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Runge 4th Order Method
Civil Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
1/10/2010
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
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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
C i +1 = C i +
5
1
(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, 10 7 + (− 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, 10 7 + (− 537000 )3.5 
2
2
2
2




= f (1.75, 9060300 ) = −0.06(9060300 ) = −543620
(
k 4 = f (t0 + h, C0 + k3 h ) = 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
7
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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 × 10 6 = −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




(
)
,
3
.
5
3
.
5
,
8105900
+ (− 435290 )3.5 
k3 = f  t1 + h C1 + k 2 h  = f  +
2
2
2
2




= f (5.25, 7344100 ) = −0.06(7344100 ) = −440648
k 4 = f (t1 + h, C1 + k3 h ) = f (3.5 + 3.5, 8105900 + (− 440648)3.5)
= f (7, 6563600 ) = −0.06(6563600 ) = −393820
8
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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
C 2 is the approximate concentration of bacteria at
t 2 = t1 + h = 3.5 + 3.5 = 7 weeks
C (7 ) ≈ C2 = 6.5705 ×106 parts/m 3
9
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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
10
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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
Step size
h
7
3.5
1.75
0.875
0.4375
C (7 )
6.5715
6.5705
6.5705
6.5705
6.5705
|∈t | %
Et
106
106
106
106
106
−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.
14
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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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