1. CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36
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Read , 26-2, 27-1
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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
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3. Lecture 31 Lesson 4: Runge-Kutta Methods
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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.
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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.
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6. Runge-Kutta Method
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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.
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8. Taylor Series in One Variable
Approximation
Error
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9. Taylor Series in One Variable - Another Look -
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10. Definitions
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11. Taylor Series Expansion
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12. Taylor Series in Two Variables
y+k y x
x+h
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13. Runge-Kutta Method
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14. Runge-Kutta Method
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15. Runge-Kutta Method
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16. Runge-Kutta Method
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17. Runge-Kutta Method Alternative Formula
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18. Runge-Kutta Method Alternative Formula
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19. Runge-Kutta Method Alternative Formulas
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20. Runge-Kutta Method
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21. Second order Runge-Kutta Method Example
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22. Second order Runge-Kutta Method Example
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23. Second order Runge-Kutta Method Example
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24. CISE301_Topic8L4&5
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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.
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26. Lecture 32 Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs
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27. Learning Objectives of Lesson 5
Use Runge-Kutta methods of different orders to solve first order ODEs.
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28. Runge-Kutta Method
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29. Runge-Kutta Methods
RK2
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30. Runge-Kutta Methods
RK3
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31. Runge-Kutta Methods
RK4
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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.
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33. Fifth Order Runge-Kutta Methods
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34. Second Order Runge-Kutta Method
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35. Second Order Runge-Kutta Method
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36. Second Order Runge-Kutta Method
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37. Example 1 Second Order Runge-Kutta Method
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38. Example 1 Second Order Runge-Kutta Method
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39. Example 1 Second Order Runge-Kutta Method
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40. Example 1 Second Order Runge-Kutta Method
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41. Example 1 Second Order Runge-Kutta Method
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42. Example 1 Summary of the solution
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43. Solution after 100 steps
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44. Example 2 4th-Order Runge-Kutta Method
See RK4 Formula
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45. Example 2 Fourth Order Runge-Kutta Method
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46. Example 2 Fourth Order Runge-Kutta Method
See RK4 Formula
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47. Runge-Kutta Methods
RK4
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48. Example 2 Fourth Order Runge-Kutta Method
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49. Example 2 Summary of the solution
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50. Remaining Lessons in Topic 8
Solving Systems of high order ODE
Lesson 7:
Multi-step methods
Lessons 8-9:
Methods to solve Boundary Value Problems
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