2.29 Numerical Fluid Mechanics Spring 2015 Lecture 12
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1 REVIEW Lctur 11: Numrical Fluid Mchaics Sprig 2015 Lctur 12 Fiit Diffrcs basd Polyomial approximatios Obtai polyomial (i gral u-qually spacd), th diffrtiat as dd Nwto s itrpolatig polyomial formulas Triagular Family of Polyomials (cas of Equidistat Samplig, similar if ot quidistat) Lagrag polyomial (Rformulatio of Nwto s polyomial) 0 0, x x f ( x) L ( x) f ( x ) with L ( x) x x Hrmit Polyomials ad Compact/Pad s Diffrc schms (Us th valus of th fuctio ad its drivativ(s) at ods) Fiit Diffrc: Boudary coditios s Diffrt approx. at ad ar th boudary => impacts global ordr of accuracy ad liar systm to b solvd b m u i m i i x ir x i i p q a u Numrical Fluid Mchaics PFJL Lctur 12, 1
2 Numrical Fluid Mchaics Sprig 2015 Lctur 12 REVIEW Lctur 11: Fiit Diffrc: Boudary coditios Diffrt approx. at ad ar th boudary => impacts liar systm to b solvd Fiit-Diffrcs o No-Uiform Grids ad Uiform Errors: 1-D If o-uiform grid is rfid, rror du to th 1 st ordr trm dcrass fastr tha that of 2 d ordr trm Covrgc bcoms asymptotically 2 d ordr (1 st ordr trm cacls) Grid-Rfimt ad Error stimatio Estimatio of th ordr of covrgc ad of th discrtizatio rror Richardso s xtrapolatio ad Itrativ improvmts usig Roombrg s algorithm Numrical Fluid Mchaics PFJL Lctur 12, 2
3 FINITE DIFFERENCES Outli for Today Fiit-Diffrcs o No-Uiform Grids ad Uiform Errors: 1-D Grid Rfimt ad Error Estimatio Fourir Aalysis ad Error Aalysis Diffrtiatio, dfiitio ad smoothss of solutio for ordr of spatial oprators Stability Huristic Mthod Ergy Mthod Vo Numa Mthod (Itroductio) : 1st ordr liar covctio/wav q. Hyprbolic PDEs ad Stability Exampl: 2d ordr wav quatio ad wavs o a strig Effctiv umrical wav umbrs ad disprsio CFL coditio: Dfiitio Exampls: 1st ordr liar covctio/wav q., 2d ordr wav q., othr FD schms Vo Numa xampls: 1st ordr liar covctio/wav q. Tabls of schms for 1st ordr liar covctio/wav q. Numrical Fluid Mchaics PFJL Lctur 12, 3
4 Rfrcs ad Radig Assigmts Lapidus ad Pidr, 1982: Numrical solutios of PDEs i Scic ad Egirig. Sctio 4.5 o Stability. Chaptr 3 o Fiit Diffrc Mthods of J. H. Frzigr ad M. Pric, Computatioal Mthods for Fluid Dyamics. Sprigr, NY, 3 rd ditio, 2002 Chaptr 3 o Fiit Diffrc Approximatios of H. Lomax, T. H. Pulliam, D.W. Zigg, Fudamtals of Computatioal Fluid Dyamics (Scitific Computatio). Sprigr, 2003 Chaptr 29 ad 30 o Fiit Diffrc: Elliptic ad Parabolic quatios of Chapra ad Caal, Numrical Mthods for Egirs, 2014/2010/2006. Numrical Fluid Mchaics PFJL Lctur 12, 4
5 Grid-Rfimt ad Error stimatio W foud that for a covrgt schm, th discrtizatio rror ε is of th form: p (rcall: ˆ, ( ) 0, ˆ ( ˆ ) 0 ) O( x ) R whr R is th rmaidr Th dgr of accuracy ad discrtizatio rror ca b stimatd btw solutios obtaid o systmatically rfid/coarsd grids p uu x x R -Tru solutio u ca b xprssd ithr as: p uu x R -Thus, th xpot p ca b stimatd: 2 ' (2 ) ' (d to limiat u ad th d 2 qs. to limiat both Δx ad p, hc u 4Δx ) -Th discrtizatio rror o th grid Δx ca b stimatd by: -Good ida: stimat p to chc cod. Is it qual to what it is supposd to b? -Wh solutios o svral grids ar availabl, a approximatio of highr accuracy ca b obtaid from th rmaidr: Richardso Extrapolatio! Numrical Fluid Mchaics PFJL Lctur 12, 5 x u u p 2x 4x log log 2 u x u2 x x x x ) u u 2 1 x 2x x p
6 Richardso Extrapolatio ad Rombrg Itgratio Richardso Extrapolatio: mthod to obtai a third improvd stimat of a itgral basd o two othr stimats Cosidr: I(h) For two diffrt grid spac h1 ad h2: Trapzoidal Rul: I 2 Exampl h 2 h 1 h (grid spac) 2 Assum: ) Richardso Extrapolatio: 2 From two O(h 2 ), w gt a O(h 4 )! Numrical Fluid Mchaics PFJL Lctur 12, 6
7 Rombrg s Itgratio: Itrativ applicatio of Richardso s xtrapolatio I(h) Rombrg Itgratio Algorithm, for ay ordr For Ordr 2 (cas of prvious slid): I h 2 h 1 h 1: O(h 2 ) 2: O(h 4 ) 3: O(h 6 ) 4: O(h 8 ) a b, Icrasig ordr c Icrasig rsolutio Numrical Fluid Mchaics PFJL Lctur 12, 7
8 Rombrg s Diffrtiatio: Itrativ applicatio of Richardso s xtrapolatio D(h) Rombrg Diffrtiatio Algorithm, for ay ordr D D D For Ordr 2 (as prvious slid, but for diffrtiatio): D h 3 h 2 h 1 D h 4D 2,1 D 1,1 Icrasig ordr 1: O(h 2 ) 2: O(h 4 ) 3: O(h 6 ) 4: O(h 8 ) a b, c Icrasig rsolutio Numrical Fluid Mchaics PFJL Lctur 12, 8
9 Fourir (Error) Aalysis: Dfiitios Ladig rror trms ad discrtizatio rror stimats ca b complmtd by a Fourir rror aalysis Fourir dcompositio: Ay arbitrary priodic fuctio ca b dcomposd ito its Fourir compots: ix f ( x) f ( itgr, wavumbr) Usig th orthog. proprty, taig th itgral/ft of f(x): 2 0 i x im x Not: rat at which f with dcays dtrmi smoothss of f (x) 2 1 i x f f ( x) dx (orthogoality proprty) Exampls draw i lctur: si(x), Gaussia xp(-πx 2 ), multi-frqucy fuctios, tc m Numrical Fluid Mchaics PFJL Lctur 12, 9
10 Fourir (Error) Aalysis: Diffrtiatios Cosidr th dcompositios: ix f ( x) f or f ( x, t) f ( t) ix Taig spatial drivativs givs: f x f () t i ix Taig tmporal drivativs givs: r r f d f () t r r t dt ix Hc, i particular, for v or odd spatial drivativs: q 2q 2 ( 1) (ral) q i q 2q ( 1) (imagiary) q i i Numrical Fluid Mchaics PFJL Lctur 12, 10
11 Fourir (Error) Aalysis: Gric quatio Cosidr th gric PDE: f t f x Fourir Aalysis: f ( x, t) f ( t) ix Hc: d f () t dt ix f () t i ix Thus: Ad: d f () t dt i f ( t) f ( t) for i t ixt f ( t) f (0), f ( x, t) f(0) Phas spd : c / i Numrical Fluid Mchaics PFJL Lctur 12, 11
12 Fourir (Error) Aalysis: Gric quatio Gric PDE, FT: Hc: Etc ( x, t) f (0) ixt d f () t dt f f Propagatio: c / i 1, 1 i t x No disprsio f f t x Dcay 2 3 f f 3 i 3 t x f ( t) for i 3 2 Propagatio: /, With disprsio f f t x c i : (Fast) Growth, : (Fast) Dcay 4 q 2q 2 ( 1) (ral) q i q 2q ( 1) (imagiary) q i i Numrical Fluid Mchaics PFJL Lctur 12, 12
13 Fourir Error Aalysis: 1 st drivativs f x I th dcompositio: f ( x, t) f ( t) ix All compots ar of th form: Exact 1 st ordr spatial drivativ: Howvr, if w apply th ctrd fiit-diffrc (2 d ordr accurat): f x 1 1 2x ff = ffctiv wavumbr () t ix 3 2 si( x ) ff x... x 6 Shows th 2 d ordr atur of ctr-diffrc approx. (hr, of by ff ) For low wavumbrs (smooth fuctios): f f () t x ix ix f ( t) i f ( t) i ix i ( x x) i ( x x) ix ix si( x) ix i i ff x 2x 2x x f f si( x) whr ff (uiform grid rsolutio x) x ix ix ix Numrical Fluid Mchaics PFJL Lctur 12, 13
14 Diffrt approximatios CDS, 2 d ordr: CDS, 4 th ordr: Pad schm, 4 th ordr: Fourir Error Aalysis, Cot d: Effctiv Wav umbrs ff ff x ix hav diffrt ffctiv wavumbrs 3 2 si( x ) x... x 6 si( x) 4 cos( x) 3x 3i si( x) iff 2 cos( x) x max Δx Sprigr. All rights rsrvd. This cott is xcludd from our Crativ Commos lics. For mor iformatio, s Sourc: Lomax, H., T. Pulliam, D. Zigg. Fudamtals of Computatioal Fluid Dyamics. Sprigr, Not that ff is boudd: Numrical Fluid Mchaics 0 ff max x max PFJL Lctur 12, 14
15 Diffrt approximatios Fourir Error Aalysis, Cot d Effctiv Wav Spds Cosidr liar covctio quatios: For th xact solutio: For th umrical sol.: if x ix also lad to diffrt ffctiv wav spds: which w ca solv xactly (our itrst hr is oly rror du to spatial approx.) um f. ( t) f (0) i ff c t umrical ixi ff c t i ( xcff t) (, ) (0) (0) f x t f f cff ff ff (dfiig ff i ff c i cff ) c f t f c x 0 ixt i ( xct) f ( x, t) f(0) f(0) (sic i c) um. ix um. ix d f ix um. um. ( ) ( ) ( ) ff f f t f t c f t c i dt x c ff c ix Oft, c ff / c < 1 => umrical solutio is too slow. Sic, c ff is a fuctio of th ffctiv wavumbr th schm is disprsiv (v though th PDE is ot) Sprigr. All rights rsrvd. This cott is xcludd from our Crativ Commos lics. For mor iformatio, s Sourc: Lomax, H., T. Pulliam, D. Zigg. Fudamtals of Computatioal Fluid Dyamics. Sprigr, Numrical Fluid Mchaics PFJL Lctur 12, 15
16 Evaluatio of th Stability of a FD Schm: Thr mai approachs Rcall: ( ) ˆ ( ˆ ) ˆ ( ) Stability: ˆ 1 Cost. (for liar systms) Huristic stability: Stability is dfid with rfrc to a rror (.g. roud-off) mad i th calculatio, which is dampd (stability) or grows (istability) Huristic Procdur: Try it out Itroduc a isolatd rror ad obsrv how th rror bhavs Rquirs a xhaustiv sarch to sur full stability, hc maily iformatioal approach Ergy Mthod Basic ida: x x x x Fid a quatity, L 2 orm.g. 2 Shows that it rmais boudd for all Lss usd tha Vo Numa mthod, but ca b applid to oliar quatios ad to o-priodic BCs x x Vo Numa mthod (Fourir Aalysis mthod) Numrical Fluid Mchaics PFJL Lctur 12, 16
17 Evaluatio of th Stability of a FD Schm Ergy Mthod Exampl Cosidr agai: c 0 t x 1 c t x A possibl FD formula ( upwid schm for c>0): (t = Δt, x = Δx) which ca b rwritt: 1 c t (1 ) 1 with x t x Drivatio rmovd du to copyright rstrictios. For th rst of this drivatio,plas s quatios 2.18 through 2.22 i Durra, D. Numrical Mthods for Wav Equatios i Gophysical Fluid Dyamics. Sprigr, ISBN: Numrical Fluid Mchaics PFJL Lctur 12, 17
18 Evaluatio of th Stability of a FD Schm Ergy Mthod Exampl Drivatio rmovd du to copyright rstrictios. For th rst of this drivatio, plas s quatios 2.18 through 2.22 i Durra, D. Numrical Mthods for Wav Equatios i Gophysical Fluid Dyamics. Sprigr, ISBN: Numrical Fluid Mchaics PFJL Lctur 12, 18
19 Widly usd procdur Vo Numa Stability Assums iitial rror ca b rprstd as a Fourir Sris ad cosidrs growth or dcay of ths rrors I thortical ss, applis oly to priodic BC problms ad to liar problms Suprpositio of Fourir mods ca th b usd Agai, us, but for th rror: i x ( x, t) ( t) Big itrstd i rror growth/dcay, cosidr oly o mod: i x t i x ( t) whr is i gral complx ad fuctio of : ( ) Strict Stability: Th rror will ot to grow i tim if t 1 i othr words, for t = Δt, th coditio for strict stability ca b writt: t t 1 or for, 1 vo Numa coditio Norm of amplificatio factor ξ smallr or qual to1 Numrical Fluid Mchaics PFJL Lctur 12, 19
20 MIT OpCoursWar Numrical Fluid Mchaics Sprig 2015 For iformatio about citig ths matrials or our Trms of Us, visit:
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