1. CSE245:Lec4 02/24/2003

2. Integration Method Problem formulation

3. Integration Method

4. An example (1) (2) From (1)   

5. Integration Method

6. (3)

7. Adams-Bashforth formula  0 =0 The first order Adams-Bashforth formula (forward Euler) The second order Adams-Bashforth formula

8. Adams-Moulton formula 0 00 0 The first order Adams-Moulton formula (backward Euler) The second order Adams-Moulton formula (trapezoidal)

9. Properties Definition 1: Consistency If M=T/h is the number of steps, Global Error can be estimated by

10. Properties Definition 2: Convergence Definition 3: Stability If  h 0 &k , for any two initial condition x 0, y 0 and h=T/M<h 0

11. Stabilities Backward Euler

12. Stabilities Froward Euler

13. Stabilities Trapezoidal

14. Stabilities A General Integration Equation (4)

15. Stabilities Assume (4) stable M i =1, stable M i >1, unstable

16. A-stable Definition: a method is A-stable, if the region of absolute stability includes the entire left-hand plane. Dahlquist’s theorem: 1.An A-stable method cannot exceed 2 nd order accuracy. 2.The most accurate A-stable method (smallest LTE) is the trapezoidal formula.

17. Non-linear Circuit Newton-Ralphson (5) (6) (6)+(5)(7) Newton-Ralphson iteration (8)

18. Non-linear Circuit Newton-Ralphson (8)+(7) Rewrite (4) (9) (10) Compare (9) with (10)

19. Oscillation Reason1: x (0) is not close to x *. Reason2: error in derivative cause oscillation.

20. Oscillation Implications 1.Model equations must be continuous, and continuous derivatives. 2.Watch out the floating nodes. 3.Provide good initial guess for x (0). 4.Make sure, has  greater than model error.