RUNGE-KUTTA 2^nd ORDER METHOD

INTERACTIVE E-BOOK

Ordinary Differential Equations

MAJOR

GENERAL

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Title	An interactive e-book for illustrating Runge-Kutta 2nd Order Method of solving ordinary differential equations
Creator	Autar K Kaw
Subject and Keywords	Runge-Kutta 2^nd Order Method, Ordinary Differential Equations, Mathcad, Maple, Mathematica, Matlab, Simulations.
Description	This is an interactive E-book for illustrating Runge-Kutta 2^nd 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/runge2nd_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 2^nd 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 2^nd Order Method

Methods

Heun’s Method

Midpoint Method

Ralston’s Method

Simulation of Runge-Kutta 2nd Order Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Convergence of Runge-Kutta 2nd Order Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

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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 2^nd Order Method for Ordinary Differential Equations

Runge-Kutta 2^nd order method is a numerical technique to solve ordinary differential equation of the form

Only first order ordinary differential equations can be solved by using Runge-Kutta 2^nd 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

Euler’s method is given by

(1)

where

To understand Runge-Kutta 2^nd order method, we need to derive Euler’s method from Taylor series.

(2)

As you can see the first two terms of the Taylor series

are the Euler’s method and hence can be considered to be Runge-Kutta 1^st order method.

The true error in the approximation is given by

(3)

So how would a 2^nd order method formula look like. It would include one more term of the Taylor series as follows.

(4)

Let us take a generic example of a first ordinary differential equation

Now since y is a function of x,

(5)

The 2^nd order formula for the above example would be

However, we already see the difficulty of having to find in the above method. What Runge and Kutta did was write the 2^nd order method as

(6)

where

(7)

This form allows one to take advantage of the 2^nd order method without having to calculate.

So how do we find the unknowns , , and . Without proof, equating Equation (4) and (6) , gives three equations.

Since we have 3 equations and 4 unknowns, we can assume the value of one of the unknowns. The other three will then be determined from the three equations. Generally the value of is chosen to evaluate the other three constants. The three values generally used for are, and, and are known as Heun’s Method, Midpoint method and Ralston’s method, respectively.

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Heun’s method

Here is chosen, giving

resulting in

(8)

where

(9a)

(9b)

This method is graphically explained in Figure 1.

Figure 1. Runge-Kutta 2^nd order method (Heun’s method)

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Midpoint method

Here is chosen, giving

resulting in

(10)

where

(11a)

(11b)

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Ralston’s method

Here is chosen, giving

resulting in

(12)

where

(13a)

(13b)

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Simulation of Runge-Kutta 2nd Order Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Convergence of Runge-Kutta 2nd Order Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

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