1. 5/19/2015 http://numericalmethods.eng.usf.edu 1 Runge 4 th Order Method Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

2. Runge-Kutta 4 th Order Method http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

3. 3 Runge-Kutta 4 th Order Method where For Runge Kutta 4 th order method is given by

4. http://numericalmethods.eng.usf.edu4 How to write Ordinary Differential Equation Example is rewritten as In this case How does one write a first order differential equation in the form of

5. http://numericalmethods.eng.usf.edu5 Example A solid steel shaft at room temperature of 27°C is needed to be contracted so that it can be shrunk-fit into a hollow hub. It is placed in a refrigerated chamber that is maintained at −33°C. The rate of change of temperature of the solid shaft  is given by Find the temperature of the steel shaft after 24 hours. Take a step size of h=43200 seconds.

6. http://numericalmethods.eng.usf.edu6 Solution Step 1:

7. http://numericalmethods.eng.usf.edu7 Solution Cont is the approximate temperature at

8. http://numericalmethods.eng.usf.edu8 Solution Cont Step 2:

9. http://numericalmethods.eng.usf.edu9 Solution Cont is the approximate temperature at

10. http://numericalmethods.eng.usf.edu10 Solution Cont The solution to this nonlinear equation at t=86400s is

11. http://numericalmethods.eng.usf.edu11 Comparison with exact results Figure 1. Comparison of Runge-Kutta 4th order method with exact solution

12. Step Size, 86400 43200 21600 10800 5400 −5.3468×10 28 −1.0105×10 23 −26.061 −26.094 −26.097 −5.2468×10 28 −0.034808×10 23 −0.038680 −0.0050630 −0.0015763 2.0487×10 29 3.8718×10 23 0.14820 0.019400 0.0060402 http://numericalmethods.eng.usf.edu12 Effect of step size (exact) Table 1. Value of temperature at 86400 seconds for different step sizes

13. http://numericalmethods.eng.usf.edu13 Effects of step size on Runge- Kutta 4 th Order Method Figure 2. Effect of step size in Runge-Kutta 4th order method

14. http://numericalmethods.eng.usf.edu14 Comparison of Euler and Runge- Kutta Methods Figure 3. Comparison of Runge-Kutta methods of 1st, 2nd, and 4th order.

15. 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

16. THE END http://numericalmethods.eng.usf.edu