1. MATH 175: NUMERICAL ANALYSIS II
Lecturer: Jomar F. Rabajante
2nd Sem AY
IMSP, UPLB

2. 2nd Method: Taylor Method of Order k
Consider Taylor expansion: From the Taylor Series we can have the Taylor Method of Order k:

3. 2nd Method: Taylor Method of Order k
The Taylor Method of order k has global error of order k. If a method is of order p, then if we reduce h by half, the global error will be divided by 2p. This means that ODE methods of arbitrary order exist. But, Taylor Method is only used for specialized purposes since it requires extra effort in computing derivatives.

4. 3rd & 4th Method: Trapezoid Method O(h2)
Implicit Trapezoid Method:
Explicit Trapezoid Method (also called improved or modified Euler’s Method):

5. We call these methods as PREDICTOR-CORRECTOR METHODS.
Explicit Trapezoid Method (also called improved or modified Euler’s Method):
Implicit Trapezoid Method is the Corrector
Predictor
We call these methods as PREDICTOR-CORRECTOR METHODS.

6. 5th & 6th Method: Midpoint Method O(h2)
Implicit Midpoint Method (similar to Euler’s):
Explicit Midpoint Method:
where

7. 7th Method: Runge-Kutta Methods O(hm) for m<4; we need more stages to get higher order (for m>4)
Euler, Trapezoid and Midpoint Methods are R-K methods
Explicit R-K method:
R-K stages

8. 7th Method: Runge-Kutta Methods
The famous R-K 4: (4th order R-K)
R-K 4 offers a good balance between discretization and roundoff error.

9. OTHER METHODS (to be discussed later)
Variable Step-size methods (Adaptive)
Methods for Stiff ODEs
Multistep Methods
We will now first focus on solving system of ODEs and higher-order ODEs.

10. SYSTEM OF ODEs
Example: 1. Using Euler’s Method:

11. SYSTEM OF ODEs Using Euler’s Method: let h=0.1
(Solution with crude sensitivity/well-posedness analysis)
I also perturbed the parameter: t z w
pert z
pert w
diff z
diff w 5 2
5.01
2.01
0.01
0.1
5.2
3.1
5.211
0.011
0.2
5.51
4.55
0.3
5.965
6.466
0.4
6.6116
9.0023
0.5
0.6
0.7
0.8
0.9 1
0.1301

12. t z w
pert z
pert w
diff z
diff w 5 2
5.01
2.01
0.01
0.1
5.2
3.1
5.211
3.2145
0.011
0.1145
0.2
5.51
4.55
0.3
5.965
6.466
0.4
6.6116
9.0023
0.5
0.6
0.7
0.8
0.9 1
Do sensitivity analysis since our solution is prone to round-off error and HUMAN error.
Perturbed parameter:

13. However, even though we got a solution, we should still improve this by making our h smaller. (and use a more sophisticated method)

14. Using h=0.001 w z

15. The famous R-K 4 for system of ODEs

16. SYSTEM OF ODEs
Example: 2. Using R-K 4: To be done in the laboratory.

17. Using R-K 4 in Berkeley Madonna

18. HIGHER-ORDER ODEs
Example 1: Convert this in to a system of ODEs:

19. HIGHER-ORDER ODEs
Example 2: Convert this in to a system of ODEs: