1 / 19

Runge 2 nd Order Method

Runge 2 nd Order Method. Electrical 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.

heinz
Download Presentation

Runge 2 nd Order Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Runge 2nd Order Method Electrical 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 rectifier-based power supply requires a capacitor to temporarily store power when the rectified waveform from the AC source drops below the target voltage. To properly size this capacitor a first-order ordinary differential equation must be solved. For a particular power supply, with a capacitor of 150 μF, the ordinary differential equation to be solved is Find voltage across the capacitor at t= 0.00004s. Use step size h=0.00002 http://numericalmethods.eng.usf.edu

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

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

  11. Solution Continued The solution to this nonlinear equation at t=0.00004 seconds 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

More Related