MATH 175: NUMERICAL ANALYSIS II

1. MATH 175: NUMERICAL ANALYSIS II Lecturer: Jomar F. Rabajante 2ndSemAY2012-2013 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. 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 MethodsO(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 I also perturbed the parameter: • Using Euler’s Method: let h=0.1 (Solution with crude sensitivity/well-posedness analysis)

12. 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 4for 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: