Runge 2 nd Order Method

04/24/23http://

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Runge 2nd Order Method

Civil 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 polluted lake with an initial concentration of a bacteria is 107 parts/m3, while the acceptable level is only 5×106 parts/m3. The concentration of the bacteria will reduce as fresh water enters the lake. The differential equation that governs the concentration C of the pollutant as a function of time (in weeks) is given by

710)0(,006.0 CCdtdC

Find the concentration of the pollutant after 7 weeks. Take a step size of 3.5 weeks.

CdtdC 06.0

CCtf 06.0,

hkkCC ii

211 2

121

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SolutionStep 1:

700 10,0,0 Cti

6000001006.010,0, 77001 fCtfk

474000109.706.0109.7,5.3

5.360000010,5.30,66

71002

f

fhkChtfk

36

7

7

2101

parts/m101205.8

5.353700010

5.347400021600000

2110

21

21

hkkCC

C1 is the approximate concentration of bacteria at weeks5.35.3001 httt

361 parts/m101205.85.3 CC

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Solution ContStep 2:

36101 parts/m101205.8,5.35.30,1 Chtti

487230101205.806.0101205.8,5.3, 66111 fCtfk

384910641520006.06415200,7

5.3487230101205.8,5.35.3, 61112

ffhkChtfk

36

6

6

2112

parts/m105943.6

5.3436070101205.8

5.338491021487230

21101205.8

21

21

hkkCC

C2 is the approximate concentration of bacteria at weeks75.35.312 httt

362 parts/m105943.67 CC

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

The solution to this nonlinear equation at t=7 weeks is

503

7101)(t

etC

36 parts/m105705.6)7( C

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Comparison with exact results

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

Step size,

73.51.75

0.8750.4375

6.6520×106

6.5943×106

6.5760×106

6.5718×106

6.5708×106

−111530−23784−5489.1−1318.8−323.24

1.69750.36198

0.0835420.0200710.0049195

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Effect of step size

h tE %|| t

(exact)6105705.6)7( C

7C

Table 1. Effect of step size for Heun’s method

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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

73.5

1.750.875

0.4375

5.8000×106

6.2410×106

6.4160×106

6.4960×106

6.5340×106

6.6820×106

6.5943×106

6.5760×106

6.5718×106

6.5708×106

6.6820×106

6.5943×106

6.5760×106

6.7518×106

6.5708×106

6.6820×106

6.5943×106

6.5760×106

6.5718×106

6.5708×106

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Comparison of Euler and Runge-Kutta 2nd Order

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

(exact)6105705.6)7( C

)7(C

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Comparison of Euler and Runge-Kutta 2nd Order

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

(exact)

%t

6105705.6)7( C

Step size,h Euler Heun Midpoint Ralston

73.5

1.750.8750.4375

11.7265.01442.34471.1362

0.55952

1.69750.36198

0.0835420.020071

0.0049195

1.69750.36198

0.0835420.020071

0.0049195

1.69750.36198

0.0835420.020071

0.0049195

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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

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