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Runge 2 nd Order Method. Civil 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. For. - PowerPoint PPT Presentation
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04/24/23http://
numericalmethods.eng.usf.edu 1
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
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 ,
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
http://numericalmethods.eng.usf.edu/topics/runge_kutta_2nd_method.html
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