runge 2 nd order method
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Runge 2 nd Order Method. Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Runge-Kutta 2 nd Order Method http://numericalmethods.eng.usf.edu. Runge-Kutta 2 nd Order Method. - PowerPoint PPT PresentationTRANSCRIPT
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04/20/23http://
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Runge 2nd Order Method
Mechanical Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
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Runge-Kutta 2nd Order Method
http://numericalmethods.eng.usf.edu
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Runge-Kutta 2nd Order Method
Runge Kutta 2nd order method is given by
hkakayy ii 22111
where
ii yxfk ,1
hkqyhpxfk ii 11112 ,
For0)0(),,( yyyxf
dx
dy
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Heun’s Method
x
y
xi xi+1
yi+1, predicted
yi
Figure 1 Runge-Kutta 2nd order method (Heun’s method)
hkyhxfSlope ii 1,
iiii yxfhkyhxfSlopeAverage ,,2
1 1
ii yxfSlope ,
Heun’s method
2
11 a
11 p
111 q
resulting in
hkkyy ii
211 2
1
2
1
where
ii yxfk ,1
hkyhxfk ii 12 ,
Here a2=1/2 is chosen
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Midpoint MethodHere 12 a is chosen, giving
01 a
2
11 p
2
111 q
resulting in
hkyy ii 21
where
ii yxfk ,1
hkyhxfk ii 12 2
1,
2
1
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Ralston’s MethodHere
3
22 a is chosen, giving
3
11 a
4
31 p
4
311 q
resulting in
hkkyy ii
211 3
2
3
1
where ii yxfk ,1
hkyhxfk ii 12 4
3,
4
3
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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,
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ExampleA 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 is given by
C270
33588510425
1035110332106931033.5
2
2335466
θ.θ.
θ.θ.θ.
dt
dθ
Find the temperature of the steel shaft after 24 hours. Take a step size of h = 43200 seconds.
335885104251035110332106931033.5 22335466 θ.θ.θ.θ.θ.dt
dθ
335885104251035110332106931033.5 22335466 θ.θ.θ.θ.θ.t,θf
hkkii
211 2
1
2
1
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SolutionStep 1: For 27,0,0 00 ti
0020893.0
3327588.5271042.5271035.1
271033.2271069.31033.527,0,
223
35466
01
ftfk o
0092607.0
33278.63588.5278.631042.5278.631035.1
278.631033.2278.631069.31033.5
278.63,43200432000020893.027,432000,
223
35466
1002
ffhkhtfk
C
hkk
16.218432000056750.027
432000092607.02
10020893.0
2
127
2
1
2
12101
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Solution ContStep 2: Cti 16.218,43200,1 11
4304.8
3316.218588.516.2181042.516.2181035.1
16.2181033.216.2181069.31033.5
16.218,43200,
223
35466
111
ftfk
17
223
35466
1112
102638.1
33364410588.53644101042.53644101035.1
)364410(1033.2)364410(1069.31033.5
364410,86400432004304.816.218,4320043200,
ffhkhtfk
C107298.243200103190.616.218
43200102638.12
14304.8
2
116.218
2
1
2
1
2116
172112
hkk
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Solution Cont
The solution to this nonlinear equation at t=86400s is
C 099.2686400
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Comparison with exact results
Figure 2. Heun’s method results for different step sizes
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Step size,
864004320021600108005400
−58466−2.7298×1021
−24.537−25.785−26.027
584402.7298×1021
−1.5619−0.31368−0.072214
2239201.0460×1011
5.98451.2019
0.27670
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Effect of step size
h tE %|| t
(exact) C 099.2686400
Table 1. Effect of step size for Heun’s method
86400
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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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Step size,h Euler Heun Midpoint Ralston
864004320021600108005400
−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
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Comparison of Euler and Runge-Kutta 2nd Order
MethodsTable 2. Comparison of Euler and the Runge-Kutta methods
(exact) C 099.2686400
)86400(
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Stepsize,
h Euler Heun Midpoint Ralston
864004320021600108005400
448.34737.9714.4267.09573.5755
122160
5.72921.1993
0.27435
1027.676.3607.05081.0707
0.22604
2384442.5716.63051.2135
0.25776
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Comparison of Euler and Runge-Kutta 2nd Order
Methods
C 217.25)86400(
Table 2. Comparison of Euler and the Runge-Kutta methods
(exact)
%t
11109064.7
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Comparison of Euler and Runge-Kutta 2nd Order
Methods
Figure 4. Comparison of Euler and Runge Kutta 2nd order methods with exact results.
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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_2nd_method.html