6/1/2014 1 Runge 4 th Order Method Industrial Engineering Majors Authors: Autar Kaw, Charlie Barker...

04/10/23http://

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Runge 4th Order Method

Industrial Engineering Majors

Authors: Autar Kaw, Charlie Barker

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

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Runge-Kutta 4th Order Method

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Runge-Kutta 4th Order Method

where

hkkkkyy ii 43211 226

1

ii yxfk ,1

hkyhxfk ii 12 2

1,

2

1

hkyhxfk ii 23 2

1,

2

1

hkyhxfk ii 34 ,

For0)0(),,( yyyxf

dx

dy

Runge Kutta 4th order method is given by


How to write Ordinary Differential Equation

Example

50,3.12 yeydx

dy x

is rewritten as

50,23.1 yyedx

dy x

In this case

yeyxf x 23.1,

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

yxfdx

dy,


ExampleThe open loop response, that is, the speed of the motor to a voltage input of 20 V, assuming a system without damping is

wdt

dw06.002.020

If the initial speed is zero , and using the Runge-Kutta 4th order method, what is the speed at t = 0.8 s? Assume a step size of h = 0.4 s.

wdt

dw31000

wwtf 31000,

hkkkkww ii 43211 226

1

00 w


SolutionStep 1: For 00,0 00 wti

10000310000,0, 001 fwtfk

40020031000200,2.04.010002

10,4.0

2

10

2

1,

2

11002

ffhkwhtfk

760803100080,2.04.04002

10,4.0

2

10

2

1,

2

12003

ffhkwhtfk

8830431000304,4.04.07600,4.00, 3004 ffhkwhtfk


Solution Cont

rad/s2.227

4.034086

10

4.0887602400210006

10

)22(6

1432101

hkkkkww

is the approximate speed of the motor at1w

s4.04.0001 httt

rad/s2.2274.0 1 ww


Solution ContStep 2: 2.227,8.0,1 11 wti

4.3182.227310002.227,4.0, 111 fwtfk

36.12788.2903100088.290,6.0

4.04.3182

12.227,4.0

2

14.0

2

1,

2

11112

f

fhkwhtfk

98.24167.2523100067.252,6.0

4.036.1272

12.227,4.0

2

14.0

2

1,

2

12113

f

fhkwhtfk

019.2899.3233100099.323,8.0

4.098.2412.227,4.04.0, 3114

f

fhkwhtfk


Solution Cont

rad/s54.299

4.01.10856

12.227

4.0019.2898.241236.12724.3186

12.227

)22(6

1432112

hkkkkww

is the approximate speed of the motor at2w

s8.04.04.012 httt

rad/s54.2998.0 2 ww


Solution Cont

The exact solution of the ordinary differential equation is given by

The solution to this nonlinear equation at t=0.8 seconds is

tetw 3

3

1000

3

1000)(

rad/s09.3038.0 w


Comparison with exact results

Figure 1. Comparison of Runge-Kutta 4th order method with exact solution

Step size,

0.80.40.20.1

0.05

147.20299.54302.96303.09303.09

155.893.5535

0.129880.0062962

0.00034702

51.4341.1724

0.0428520.0020773

0.00011449


Effect of step size

h tE %|| t

(exact)

Table 1 Values of speed of the motor at 0.8 seconds for different step sizes

09.3038.0 w

8.0w


Effects of step size on Runge-Kutta 4th Order Method

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


Comparison of Euler and Runge-Kutta Methods

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

Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/runge_kutta_4th_method.html



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6/1/2014 1 Runge 4 th Order Method Industrial Engineering Majors Authors: Autar Kaw, Charlie Barker...

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