1 / 55

EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations

EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations. Lesson 3: Midpoint and Heun’s Predictor corrector Methods. Lessons in Topic 8. Lesson 1: Introduction to ODE Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method

Download Presentation

EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations

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. EE 3561 : Computational MethodsUnit 8Solution of Ordinary Differential Equations Lesson 3: Midpoint and Heun’s Predictor corrector Methods Al-Dhaifallah1435

  2. Lessons in Topic 8 • Lesson 1: Introduction to ODE • Lesson 2: Taylor series methods • Lesson 3: Midpoint and Heun’s method • Lessons 4-5: Runge-Kutta methods • Lesson 6: Solving systems of ODE Al-Dhaifallah1435

  3. Learning Objectives of Lesson 3 • To be able to solve first order differential equation using Midpoint Method • To be able to solve first order differential equation using Heun’s Predictor Corrector method Al-Dhaifallah1435

  4. Outlines of Lesson 3 • Lesson 3: Midpoint and Heun’s • Predictor-corrector methods • Review Euler Method • Heun’s Method • Midpoint method Al-Dhaifallah1435

  5. Euler Method Al-Dhaifallah1435

  6. Introduction • We have seen Taylor series method • Euler method is simple but not accurate • Higher order Taylor series methods are accurate • but require calculating higher order derivatives • analytically Al-Dhaifallah1435

  7. Introduction • The methods proposed in this lesson have the general form • For the case of Euler • Different forms of will be used for the midpoint and Heun’s methods Al-Dhaifallah1435

  8. Midpoint Method Al-Dhaifallah1435

  9. Motivation • The midpoint can be summarized as • Euler method is used to estimate the solution at the midpoint. • The value of the rate function f(x,y) at the mid point is calculated • This value is used to estimate yi+1. • Local Truncation error of order O(h3) • Comparable to Second order Taylor series method Al-Dhaifallah1435

  10. Midpoint Method Al-Dhaifallah1435

  11. Midpoint Method Al-Dhaifallah1435

  12. Midpoint Method Al-Dhaifallah1435

  13. Midpoint Method Al-Dhaifallah1435

  14. Midpoint Method Al-Dhaifallah1435

  15. Midpoint Method Al-Dhaifallah1435

  16. Example 1 Al-Dhaifallah1435

  17. Example 1 Al-Dhaifallah1435

  18. Summary • The midpoint can be summarized as • Euler method is used to estimate the solution at the midpoint. • The value of the rate function f(x,y) at the mid point is calculated • This value is used to estimate yi+1. • Local Truncation error of order O(h3) • Comparable to Second order Taylor series method Al-Dhaifallah1435

  19. Heun’s Predictor Corrector Al-Dhaifallah1435

  20. Heun’s Predictor Corrector Method Al-Dhaifallah1435

  21. Heun’s Predictor Corrector(Prediction) Al-Dhaifallah1435

  22. Heun’s Predictor Corrector(Prediction) Al-Dhaifallah1435

  23. Heun’s Predictor Corrector(Prediction) Al-Dhaifallah1435

  24. Heun’s Predictor Corrector Al-Dhaifallah1435

  25. Heun’s Predictor Corrector Al-Dhaifallah1435

  26. Example 2 Al-Dhaifallah1435

  27. Example 2 Al-Dhaifallah1435

  28. Summary • Euler, Midpoint and Heun’s methods are similar in the following sense: • Different methods use different estimates of the slope • Both Midpoint and Heun’s methods are comparable in accuracy to second order Taylor series method. Al-Dhaifallah1435

  29. Comparison Al-Dhaifallah1435

  30. More in this Unit • Lessons 4-5: Runge-Kutta Methods • Lesson 6: Systems of High order ODE • Lesson 7: Multi-step methods • Lessons 8-9: Boundary Value Problems Al-Dhaifallah1435

  31. EE 3561 : Computational MethodsTopic 8Solution of Ordinary Differential Equations Lesson 4: Runge-Kutta Methods Al-Dhaifallah1435

  32. Lessons in Topic 8 • Lesson 1: Introduction to ODE • Lesson 2: Taylor series methods • Lesson 3: Midpoint and Heun’s method • Lessons 4-5: Runge-Kutta methods • Lesson 6: Solving systems of ODE Al-Dhaifallah1435

  33. Learning Objectives of Lesson 4 • To understand the motivation for using Runge Kutta method and basic idea used in deriving them. • To Familiarize with Taylor series for functions of two variables • Use Runge Kutta of order 2 to solve ODE Al-Dhaifallah1435

  34. Motivation • We seek accurate methods to solve ODE that does not require calculating high order derivatives. • The approach is to suggest a formula involving unknown coefficients then determine these coefficients to match as many terms of the Taylor series expansion Al-Dhaifallah1435

  35. Runge-Kutta Method Al-Dhaifallah1435

  36. Lecture Taylor Series in Two Variables The Taylor Series discussed in Chapter 4 is extended to the 2-independent variable case. This is used to prove RK formula Al-Dhaifallah1435

  37. Taylor Series in One Variable Approximation Error Al-Dhaifallah1435

  38. Taylor Series in One Variableanother look Al-Dhaifallah1435

  39. Definitions Al-Dhaifallah1435

  40. Taylor Series Expansion Al-Dhaifallah1435

  41. Taylor Series in Two Variables y+k y x x+h Al-Dhaifallah1435

  42. Runge-Kutta Method Al-Dhaifallah1435

  43. Runge-Kutta Method Al-Dhaifallah1435

  44. Runge-Kutta Method Al-Dhaifallah1435

  45. Runge-Kutta Method Al-Dhaifallah1435

  46. Runge-Kutta MethodAlternative Formula Al-Dhaifallah1435

  47. Runge-Kutta MethodAlternative Formula Al-Dhaifallah1435

  48. Runge-Kutta MethodAlternative Formulas Al-Dhaifallah1435

  49. Runge-Kutta Method Al-Dhaifallah1435

  50. Second order Runge-Kutta Method Example Al-Dhaifallah1435

More Related