Mathematical Formulation

Transcription

1 Mahemacal Formulao The purpose of a fe fferece equao s o appromae he paral ffereal equao (PE) whle maag he physcal meag. Eample PE: p c k FEs are usually formulae by Taylor Seres Epaso abou a po a eglecg hgher orer erms p () ( where h) () h h h 6 h )! s a arbrary fuco a h s sace bewee pos.... ( ()

2 Mahemacal Formulao Taylor seres epaso abou po -h h - + efe h = + =, he ( ) ( ) 6... () For po, 6... (4)

3 Mahemacal Formulao Taylor seres epaso abou po -h h - + Equao 4 ca be rearrage as: 6... (5) Trucao error Kow as frs orer accuracy O( ) sce s he frs erm of TE

4 Mahemacal Formulao Taylor seres epaso abou po -h h - + Kow as forwar fferece, upw or upsream (6)

5 Mahemacal Formulao Taylor seres epaso abou po -h h - + efe h = - = -, he ( ) ( ) 6... (7) For po, 6... (8)

6 Mahemacal Formulao Taylor seres epaso abou po -h h - + Equao 4 ca be rearrage as: 6... (9) Trucao error Kow as frs orer accuracy O( ) sce s he frs erm of TE

7 Mahemacal Formulao Taylor seres epaso abou po -h h - + Kow as backwar fferece, oww or owsream (0) Forwar a backwar scheme sgfca boh accuracy of appromao a mobly weghg scheme

8 Mahemacal Formulao Sce he Taylor seres epaso s oly val whe 0, he esre s o have small. Therefore hgher orer accuracy s preferre. > ) > ( )... How o crease he accuracy of FE of ( /)? Average Equao (5) a (9) 6... () Whch s orer accuracy Kow as ceral fferece

9 Mahemacal Formulao backwar rue ceral forwar omparso of forwar, backwar a ceral fferece appromao for a smooh, couous fuco

10 Mahemacal Formulao How o appromae ( / )? ombe Equaos (4) a (8) () Trucao error Whch s orer accuracy O( )

11 Mahemacal Formulao Trucao Error For a fuco wha s he rucao error, TE, of he FE for = a 0.0? () Trucao error

12 Mahemacal Formulao Trucao Error (Soluo) 8 4 For a fuco 7 wha s he rucao error, TE, of he FE for = a 0.0? '''' For = For = 0.0 TE TE 0.0 *9 4 4 Eample shows ha a ecrease gr sze wll reuce he rucao error, TE. TE *9 TE *9

13 Mahemacal Formulao Trase Problem ffusvy equao s epresse by: Where s he sperso coeffce a s cocerao Is screze fel or oma s, () +, fuure, ukow, curre, kow -, pas, kow - +

14 Mahemacal Formulao Trase Problem Nomeclaure me, j,k (4) Epa Eq. (), reco y reco z reco A me level, A me level + (5) (6)

15 Mahemacal Formulao Trase Problem Eqs (5) a (6) ca be epae space usg ceral fferece, Eq. () (7) (8)

16 Mahemacal Formulao Trase Problem I me reco: a be appromae by? a? Oe-way vs wo-way coorae oes wha happes a me + oly be affece by values a me level, a o a me level, +? Tme cosere a oe-way coorae Space s referre o as a wo-way coorae

17 Mahemacal Formulao Trase Problem I me reco, sugges bewee a +, use * How o relae Eq (9) wh Eqs (7) a (8)? (9) * +

18 Mahemacal Formulao Trase Problem Le s efe a weghg scheme, ombe Eqs (7-9), we have a complee algebrac equao, ) ( * (0) ) ( () O(, )

19 Mahemacal Formulao Trase Problem For =, Fully Implc Meho ( ) Spaal evaluao po Nee mar solver - + +

20 Mahemacal Formulao Trase Problem For = /, rak Ncholso Meho ( ) Implc Nees mar solver - Spaal evaluao po + +

21 Mahemacal Formulao Trase Problem For = 0, Eplc Meho ( ) Spaal evaluao po + - +

22 Mahemacal Formulao Trase Problem Eercse Gve a oe, wha s he opmal value of? Use = 0.,.0, 0, 00 a =. Soluo e 0.05 (0) 0.05e 0.05 (0.) () (0) (0) (00) (.) ( )

23 Mahemacal Formulao Trase Problem Eercse Gve a oe, wha s he opmal value of? Use = 0.,.0, 0, 00 a =. Soluo e 0.05 ( ). 0. ( ) ( ) ( )( 0.05)

24 Mahemacal Formulao Trase Problem Eercse Gve a oe, wha s he opmal value of? Use = 0.,.0, 0, 00 a =. Soluo e 0.05 urg smulao, a are ukow, ral a error s requre. Wha s he bes guess for? Suppose we choose =

25 Mahemacal Formulao Trase Problem Eercse Gve a b c wha s he opmal value of Soluo