02/11/10 http://numericalmethods.eng.usf.edu. 1. Solving ODEs. Euler Method &
RK2/4. Major: All Engineering Majors. Authors: Autar Kaw, Charlie Barker.
Solving ODEs Euler Method & RK2/4
Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM 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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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) http:// numericalmethods.eng.usf.edu
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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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
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
seconds using Heun’s method. Assume a step size of
seconds.
8
http:// numericalmethods.eng.usf.edu
Solution Step 1:
9
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Solution Cont Step 2:
10
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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
11
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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. http:// 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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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.
http:// numericalmethods.eng.usf.edu
Solution Step 1:
6
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Solution Cont
is the approximate temperature at
7
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Solution Cont Step 2:
8
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Solution Cont
θ2 is the approximate temperature at
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
The solution to this nonlinear equation at t=480 seconds is
10
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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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