04/24/23http://

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Runge 2nd Order Method

Major: All 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 ,

For 0)0(),,( yyyxfdxdy

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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 ,,21 1

ii yxfSlope ,

Heun’s method

21

1 a

11 p

111 q

resulting inhkkyy ii

211 2

121

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

21

1 p

21

11 q

resulting inhkyy ii 21

where ii yxfk ,1

hkyhxfk ii 12 2

1,21

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Ralston’s MethodHere

32

2 a is chosen, giving

31

1 a

43

1 p

43

11 q

resulting inhkkyy ii

211 3

231

where ii yxfk ,1

hkyhxfk ii 12 4

3,43

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How to write Ordinary Differential Equation

Example

50,3.12 yeydxdy x

is rewritten as

50,23.1 yyedxdy x

In this case

yeyxf x 23.1,

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

yxfdxdy ,

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ExampleA ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by

Kdtd 12000,1081102067.2 8412

Find the temperature at 480t seconds using Heun’s method. Assume a step size of

240h seconds.

8412 1081102067.2

dtd

8412 1081102067.2, tf

hkkii

211 2

121

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SolutionStep 1: Kti 1200)0(,0,0 00

5579.4

10811200102067.2

1200,0,

8412

01

ftfk o

017595.0

108109.106102067.2

09.106,2402405579.41200,2400

,

8412

1002

ff

hkhtfk

K

hkk

16.6552402702.21200

240017595.0215579.4

211200

21

21

2101

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Solution ContStep 2: Khtti 16.655,2402400,1 101

38869.0

108116.655102067.2

16.655,240,

8412

111

ftfk

20206.0

108187.561102067.2

87.561,48024038869.016.655,240240

,

8412

1112

ff

hkhtfk

K

hkk

27.58424029538.016.655

24020206.02138869.0

2116.655

21

21

2112

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Solution Cont

The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as

9282.21022067.00033333.0tan8519.1300300ln92593.0 31

t

The solution to this nonlinear equation at t=480 seconds is

K57.647)480(

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Comparison with exact results

Figure 2. Heun’s method results for different step sizes

-400

0

400

800

1200

0 100 200 300 400 500

Time, t(sec)

Tem

pera

ture

,θ(K

) Exact h=120

h=240

h=480

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Effect of step sizeTable 1. Temperature at 480 seconds as a function of step size, h

Step size, h (480) Et |єt|%

4802401206030

−393.87584.27651.35649.91648.21

1041.463.304

−3.7762−2.3406

−0.63219

160.829.7756

0.583130.36145

0.097625

K57.647)480( (exact)

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Effects of step size on Heun’s Method

Figure 3. Effect of step size in Heun’s method

-400

-200

0

200

400

600

800

0 100 200 300 400 500Step size, h

Tem

pera

ture

,θ(

480)

Step size,h

(480)Euler Heun Midpoint Ralston

4802401206030

−987.84110.32546.77614.97632.77

−393.87584.27651.35649.91648.21

1208.4976.87690.20654.85649.02

449.78690.01667.71652.25648.61

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Comparison of Euler and Runge-Kutta 2nd Order

MethodsTable 2. Comparison of Euler and the Runge-Kutta methods

K57.647)480( (exact)

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Comparison of Euler and Runge-Kutta 2nd Order

MethodsTable 2. Comparison of Euler and the Runge-Kutta methods

Step size,h Euler Heun Midpoint Ralston

4802401206030

252.5482.96415.5665.03522.2864

160.829.7756

0.583130.361450.097625

86.61250.8516.58231.1239

0.22353

30.5446.55373.10920.722990.15940

K57.647)480( (exact)

%t

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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.

500

600

700

800

900

1000

1100

1200

0 100 200 300 400 500 600

Time, t (sec)

Tem

pera

ture

,

Analytical

Ralston

Midpoint

Euler

Heun

θ(K

)

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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