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

Runge-Kutta 2nd Order Method

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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

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

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(

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.

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

THE END

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