 (100) ◽  
pp. 75-96 ◽  
Author(s):  
Truong Nguyen-Ba ◽  
Vladan Bozic ◽  
Emmanuel Kengne ◽  
Rémi Vaillancourt
Keyword(s):  
Order Conditions ◽  
High Order ◽  
Kutta Method ◽  
True Solution ◽  
First Order ◽  
Cpu Time ◽  
Runge Kutta Method ◽  
Higher Derivatives ◽  
Runge Kutta ◽  
Taylor Methods

A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y'= f(x, y), y(x0) = y0. The method uses y' and higher derivatives y(2) to y(6) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y(3) to y(6). The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7)13M and is superior to DP(8,7)13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

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 PENGARUH PERUBAHAN NILAI PARAMETER TERHADAP NILAI ERROR PADA METODE RUNGE-KUTTA ORDE 3

2017 ◽  
Vol 3 (2) ◽  
Author(s):  
Tornados P Silaban ◽  
Faiz . Ahyaningsih
Keyword(s):  
Linear Equations ◽  
Research Process ◽  
Kutta Method ◽  
System Of Linear Equations ◽  
Runge Kutta Method ◽  
A Value ◽  
Fixed Parameter ◽  
Higher Derivatives ◽  
Runge Kutta ◽  
Parameter Values

ABSTRACTRunge-Kutta method is a numerical method used to find the solution of an equation. This method seeks to obtain a higher degree of precision, and at the same time seeking to avoid the need of higher derivatives by evaluating the function f (x, y) at the selected point in each interval step. In this paper discussed the effect of changes in the value of the parameter (h) to the value of the error in the Runge-Kutta method Order-3. The equation to be discussed is a linear ordinary differential equation of the two levels that have been changed into a system of linear equations. In the research process was not found fixed parameter values to get the minimum error value, because each parameter has a value of error varied for each equation.Keywords: Runge-Kutta, parameters, error.

scholarly journals Optimum Integration Procedure for Connectionist and Dynamic Field Equations

2021 ◽  
Vol 15 ◽  
Author(s):  
Andrés Rieznik ◽  
Rocco Di Tella ◽  
Lara Schvartzman ◽  
Andrés Babino
Keyword(s):  
Euler Method ◽  
Connectionist Model ◽  
Kutta Method ◽  
Field Equations ◽  
Single Digit ◽  
First Order ◽  
Runge Kutta Method ◽  
Dynamic Field ◽  
Runge Kutta ◽  
First Order Differential Equations

Connectionist and dynamic field models consist of a set of coupled first-order differential equations describing the evolution in time of different units. We compare three numerical methods for the integration of these equations: the Euler method, and two methods we have developed and present here: a modified version of the fourth-order Runge Kutta method, and one semi-analytical method. We apply them to solve a well-known nonlinear connectionist model of retrieval in single-digit multiplication, and show that, in many regimes, the semi-analytical and modified Runge Kutta methods outperform the Euler method, in some regimes by more than three orders of magnitude. Given the outstanding difference in execution time of the methods, and that the EM is widely used, we conclude that the researchers in the field can greatly benefit from our analysis and developed methods.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

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