Runge 4th Order Method
Civil Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
7/12/2016
http://numericalmethods.eng.usf.edu
1
Runge-Kutta 4th Order Method
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Runge-Kutta 4th Order Method
For
dy
 f ( x, y ), y (0)  y0
dx
Runge Kutta 4th order method is given by
1
yi 1  yi  k1  2k2  2k3  k4 h
6
where
k1  f xi , yi 
1
1


k2  f  xi  h, yi  k1h 
2
2 

1
1


k3  f  xi  h, yi  k2 h 
2
2


k4  f xi  h, yi  k3h
3
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How to write Ordinary Differential
Equation
How does one write a first order differential equation in the form of
dy
 f  x, y 
dx
Example
dy
 2 y  1.3e  x , y 0   5
dx
is rewritten as
dy
 1.3e  x  2 y, y 0   5
dx
In this case
f x, y   1.3e  x  2 y
4
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Example
A polluted lake with an initial concentration of a bacteria is 107 parts/m3,
while the acceptable level is only 5x106 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
dC
 0.06C  0, C (0)  107
dt
Find the concentration of the pollutant after 7 weeks. Take a step size of
3.5 weeks.
dC
 0.06C
dt
f t, C   0.06C
Ci 1
5
1
 Ci  k1  2k 2  2k 3  k 4 h
6
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Solution
7
3
Step 1: i  0, t0  0, C0  10 parts / m


 
k1  f t0 , C0   f 0, 107  0.06 107  600000
1
1
1
1




k 2  f  t0  h, C0  k1h   f  0  3.5, 107   600000 3.5 
2
2
2
2




 f 1.75, 8950000   0.068950000   537000
1
1
1
1




k3  f  t0  h, C0  k 2 h   f  0  3.5, 107   5370003.5 
2
2
2
2




 f 1.75, 9060300  0.069060300  543620

k4  f t0  h, C0  k3h   f 0  3.5, 107   5436203.5
 f 1.75, 8097300  0.068097300  485840
6

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Solution Cont
1
k1  2k2  2k3  k4 
6
1
 107   600000  2 537000   2 543620    485840 3.5
6
1
7
 10   3247100 3.5
6
 8.1059 106
C1  C0 
C1 is the approximate concentration of bacteria at
t  t1  t0  h  0  3.5  3.5 weeks
C 3.5  C1  8.1059 106 parts/m 3
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Solution Cont
Step 2:
i  1, t1  3.5, C1  8.1059 106




k1  f t1 , C1   f 3.5, 8.1059 106  0.06 8.1059 106  486350
1
1
1
1




k 2  f  t1  h, C1  k1h   f  3.5  3.5, 8105900   4863503.5 
2
2
2
2




 f 5.25, 7254800  0.067254800  435290
1
1
1
1




k3  f  t1  h, C1  k 2 h   f  3.5  3.5, 8105900   4352903.5 
2
2
2
2




 f 5.25, 7344100  0.067344100  440648
k4  f t1  h, C1  k3h   f 3.5  3.5, 8105900   4406483.5
 f 7, 6563600  0.066563600  393820
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Solution Cont
1
C2  C1  (k1  2k 2  2k3  k 4 )h
6
1
 8105900   484350  2 435290  2 440648   398203.5
6
1
 8105900   26320003.5
6
 6.5705 106 parts/m 3
C2 is the approximate concentration of bacteria at
t2  t1  h  3.5  3.5  7weeks
C 7   C2  6.5705 106 parts/m 3
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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
C (t )  1 107 e
 3t 


 50 
The solution to this nonlinear equation at t=7 weeks is
C (7)  6.5705 106 parts/m 3
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Comparison with exact results
Figure 1. Comparison of Runge-Kutta 4th order method with exact solution
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Effect of step size
Table 1 Value of concentration of bacteria at 3 minutes for
different step sizes
h
C7
7
3.5
1.75
0.875
0.4375
6.5715×106
6.5705×106
6.5705×106
6.5705×106
6.5705×106
Step size
|t | %
Et
−1017.2
−53.301
−3.0512
−0.18252
−0.011161
0.015481
8.1121×10−4
4.6438×10−5
2.7779×10−6
1.6986×10−7
C (7)  6.5705  10 6 (exact)
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Effects of step size on RungeKutta 4th Order Method
Figure 2. Effect of step size in Runge-Kutta 4th order method
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Comparison of Euler and RungeKutta Methods
Figure 3. Comparison of Runge-Kutta methods of 1st, 2nd, and 4th order.
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Additional Resources
For 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_kutt
a_4th_method.html
THE END
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