Runge-Kutta 2nd Order Method for Solving Ordinary
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Transcript Runge-Kutta 2nd Order Method for Solving Ordinary
Runge 2nd Order Method
Mechanical Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
7/20/2015
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1
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
3
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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
yi+1, predicted
Slope f xi , yi
p1 1
q11 1
resulting in
1
1
yi 1 yi k1 k2 h
2
2
where
k1 f xi , yi
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
4
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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
5
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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 k2 h
3
3
where
k1 f xi , yi
3
3
k 2 f xi h, yi k1h
4
4
6
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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
A solid steel shaft at room temperature of 27°C is needed to be
contracted so it can be shrunk-fit into a hollow hub. It is placed in
a refrigerated chamber that is maintained at −33°C. The rate of
change of temperature of the solid shaft q is given by
6 4
5 3
3 2
3
.
69
10
θ
2
.
33
10
θ
1
.
35
10
θ
dθ
6
θ 33
5.3310
2
dt
5.4210 θ 5.588
q 0 27C
Find the temperature of the steel shaft after 24 hours. Take
a step size of h = 43200 seconds.
dθ
5.33 10 6 3.69 10 6 θ 4 2.33 10 5 θ 3 1.35 10 3 θ 2 5.42 10 2 θ 5.588 θ 33
dt
f t,θ 5.33106 3.69106 θ 4 2.33105 θ 3 1.35103 θ 2 5.42102 θ 5.588 θ 33
1
2
1
2
qi 1 qi k1 k2 h
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Step 1: For i 0, t0 0,
Solution
q0 27
4
3
6
5
3
.
69
10
27
2
.
33
10
27
6
k1 f t0 , q o f 0, 27 5.3310
27 33
2
1.35103 27 5.42102 27 5.588
0.0020893
k 2 f t0 h, q 0 k1h f 0 43200, 27 0.002089343200 f 43200,63.278
3.69106 63.2784 2.33105 63.2783
6
5.3310
63
.
278
33
1.35103 63.2782 5.4210 2 63.278 5.588
0.0092607
1
1
1
0.0020893 1 0.009260743200
2
2
2
2
27 0.005675043200 218.16C
q1 q 0 k1 k2 h 27
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Solution Cont
Step 2: i 1, t1 43200, q1 218.16C
k1 f t1 ,q1 f 43200,218.16
4
3
6
5
3
.
69
10
218
.
16
2
.
33
10
218
.
16
6
5.3310
218
.
16
33
1.35103 218.162 5.4210 2 218.16 5.588
8.4304
k 2 f t1 h, q1 k1h f 43200 43200,218.16 8.430443200 f 86400,364410
3.69106 (364410) 4 2.33105 (364410)3
6
5.3310
364410 33
2
3
2
1.3510 364410 5.4210 364410 5.588
1.26381017
1
1
1
8.4304 1 1.26381017 43200
2
2
2
2
218.16 6.31901016 43200 2.72981021C
q 2 q1 k1 k2 h 218.16
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Solution Cont
The solution to this nonlinear equation at t=86400s is
q 86400 26.099C
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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
86400
43200
21600
10800
5400
q 86400
|t | %
Et
−58466
58440
−2.7298×1021 2.7298×1021
−24.537
−1.5619
−25.785
−0.31368
−26.027
−0.072214
223920
1.0460×1011
5.9845
1.2019
0.27670
q 86400 26.099C (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
q (86400)
Step size,
h
Euler
Heun
Midpoint
Ralston
86400
43200
21600
10800
5400
−153.52
−464.32
−29.541
−27.795
−26.958
−58466
−2.7298×1021
−24.537
−25.785
−26.027
−774.64
−0.33691
−24.069
−25.808
−26.039
−12163
−19.776
−24.268
−25.777
−26.032
q 86400 26.099C
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(exact)
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Comparison of Euler and RungeKutta 2nd Order Methods
Table 2. Comparison of Euler and the Runge-Kutta methods
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t %
Step
size,
h
Euler
Heun
Midpoint
Ralston
86400
43200
21600
10800
5400
448.34
737.97
14.426
7.0957
3.5755
12216011
5.7292
1.1993
0.27435
1027.6
76.360
7.0508
1.0707
0.22604
23844
42.571
6.6305
1.2135
0.25776
q (86400) 25.217C
(exact)
7.906410
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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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