Solving ODEs
Euler Method & RK2/4
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
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02/11/10
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1
Euler Method
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Euler’s Method
y
True value
Slope
x0,y0
Φ
y1, Predicted
value
Step size, h
x
Figure 1 Graphical interpretation of the first step of Euler’s method
3
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Euler’s Method
y
True Value
yi+1, Predicted value
Φ
yi
h
Step size
xi
xi+1
x
Figure 2. General graphical interpretation of Euler’s method
4
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How to write Ordinary Differential
Equation
How does one write a first order differential equation in the form of
Example
is rewritten as
In this case
5
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Example
A 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
Find the temperature at
seconds using Euler’s method. Assume a step size of
seconds.
6
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Solution
Step 1:
is the approximate temperature at
7
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Solution Cont
Step 2:
For
is the approximate temperature at
8
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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
The solution to this nonlinear equation at t=480 seconds is
9
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Comparison of Exact and
Numerical Solutions
1500,0000
θ(K)
Temperature,
1125,0000
Exact Solution
750,0000
h=240
375,0000
0
0
125
250
375
500
Time, t(sec)
Figure 3. Comparing exact and Euler’s method
10
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Effect of step size
Table 1. Temperature at 480 seconds as a function of step size, h
11
Step, h
θ(480)
Et
|єt|%
480
240
120
60
30
−987.8
1
110.32
546.77
614.97
1635.4
537.26
100.80
32.607
14.806
252.54
82.964
15.566
5.0352
2.2864
(exact)
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Comparison with exact results
1500,0000
Exact solution
Temperature,
θ(K)
1125,0000
750,0000
h=120
h=240
375,0000
0
h=480
-375,0000
-750,0000
-1125,0000
0
125
250
375
500
Time, t (sec)
Figure 4. Comparison of Euler’s method with exact solution for different step sizes
12
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Effects of step size on Euler’s
Method
750,0000
Temperature,
θ(K)
500,0000
250,0000
0
-250,0000
-500,0000
-750,0000
-1000,0000
0
125
250
375
500
Step size, h (s)
Figure 5. Effect of step size in Euler’s method.
13
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Errors in Euler’s Method
It can be seen that Euler’s method has large errors. This can be illustrated using
Taylor series.
As you can see the first two terms of the Taylor series
are the Euler’s method.
The true error in the approximation is given by
14
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Runge 2nd Order Method
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
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Transforming Numerical Methods Education for STEM
Undergraduates
02/11/10
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1
Runge-Kutta
nd
2
Order Method
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Runge-Kutta 2nd Order Method
For
Runge Kutta 2nd order method is given by
where
3
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Heun’s Method
Heun’s method
y
Here a2=1/2 is chosen
resulting in
yi+1, predicted
yi
xi
where
4
xi+1
x
Figure 1 Runge-Kutta 2nd order method (Heun’s method)
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Midpoint Method
Here
is chosen, giving
resulting in
where
5
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Ralston’s Method
Here
is chosen, giving
resulting in
where
6
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numericalmethods.eng.usf.edu
How to write Ordinary Differential
Equation
How does one write a first order differential equation in the form of
Example
is rewritten as
In this case
7
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numericalmethods.eng.usf.edu
Example
A 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
Find the temperature at
seconds using Heun’s method. Assume a step size of
seconds.
8
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numericalmethods.eng.usf.edu
Solution
Step 1:
9
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Solution Cont
Step 2:
10
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numericalmethods.eng.usf.edu
Solution Cont
The 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=480 seconds is
11
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numericalmethods.eng.usf.edu
Comparison with exact results
Figure 2. Heun’s method results for different step sizes
12
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Effect of step size
Table 1. Temperature at 480 seconds as a function of step size, h
Step size, h θ(480)
Et
|єt|%
480
240
120
60
30
1041.4
63.304
−3.7762
−2.3406
−0.63219
160.82
9.7756
0.58313
0.36145
0.097625
−393.87
584.27
651.35
649.91
648.21
(exact)
13
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Effects of step size on Heun’s
Method
Figure 3. Effect of step size in Heun’s method
14
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Comparison of Euler and RungeKutta 2nd Order Methods
Table 2. Comparison of Euler and the Runge-Kutta methods
Step
size,
h
480
240
120
60
30
15
θ(480)
Euler
Heun
Midpoint
Ralston
−987.84
110.32
546.77
614.97
632.77
−393.87
584.27
651.35
649.91
648.21
1208.4
976.87
690.20
654.85
649.02
(exact)
449.78
690.01
667.71
652.25
648.61
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Comparison of Euler and RungeKutta 2nd Order Methods
Table 2. Comparison of Euler and the Runge-Kutta methods
Step size,
h
Euler
Heun
480
240
120
60
30
252.54
82.964
15.566
5.0352
2.2864
160.82
9.7756
0.58313
0.36145
0.097625
Midpoin
t
86.612
50.851
Ralston
6.5823
3.1092
1.1239
0.7229
9
0.22353
30.544
6.5537
0.1594
(exact)
16
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Comparison of Euler and RungeKutta 2nd Order Methods
1200,0000
Midpoint
Temperature,
θ(K)
1025,0000
Ralston
Heun
850,0000
Analytical
675,0000
Euler
500,0000
0
125
250
375
500
Time, t (sec)
Figure 4. Comparison of Euler and Runge Kutta 2nd order methods with
exact results.
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17
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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_kutta_2nd_method.html
Runge
th
4
Order Method
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
02/11/10
http://numericalmethods.eng.usf.edu
1
Runge-Kutta
th
4
Order Method
http://numericalmethods.eng.usf.edu
Runge-Kutta 4th Order Method
For
Runge Kutta 4th order method is given by
where
3
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numericalmethods.eng.usf.edu
How to write Ordinary Differential
Equation
How does one write a first order differential equation in the form of
Example
is rewritten as
In this case
4
http://
numericalmethods.eng.usf.edu
Example
A 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
Find the temperature at
Assume a step size of
5
seconds using Runge-Kutta 4th order method.
seconds.
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numericalmethods.eng.usf.edu
Solution
Step 1:
6
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Solution Cont
is the approximate temperature at
7
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numericalmethods.eng.usf.edu
Solution Cont
Step 2:
8
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numericalmethods.eng.usf.edu
Solution Cont
θ2 is the approximate temperature at
9
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numericalmethods.eng.usf.edu
Solution Cont
The 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=480 seconds is
10
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numericalmethods.eng.usf.edu
Comparison with exact results
Figure 1. Comparison of Runge-Kutta 4th order method with exact solution
11
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Effect of step size
Table 1. Temperature at 480 seconds as a function of step size, h
Step size,
θ (480)
h
480
−90.278
240
594.91
120
646.16
60
647.54
30
647.57
Et
|єt|%
737.85
52.660
1.4122
0.033626
0.00086900
113.94
8.1319
0.21807
0.0051926
0.00013419
(exact)
12
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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
13
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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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THE END