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SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

Learn about Runge-Kutta methods for solving Ordinary Differential Equations (ODEs), including motivation, Taylor series, and applications. Explore second and higher-order Runge-Kutta methods with examples and formulas.

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SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

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  1. SE301: Numerical MethodsTopic 8Ordinary Differential Equations (ODEs)Lecture 28-36 KFUPM (Term 101) Section 04 Read 25.1-25.4, 26-2, 27-1

  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

  3. Lecture 31Lesson 4: Runge-Kutta Methods

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

  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.

  6. Second Order Runge-Kutta Method

  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.

  8. Taylor Series in One Variable Error Approximation

  9. Derivation of 2nd OrderRunge-Kutta Methods – 1 of 5

  10. Derivation of 2nd OrderRunge-Kutta Methods – 2 of 5

  11. Taylor Series in Two Variables

  12. Derivation of 2nd OrderRunge-Kutta Methods – 3 of 5

  13. Derivation of 2nd OrderRunge-Kutta Methods – 4 of 5

  14. Derivation of 2nd OrderRunge-Kutta Methods – 5 of 5

  15. 2nd Order Runge-Kutta Methods

  16. Alternative Form

  17. Choosing , , w1 and w2

  18. Choosing , , w1 and w2

  19. 2nd Order Runge-Kutta MethodsAlternative Formulas

  20. Second order Runge-Kutta Method Example

  21. Second order Runge-Kutta Method Example

  22. Lecture 32Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs Using Runge-Kutta methods of different orders to solve first order ODEs

  23. 2nd Order Runge-Kutta RK2

  24. Higher-Order Runge-Kutta Higher order Runge-Kutta methods are available. Derived similar to second-order Runge-Kutta. Higher order methods are more accurate but require more calculations.

  25. 3rd Order Runge-Kutta RK3

  26. 4th Order Runge-Kutta RK4

  27. Higher-Order Runge-Kutta

  28. Example4th-Order Runge-Kutta Method RK4

  29. Example: RK4

  30. 4th Order Runge-Kutta RK4

  31. Example: RK4 See RK4 Formula Step 1

  32. Example: RK4 Step 2

  33. Example: RK4 Summary of the solution

  34. Summary • Runge Kutta methods generate an accurate solution without the need to calculate high order derivatives. • Second order RK have local truncation error of order O(h3) and global truncation error of order O(h2). • Higher order RK have better local and global truncation errors. • N function evaluations are needed in the Nth order RK method.

  35. 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

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