NUMERICAL DIFFERENTIAL 1

Transcription

1 NUMERICAL DIFFERENTIAL

2 Ruge-Kutta Metods Ruge-Kutta metods are ver popular ecause o teir good eiciec; ad are used i most computer programs or dieretial equatios. Te are sigle-step metods as te Euler metods.

3 Ruge-Kutta Metods To cove some idea o ow te Ruge-Kutta is developed let s loo at te derivatio o te d order. Two estimates a

4 Ruge-Kutta Metods Te iitial coditios are: d d 0 Te Talor series epasio 0 d d d! d

5 Ruge-Kutta Metods 5 From te Ruge-Kutta Te deiitio o te uctio Epad te et step a a a

6 Ruge-Kutta Metods 6 From te Ruge-Kutta Compare wit te Talor series a a Uows

7 Ruge-Kutta Metods 7 Te Talor series coeiciets ( equatios/ uows) I ou select a as I ou select a as Note: Tese coeiciet would result i a modiied Euler or Midpoit Metod a a a

8 Ruge-Kutta Metod ( d Order) Eample Cosider d d Te iitial coditio is: Te step size is: Use te coeiciets 0. a 0 Eact Solutio 8

9 Ruge-Kutta Metod ( d Order) Eample 9 Te values are i i i i i i

10 Ruge-Kutta Metod ( d Order) Eample Te values are equivalet o Modiied Euler Estimate Solutio Eact Error ' ' * + *' + (*' + )

11 Ruge-Kutta Metod ( d Order) Eample [] Te values are i i i i i i a a

12 Ruge-Kutta Metod ( d Order) Eample [] Te values are Estimate Solutio Eact Error ' ' * + *' + (*' + ) Eact

13 Ruge-Kutta Metods Te Ruge-Kutta metods are iger order approimatio o te asic orward itegratio. Tese metods provide solutios wic are comparale i accurac to Talor series solutio i wic iger order derivatives are retaied. It sould e oted tat te equatios are ot eed to e liear.

14 Ruge-Kutta Metods Metod Equatios Euler (Error o te order ) Modiied Euler (Error o te order ) Heu (Error o te order ) t order Ruge Kutta (Error o te order 5 ) 6

15 Te t Order Ruge-Kutta 5 Te geeral orm o te equatios: 6

16 Te t Order Ruge-Kutta Tis is a ourt order uctio tat solves a iitial value prolems usig a our step program to get a estimate o te Talor series troug te ourt order. Tis will result i a local error o O(D 5 ) ad a gloal error o O(D ) 6

17 t -order Ruge-Kutta Metod 6 i i + / i + 7

18 Ruge-Kutta Metod ( t Order) Eample Cosider d d Te iitial coditio is: Eact Solutio e 0 Te step size is: 0. 8

19 Te t Order Ruge-Kutta 9 Te eample o a sigle step: /

20 Ruge-Kutta Metod Eample ( t Order) Te values or te t order Ruge-Kutta metod () Cage Eact

21 Y Value Ruge-Kutta Metod Eample ( t Order) t Order Ruge-Kutta Metod Te values are equivalet to tose o te eact solutio. I we were to go out to =5. (5) = -.9 (-.) Te error is small relative to te eact solutio Eact t order X Value

22 Y Value Asolute Error Ruge-Kutta Metod Eample ( t Order) A compariso etwee te d order ad te t order Ruge-Kutta metods sow a sligt dierece. Ruge Kutta Compariso Error o te Metods Error d order metod Error t order metod Eact d order t order X Value X Value

23 Te t Order Ruge-Kutta Higer order dieretial equatios ca e treated as i te were a set o irst-order equatios. Ruge-Kutta tpe orward itegratio solutios ca e otai. A more direct solutio ca e otaied repeatig te wole process used i irst-order cases.

24 Te t Order Ruge-Kutta Te geeral orm o te d order equatios:

25 5 Te step sizes are: Te et step would e:

26 6

27 THANK YOU 7