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.... ( ()
Mahemacal Formulao Taylor seres epaso abou po -h h - + efe h = + =, he ( ) ( ) 6... () For po, 6... (4)
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
Mahemacal Formulao Taylor seres epaso abou po -h h - + Kow as forwar fferece, upw or upsream (6)
Mahemacal Formulao Taylor seres epaso abou po -h h - + efe h = - = -, he ( ) ( ) 6... (7) For po, 6... (8)
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
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
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
Mahemacal Formulao backwar rue ceral forwar omparso of forwar, backwar a ceral fferece appromao for a smooh, couous fuco
Mahemacal Formulao How o appromae ( / )? ombe Equaos (4) a (8) 4 4... () Trucao error Whch s orer accuracy O( )
Mahemacal Formulao Trucao Error 8 4 7 For a fuco wha s he rucao error, TE, of he FE for = a 0.0? 4 4... () Trucao error
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 = 96 9 9 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 6 0.006 *9
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 - +
Mahemacal Formulao Trase Problem Nomeclaure me, j,k (4) Epa Eq. (), reco y reco z reco A me level, A me level + (5) (6)
Mahemacal Formulao Trase Problem Eqs (5) a (6) ca be epae space usg ceral fferece, Eq. () (7) (8)
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
Mahemacal Formulao Trase Problem I me reco, sugges bewee a +, use * How o relae Eq (9) wh Eqs (7) a (8)? (9) * +
Mahemacal Formulao Trase Problem Le s efe a weghg scheme, ombe Eqs (7-9), we have a complee algebrac equao, ) ( * (0) ) ( () O(, )
Mahemacal Formulao Trase Problem For =, Fully Implc Meho ( ) Spaal evaluao po Nee mar solver - + +
Mahemacal Formulao Trase Problem For = /, rak Ncholso Meho ( ) Implc Nees mar solver - Spaal evaluao po + +
Mahemacal Formulao Trase Problem For = 0, Eplc Meho ( ) Spaal evaluao po + - +
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.) 0.995 () 0.95 0.05 (0) (0) 0.607 (00) 0.00674 0.04975 (.) ( )
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 0.995. 0 ( 0.04975) ( )( 0.05)
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 = 0.75. 0.9 0.8 0.7 0.6 0.5 0.4 0. 0. 0. 0 0 0 40 60 80 00
Mahemacal Formulao Trase Problem Eercse Gve a b c wha s he opmal value of Soluo