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CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36. KFUPM Read 25.1-25.4, 26-2, 27-1. Outline of Topic 8. Lesson 1: Introduction to ODEs Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method Lessons 4-5: Runge-Kutta methods

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CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

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  1. CISE301: Numerical MethodsTopic 8Ordinary Differential Equations (ODEs)Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 KFUPM

  2. Outline of Topic 8 • Lesson 1: Introduction to ODEs • Lesson 2: Taylor series methods • Lesson 3: Midpoint and Heun’s method • Lessons 4-5: Runge-Kutta methods • Lesson 6: Solving systems of ODEs • Lesson 7: Multiple step Methods • Lesson 8-9: Boundary value Problems KFUPM

  3. Lecture 31Lesson 4: Runge-Kutta Methods KFUPM

  4. Learning Objectives of Lesson 4 • To understand the motivation for using Runge-Kutta (RK) method and the basic idea used in deriving them. • To get familiar with Taylor series for functions of two variables. • To use RK method of order 2 to solve ODEs. KFUPM

  5. Motivation • We seek accurate methods to solve ODEs that do not require calculating high order derivatives. • The approach is to use a formula involving unknown coefficients then determine these coefficients to match as many terms of the Taylor series expansion as possible. KFUPM

  6. Runge-Kutta Method KFUPM

  7. 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. KFUPM

  8. Taylor Series in One Variable Approximation Error KFUPM

  9. Taylor Series in One Variable- Another Look - KFUPM

  10. Definitions KFUPM

  11. Taylor Series Expansion KFUPM

  12. Taylor Series in Two Variables y+k y x x+h KFUPM

  13. Runge-Kutta Method KFUPM

  14. Runge-Kutta Method KFUPM

  15. Runge-Kutta Method KFUPM

  16. Runge-Kutta Method KFUPM

  17. Runge-Kutta MethodAlternative Formula KFUPM

  18. Runge-Kutta MethodAlternative Formula KFUPM

  19. Runge-Kutta MethodAlternative Formulas KFUPM

  20. Runge-Kutta Method KFUPM

  21. Second order Runge-Kutta Method Example KFUPM

  22. Second order Runge-Kutta Method Example KFUPM

  23. Second order Runge-Kutta Method Example KFUPM

  24. KFUPM

  25. Summary • RK methods generate an accurate solution without the need to calculate high order derivatives. • Second order RK have local truncation error of order O(h3). • Fourth order RK have local truncation error of order O(h5). • N function evaluations are needed in the Nth order RK method. KFUPM

  26. Lecture 32Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs KFUPM

  27. Learning Objectives of Lesson 5 • Use Runge-Kutta methods of different orders to solve first order ODEs. KFUPM

  28. Runge-Kutta Method KFUPM

  29. Runge-Kutta Methods RK2 KFUPM

  30. Runge-Kutta Methods RK3 KFUPM

  31. Runge-Kutta Methods RK4 KFUPM

  32. Runge-Kutta Methods Higher order Runge-Kutta methods are available. Higher order methods are more accurate but require more calculations. Fourth order is a good choice. It offers good accuracy with a reasonable calculation effort. KFUPM

  33. Fifth Order Runge-Kutta Methods KFUPM

  34. Second Order Runge-Kutta Method KFUPM

  35. Second Order Runge-Kutta Method KFUPM

  36. Second Order Runge-Kutta Method KFUPM

  37. Example 1Second Order Runge-Kutta Method KFUPM

  38. Example 1Second Order Runge-Kutta Method KFUPM

  39. Example 1Second Order Runge-Kutta Method KFUPM

  40. Example 1Second Order Runge-Kutta Method KFUPM

  41. Example 1Second Order Runge-Kutta Method KFUPM

  42. Example 1Summary of the solution Summary of the solution KFUPM

  43. Solution after 100 steps KFUPM

  44. Example 24th-Order Runge-Kutta Method See RK4 Formula KFUPM

  45. Example 2Fourth Order Runge-Kutta Method KFUPM

  46. Example 2Fourth Order Runge-Kutta Method See RK4 Formula KFUPM

  47. Runge-Kutta Methods RK4 KFUPM

  48. Example 2Fourth Order Runge-Kutta Method KFUPM

  49. Example 2Summary of the solution Summary of the solution KFUPM

  50. Remaining Lessons in Topic 8 Lesson 6: Solving Systems of high order ODE Lesson 7: Multi-step methods Lessons 8-9: Methods to solve Boundary Value Problems KFUPM

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