scholarly journals A Third Runge Kutta Method Based on a Linear Combination of Arithmetic Mean, Harmonic Mean and Geometric Mean

2014 ◽  
Vol 3 (5) ◽  
pp. 231 ◽  
Author(s):  
Rini Yanti
Keyword(s):  
Linear Combination ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

A modified Runge-Kutta method

SIMULATION ◽  
1968 ◽  
Vol 10 (5) ◽  
pp. 221-223 ◽  
Author(s):  
A.S. Chai
Keyword(s):  
Error Estimate ◽  
Linear Combination ◽  
Experimental Results ◽  
The Other ◽  
New Method ◽  
Kutta Method ◽  
Starting Values ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
A Minor

It is possible to replace k2 in a 4th-order Runge-Kutta for mula (also Nth-order 3 ≤ N ≤ 5) by a linear combination of k1 and the ki's in the last step, using the same procedure for computing the other ki's and y as in the standard R-K method. The advantages of the new method are: It re quires one less derivative evaluation, provides an error estimate at each step, gives more accurate results, and needs a minor change to switch to the RK to obtain the starting values. Experimental results are shown in verification of the for mula.

A New Efficient Ode Solver For Solving Initial Value Problem

1998 ◽  
pp. 47-56
Author(s):  
Nazeeruddin Yaacob ◽  
Bahrom Sanugi
Keyword(s):  
Stability Analysis ◽  
Explicit Formula ◽  
Fourth Order ◽  
Initial Value Problem ◽  
Harmonic Mean ◽  
Kutta Method ◽  
Initial Value ◽  
Order Accuracy ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper we develop a new three-stage,fourth order explicit formula of Runge-Kutta type based on Arithmetic and Harmonic means.The error and stability analyses of this method indicate that the method is stable and efficient for nonstiff problems.Two examples are given which illustrate the fcurth order accuracy of the method. Keywords: Runge-Kutta method, Harmonic Mean, three-stage, fourth-order, covergence and stability analysis.

scholarly journals Modification of Fourth order Runge-Kutta Method for Kutta Form With Geometric Means

2020 ◽  
Vol 4 (2) ◽  
pp. 221-230
Author(s):  
Irma Suryani ◽  
Wartono Wartono ◽  
Yuslenita Muda
Keyword(s):  
Series Expansion ◽  
Fourth Order ◽  
Taylor Series ◽  
Error Term ◽  
Geometric Mean ◽  
Taylor Series Expansion ◽  
Kutta Method ◽  
Geometric Means ◽  
Runge Kutta Method ◽  
Runge Kutta

This paper  discuss how to modified Fourth order Runge-Kutta Kutta method based on the geometric mean. Then we have parameters  and   however by re-comparing the Taylor series expansion of   and  up to the 4th order.  For make error term re-compering of  the Taylor series expansion of  and  up to the 5th order. In the error term an make substitution for the values of  and  into the Taylor seriese expansion up to the 5th order. So that we have error term modified Fourth Order Runge-Kutta Kutta based on the geometric mean.  Modified Fourth Order Runge-Kutta Kutta based on the geometric mean that usually used to solved ordinary differential equations.

scholarly journals Analysis of linear and nonlinear stiff problems using the RK-Butcher algorithm

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
S. Sekar
Keyword(s):  
Exact Solutions ◽  
Numerical Solution ◽  
Arithmetic Mean ◽  
Stiff Problems ◽  
Graphical Form ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Linear And Nonlinear ◽  
Runge Kutta

I present a numerical solution of linear and nonlinear stiff problems using the RK-Butcher algorithm. The obtained discrete solutions using the RK-Butcher algorithm are found to be very accurate and are compared with the exact solutions of the linear and nonlinear stiff problems and also with the Runge-Kutta method based on arithmetic mean (RKAM). A topic of stability for the RK-Butcher algorithm is discussed in detail. Error graphs for discrete and exact solutions are presented in a graphical form to show the efficiency of the RK-Butcher algorithm. The results obtained show that RK-Butcher algorithm is more useful for solving linear and nonlinear stiff problems and the solution can be obtained for any length of time.

Sign in / Sign up

Export Citation Format

Share Document