Transcript Document 7647617

Runge 2nd Order Method
Industrial Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
5/25/2016
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Runge-Kutta 2nd Order Method
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Runge-Kutta 2nd Order Method
For
dy
 f ( x, y ), y (0)  y0
dx
Runge Kutta 2nd order method is given by
yi 1  yi  a1k1  a2 k2 h
where
k1  f xi , yi 
k2  f xi  p1h, yi  q11k1h 
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Heun’s Method
Heun’s method
Slope  f xi  h, yi  k1h 
y
Here a2=1/2 is chosen
1
a1 
2
p1  1
Slope  f xi , yi 
q11  1
resulting in
1 
1
yi 1  yi   k1  k2 h
2 
2
where
k1  f xi , yi 
yi+1, predicted
Average Slope 
yi
xi
1
 f xi  h, yi  k1h   f xi , yi 
2
xi+1
x
Figure 1 Runge-Kutta 2nd order method (Heun’s method)
k 2  f xi  h, yi  k1h 
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Midpoint Method
Here a2  1 is chosen, giving
a1  0
p1 
1
2
q11 
1
2
resulting in
yi 1  yi  k2h
where
k1  f xi , yi 
1
1


k 2  f  xi  h, yi  k1h 
2
2


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Ralston’s Method
Here a 2  2 is chosen, giving
3
1
3
3
p1 
4
3
q11 
4
resulting in
a1 
2 
1
yi 1  yi   k1  k 2 h
3 
3
where
k1  f xi , yi 
3
3


k 2  f  xi  h, yi  k1h 
4
4


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How to write Ordinary Differential
Equation
How does one write a first order differential equation in the form of
dy
 f  x, y 
dx
Example
dy
 2 y  1.3e  x , y 0   5
dx
is rewritten as
dy
 1.3e  x  2 y, y 0   5
dx
In this case
f x, y   1.3e  x  2 y
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Example
The open loop response, that is, the speed of the motor to a
voltage input of 20 V, assuming a system without damping is
20  0.02 
dw
 0.06w
dt
If the initial speed is zero; use the Runge-Kutta 2nd order
method and a step size of h  0.4 s to find the speed at t  0.8 s.
dw
 1000  3w
dt
f t , w  1000  3w
1 
1
wi 1  wi   k1  k 2 h
2 
2
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Solution
Step 1: i  0, t0  0, w0  0
k1  f t0 , wo   f 0, 0  1000  30  1000
k2  f t0  h, w0  k1h  f 0  0.4, 0  10000.4  f 0.4, 400  1000  3400  200
1 
1
w1  w0   k1  k 2 h
2 
2
1
1

 0   1000    200 0.4 
2
2

 0  500  100 0.4
 160 rad/s
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Solution Cont
Step 2:
i  1, t1  t0  h  0  0.4  0.4, w1  160
k1  f t1 , w1   f 0.4, 160  1000  3160  520
k 2 f t1  h, w1  k1h   f 0.4  0.4, 160  5200.4
 f 0.8, 368  1000  3368  104
1 
1
w2  w1   k1  k 2 h
2 
2
1
1

 160   520    104 0.4
2
2

 160  2080.4
 243.2 rad/s
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Solution Cont
The exact solution of the ordinary differential equation is
given by
 1000   1000  3t
wt   

e
 3   3 
The solution to this nonlinear equation at t=3 minutes is
w0.8  303.09 rad/s
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Comparison with exact results
Figure 2. Heun’s method results for different step sizes
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Effect of step size
Table 1 Effect of step size for Heun’s method
Step size, h
0.8
0.4
0.2
0.1
0.05
x3
−160.00
243.20
295.61
301.70
302.79
|t | %
Et
463.09
59.894
7.4823
1.3929
0.30613
152.79
19.761
2.4687
0.45954
0.10100
w0.8  303.09 rad/s (exact)
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Effects of step size on Heun’s
Method
Figure 3. Effect of step size in Heun’s method
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Comparison of Euler and RungeKutta 2nd Order Methods
Table 2. Comparison of Euler and the Runge-Kutta methods
Step size,
h
0.8
0.4
0.2
0.1
0.05
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w0.8
Euler
Heun
Midpoint
Ralston
800
320
324.8
314.11
308.58
−160.00
243.20
295.61
301.70
302.79
−160.00
243.20
295.61
301.70
302.79
−160.00
243.20
295.61
301.70
302.79
w0.8  303.09 rad/s
(exact)
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Comparison of Euler and RungeKutta 2nd Order Methods
Table 2 Comparison of Euler and the Runge-Kutta methods
Step size,
h
0.8
0.4
0.2
0.1
0.05
t %
Euler
Heun
Midpoint
Ralston
163.94
5.5792
7.1629
3.6359
1.8113
152.79
19.760
2.4679
0.45861
0.098981
152.79
19.760
2.4679
0.45861
0.098981
152.79
19.760
2.4679
0.45861
0.098981
w0.8  303.09 rad/s
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(exact)
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Comparison of Euler and RungeKutta 2nd Order Methods
Figure 4. Comparison of Euler and Runge Kutta 2nd order
methods with exact results.
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Additional Resources
For 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_kutt
a_2nd_method.html
THE END
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