1. Today’s class Ordinary Differential Equations Runge-Kutta Methods
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

2. Ordinary Differential Equations
A differential equation is an equation made up of a function and derivatives of that function
Example
An ordinary differential equation (ODE) is a differential equation with only one independent variable
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

3. Ordinary Differential Equations
First-order - highest derivative in the ODE is a first derivative
Second-order - highest derivative in the ODE is a second derivative
Linear - ODE is of the following form
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

4. Initial-Value & Boundary-Value Conditions
IV Conditions
All conditions are given at the same value of the independent variable
BV Conditions
Conditions are given at different value of the independent variables
The numerical methods for solving Initial-value and boundary-value are different.
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

5. Runge-Kutta Methods Focus on methods to solve first order ODE
Euler’s Method
Huen and Midpoint methods
Classic 4th-order R-K method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

6. Euler’s Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

7. Euler’s Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

8. Euler’s Method Example: True: h = 0.5 Numerical Methods Prof. Jinbo Bi
Lecture 16
Prof. Jinbo Bi
CSE, UConn

9. Euler’s Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

10. Error Analysis Global truncation errors
Local truncation - from application of Euler’s method over a single step
Propagation truncation - from approximation produced during previous steps
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

11. Error Analysis Start with Taylor Series expansion
Local truncation error
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

12. Error Analysis Global Truncation Error Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

13. Euler’s Method Not very accurate
Reducing the step size can improve accuracy but will also increase computation
Can be useful for quick analysis (assuming large step size)
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

14. Euler’s Method Improvements
Higher order Taylor Series
Heun’s Method
Midpoint/Improved Polygon Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

15. Higher order Taylor Series
If f is a function of both the dependent and independent variables, you need to calculate the partial derivatives
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

16. Heun’s Method
Main drawback of Euler’s method is that it assumes that the derivative at the beginning of the interval is the same across the interval
A better estimate involves looking at the derivatives at both ends of the interval and then averaging
Heun method uses the average of the derivatives at the points (xi, yi) and (xi+1, yi+1) to compute yi+1. Since, the derivative at (xi+1, yi+1) is unknown, first to estimate yi+1, then using the average derivative to compute yi+1.
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

17. Heun’s Method
Local truncation error is O(h3) and global truncation error is O(h2)
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

18. Heun’s Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

19. Heun’s Method
Can further use yi+1 to re-estimate the derivative at (xi+1, yi+1) and then obtain a better result. The iteration can go on until an accurate result is achieved
Further iterations may not converge on the true answer but it will converge on a finite truncation error
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

20. Heun’s Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

21. Midpoint Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

22. Midpoint Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

23. Midpoint Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

24. Runge-Kutta Methods Numerical Methods Prof. Jinbo Bi Lecture 16
CSE, UConn

25. The principles of Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

26. The principles of Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

27. The principles of Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

28. The principles of Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

29. The principles of Runge-Kutta Method
Heun method
Midpoint method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

30. Third-order Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

31. Classic 4th-order Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

32. Classic 4th-order Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

33. Classic 4th-order Runge-Kutta Method
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

34. Comparison of Runge-Kutta Methods
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

35. Systems of ODEs
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

36. Higher-order ODE Numerical Methods Prof. Jinbo Bi Lecture 16
CSE, UConn

37. Systems of ODEs Must be careful in determining slopes
Can use same higher-order Runge Kutta methods with systems of ODEs
Must be careful in determining slopes
First find slopes at the initial value (k1)
Then slopes at the midpoints (k2)
Then refine slopes at the midpoint (k3)
Then find slopes at the endpoint (k4)
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

38. Systems of ODEs
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

39. Systems of ODEs
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn

40. Next class Ordinary Differential Equations Read Chapter 25, 26
Numerical Methods
Lecture 16
Prof. Jinbo Bi
CSE, UConn