1. Numerical Analysis – Differential Equation
Hanyang University
Jong-Il Park

2. Differential Equation

3. Solving Differential Equation
Ordinary D.E.
Partial D.E.
Linear eg.
Nonlinear eg.
Initial value problem
Boundary value problem
Usually no closed-form solution
linearization
numerical solution

4. Discretization in solving D.E.
Errors in Numerical Approach
Discretization error
Stability error y
Exact sol. t
Grid Points

5. Errors
Total error
truncation
round-off
increase as as
trade-off

6. Local error & global error
The error at the given step if it is assumed that all the previous results are all exact
Global error
The true, or accumulated, error

7. Useful concepts(I) Useful concepts in discretization Consistency Order
Convergence

8. Useful concepts(II) stability Consistent Converge stable unstable

9. Stability Stability condition eg. Exact sol. Euler method
Amplification factor
For stability

10. Implicit vs. Explicit Method
eg.
= f
Explicit :
Implicit :
h large y y ye
h small
h increase
explicit t
implicit t
“conditionally stable”
“stable”

11. Modification to solve D.E.
Modified Differential Eq.
Diff.
eq.
Discretization
Modified
D.E.
Discretization by Euler method
<Consistency check>
<Order>

12. Initial Value Problem: Concept

13. Initial value problem Initial Value Problem Simultaneous D.E.
High-order D.E.

14. Well-posed condition

15. Taylor series method(I)
Truncation error

16. Requiring complicated
Taylor series method(II)
High order differentiation
Implementation
Complicated computation
<Type 1>
<Type 2>
.... y t
More computation
accuracy y
Requiring complicated
source codes t
Less computation
accuracy

17. Euler method(I) Euler Method Talyor series expansion at to y .... ....

18. Euler method(II)
Error
Eg. y’ =-2x3+12x2-20x+8.5, y(0)=1

19. Euler method(III) Generalizing the relationship Error Analysis
Euler’s approx.
truncation error
Error Analysis
Accumulated truncation error
; 1st order

20. Eg. Euler method

21. Modified Euler method: Heun’s method
Modified Euler’s Method
Why a modification?
error
modify
Predictor
Average slope
Corrector

22. Heun’s method with iteration
significant
improvement

23. Error analysis Error Analysis
Taylor series
Total error
truncation
3rd order
; 2nd order method
※ Significant improvement over Euler’s method!

24. Eg. Euler vs. Modified Euler
Euler Method
improvement

25. Runge-Kutta method Runge-Kutta Method The idea Simple computation
very accurate
The idea
where

26. Second-order Runge-Kutta method①
Taylor series expansion ② ③ ④
③→①
Equating ② and ④

27. Modified Euler - revisited
set P2 P1
 Modified Euler method
Modified Euler method is a kind
of 2nd-order Runge-Kutta method.

28. Other 2nd order Runge-Kutta methods
Midpoint method
Ralston’s method

29. Comparison: 2nd order R-K method

30. Comparison: 2nd order R-K method
Eg. y’ =-2x3+12x2-20x+8.5, y(0)=1

31. 4-th order Runge-Kutta methods
Fourth-order Runge-Kutta
Taylor series expansion to 4-th order
accurate
short, straight, easy to use P4 P3 P1 P2
※ significant improvement over
modified Euler’s method

32. Runge-Kutta method

33. Eg. 4-th order R-K method
Significant
improvement

34. Discussion
Better!

35. Comparison
(5th order)