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Runge 2 nd Order Method

Runge 2 nd Order Method. Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Runge-Kutta 2 nd Order Method http://numericalmethods.eng.usf.edu. Runge-Kutta 2 nd Order Method.

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Runge 2 nd Order Method

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  1. Runge 2nd Order Method Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates http://numericalmethods.eng.usf.edu

  2. Runge-Kutta 2nd Order Methodhttp://numericalmethods.eng.usf.edu

  3. Runge-Kutta 2nd Order Method For Runge Kutta 2nd order method is given by where http://numericalmethods.eng.usf.edu

  4. y yi+1, predicted yi x xi+1 xi Heun’s Method Heun’s method Here a2=1/2 is chosen resulting in where Figure 1 Runge-Kutta 2nd order method(Heun’s method) http://numericalmethods.eng.usf.edu

  5. Midpoint Method Here is chosen, giving resulting in where http://numericalmethods.eng.usf.edu

  6. Ralston’s Method Here is chosen, giving resulting in where http://numericalmethods.eng.usf.edu

  7. 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 http://numericalmethods.eng.usf.edu

  8. Example A solid steel shaft at room temperature of 27°C is needed to be contracted so 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 q is given by Find the temperature of the steel shaft after 24 hours. Take a step size of h = 43200 seconds. http://numericalmethods.eng.usf.edu

  9. Solution Step 1: For http://numericalmethods.eng.usf.edu

  10. Solution Cont Step 2: http://numericalmethods.eng.usf.edu

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

  12. Comparison with exact results Figure 2. Heun’s method results for different step sizes http://numericalmethods.eng.usf.edu

  13. Effect of step size Table 1. Effect of step size for Heun’s method (exact) http://numericalmethods.eng.usf.edu

  14. Effects of step size on Heun’s Method Figure 3. Effect of step size in Heun’s method http://numericalmethods.eng.usf.edu

  15. Comparison of Euler and Runge-Kutta 2nd Order Methods Table 2. Comparison of Euler and the Runge-Kutta methods (exact) http://numericalmethods.eng.usf.edu

  16. Comparison of Euler and Runge-Kutta 2nd Order Methods Table 2. Comparison of Euler and the Runge-Kutta methods (exact) http://numericalmethods.eng.usf.edu

  17. Comparison of Euler and Runge-Kutta 2nd Order Methods Figure 4. Comparison of Euler and Runge Kutta 2nd order methods with exact results. http://numericalmethods.eng.usf.edu

  18. 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

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

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