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

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