RUNGE-KUTTA 4^th ORDER METHOD

INTERACTIVE E-BOOK

Ordinary Differential Equations

MAJOR

GENERAL

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Title	An interactive e-book for illustrating Runge-Kutta 4^th Order Method of solving ordinary differential equations
Creator	Autar K Kaw
Subject and Keywords	Runge-Kutta 4^th Order Method, Ordinary Differential Equations, Mathcad, Maple, Mathematica, Matlab, Simulations.
Description	This is an interactive E-book for illustrating Runge-Kutta 4^th Order Method for solving ordinary differential equations. It includes links to examples, simulations in Mathcad, Maple, Mathematica, and Matlab for the algorithm, convergence, and a PowerPoint presentation.
Publisher	Holistic Numerical Methods Institute, College of Engineering, University of South Florida, Tampa, FL 33620-5350.
Contributors	Autar Kaw
Format	Text/HTML
Last Revised	May 4, 2006
Identifier	http://numericalmethods.eng.usf.edu/ebooks/runge4th_08ode_ebook.pdf
Language	English
Rights	http://numericalmethods.eng.usf.edu/rights.htm

Table of Contents

Background

What are ordinary differential equations?

A Primer on Ordinary Differential Equations

Runge-Kutta 4^th Order Method for Ordinary Differential Equations

PowerPoint presentation

Examples

Example 1: First order differential equation rewritten in dy/dx=f(x,y) form

Example 2: First order differential equation rewritten in dy/dx=f(x,y) form

Example 3: Heat Transfer Problem Solved Using Runge-Kutta 4^th Order Method

Simulations

Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Convergence [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Multiple Choice Test

Test your knowledge of Runge-Kutta 4th order method [HTML] [PDF] [DOC]

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Background

This is an example of how an instructor of a Numerical Methods course can develop an E-book on a single topic by using resources that are only available at the Holistic Numerical Methods Institute (HNMI) Website.

Although one of the primary advantages of a web-based resource is that one can use documents outside of its own domain, we are particularly using this example as a way to illustrate the holistic nature of the resources available at HNMI. You are welcome to modify and edit this e-book to suit your purpose.

Using the textbook document written in MS Word 2000 as a foundation, it was made interactive within MS Word 2000 by putting bookmarks and hyperlinks to background information, PowerPoint presentations, Mathcad simulations, historical anecdotes, multiple choice tests, problem sets, etc. Then the MS Word 2000 file was saved as a webpage and that is what you are about to read here.

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Runge-Kutta 4^th Order Method for Ordinary Differential Equations

Runge-Kutta 4th order method is a numerical technique to solve ordinary differential equation of the form

So only first order ordinary differential equations can be solved by using Runge-Kutta 4^th order method. In other sections, we have discussed how Euler and Runge-Kutta methods are used to solve higher order ordinary differential equations or coupled (simultaneous) differential equations.

How does one write a first order differential equation in the above form?

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Example 1

is rewritten as

In this case

Another example is given as follows

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Example 2

is rewritten as

In this case

Runge-Kutta 4^th order method is based on the following

(1)

where knowing the value of at , we can find the value of y=y_i+1 at , and

Equation (1) is equated to the first five terms of Taylor series

(2)

Knowing that and

(3)

Based on equating Equation (2) and Equation (3), one of the popular solutions used is

(4)

(5a)

(5b)

(5c)

(5d)

Simulation of Runge-Kutta 4th Order Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Convergence of Runge-Kutta 4th Order Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

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Example 3

A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by

Find the temperature at seconds using Runge-Kutta 4^th order method. Assume a step size of seconds.

Solution

For , ,

is the approximate temperature at

For

is the approximate temperature at

Figure 1 compares the exact solution with the numerical solution using Runge-Kutta 4^th order method step size of h=240.

Figure 1. Comparison of Runge-Kutta 4^th order method with exact solution for different step sizes

Table 1 and Figure 2 shows the effect of step size on the value of the calculated temperature at t=480 seconds.

Table 1. Value of temperature at time, t=480s for different step sizes

Step size,

480

240

120

-90.278

594.91

646.16

647.54

647.57

737.85

52.660

1.4122

0.033626

0.00086900

113.94

8.1319

0.21807

0.0051926

0.00013419

Figure 2. Effect of step size in Runge-Kutta 4^th order method

In Figure 3, we are comparing the exact results with Euler’s method (Runge-Kutta 1^st order method), Heun’s method (Runge-Kutta 2^nd order method) and Runge-Kutta 4^th order method.

Figure 3. Comparison of Runge-Kutta methods of 1^st, 2^nd, and 4^th order.

The formula described in this chapter was developed by Runge. This formula is same as Simpson’s rd rule, if is only a function of. There are other versions of the fourth order method just like there are several versions of the second order methods. Formula developed by Kutta is