1. SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36
KFUPM
(Term 101)
Section 04
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 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.
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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.
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6. Second Order 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
Error
Approximation
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9. Derivation of 2nd Order Runge-Kutta Methods – 1 of 5
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10. Derivation of 2nd Order Runge-Kutta Methods – 2 of 5
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11. Taylor Series in Two Variables
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12. Derivation of 2nd Order Runge-Kutta Methods – 3 of 5
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13. Derivation of 2nd Order Runge-Kutta Methods – 4 of 5
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14. Derivation of 2nd Order Runge-Kutta Methods – 5 of 5
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15. 2nd Order Runge-Kutta Methods
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16. Alternative Form
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17. Choosing , , w1 and w2
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18. Choosing , , w1 and w2
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19. 2nd Order Runge-Kutta Methods Alternative Formulas
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20. Second order Runge-Kutta Method Example
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21. Second order Runge-Kutta Method Example
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22. CISE301_Topic8L4&5

23. Lecture 32 Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs
Using Runge-Kutta methods of different
orders to solve first order ODEs
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24. 2nd Order Runge-Kutta
RK2
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25. 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.
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26. 3rd Order Runge-Kutta
RK3
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27. 4th Order Runge-Kutta
RK4
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28. Higher-Order Runge-Kutta
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29. Example 4th-Order Runge-Kutta Method
RK4
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30. Example: RK4
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31. 4th Order Runge-Kutta
RK4
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32. Example: RK4
See RK4 Formula
Step 1
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33. Example: RK4
Step 2
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34. Example: RK4 Summary of the solution xi yi 0.0 0.5 0.2 0.8293 0.4
1.2141
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35. 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.
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36. 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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