If boundary values are necessary, they are called mixed initial-boundary value problems. Again, the simplest prototypes of these IV problems are:

Transcription

1 3. Iiial value problems: umerical soluio Fiie differeces - Trucaio errors, cosisecy, sabiliy ad covergece Crieria for compuaioal sabiliy Explici ad implici ime schemes Table of ime schemes Hyperbolic ad parabolic PDEs are iiial value (IV) or marchig problems: The soluio is obaied by usig he kow iiial values ad marchig or advacig i ime. If boudary values are ecessary, hey are called mixed iiial-boudary value problems. Agai, he simples prooypes of hese IV problems are: u = " c u x wave or advecio (hyperbolic) equaio, wih soluio u( x, ) = u( x c,0), showig ha he iformaio moves alog he characerisics x-c=cos, ad u " u = x he diffusio equaio, a parabolic equaio. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM

2 Fiie differece mehod: We ake discree values for x ad : x =Δx, =Δ. The soluio of he fiie differece equaio is also defied a he discree pois (Δx, Δ): = ( x, ). We will use small u o deoe he soluio of he PDE (coiuous) ad capial o deoe he soluio of he Fiie Differece Equaio (FDE), a discree soluio. Cosider agai he advecio equaio u = " c u x. Suppose ha we choose o approximae his PDE wih he followig FDE (called upsream scheme ): + c + " " x = 0 (.3) Noe ha boh differeces are o-ceered wih respec o he poi (Δx, Δ). We should ow ask wo fudameal quesios: ) Is his FDE cosise wih he PDE? ) For ay give ime >0, will he soluio of he FDE coverge o u as Δx 0, Δ 0? Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM

3 Le s clarify he quesios: ) We say ha he FDE is cosise wih PDE if, i he limi whe Δx 0, Δ 0 he FDE coicides wih he PDE. Obviously, his is a firs requireme ha he FDE should fulfill if is soluios are goig o be good approximaios of he soluios of he PDE. The differece bewee he PDE ad he FDE is he discreizaio error or local (i space ad ime) rucaio error. Cosisecy is raher simple o verify: Subsiue by u i he FDE, ad evaluae all erms usig a Taylor series expasio ceered o he poi (,), ad he subrac he PDE from he FDE. If he differece (or local rucaio error ) goes o zero as Δx 0, Δ 0, he he FDE is cosise wih he PDE. Example: We verify he cosisecy of + c + " " x u + c u x = 0 = 0 wih by performig a Taylor series expasio aroud he poi x, : Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 3

4 + " u = ( u + u " + u +...) x u = ( u u " x + u +...) " (.) x xx Replace i he FDE he Taylor series (.) + c + " " x = 0 " " x ( u + u cux cuxx +...) = Whe we subrac he PDE (u + cu x ) = 0 0 (.) we ge he (local) rucaio error " " x = u # cuxx + higher order erms = 0( " ) + 0( " x) (.3) So ha lim 0 # " # 0, " x# 0. Therefore he FDE is cosise. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 4

5 Noe ha boh he ime ad he space rucaio errors are of firs order, because he fiie differeces are uceered i boh space ad i ime (o ceered aroud aroud x, ). Trucaio errors for ceered differeces are secod order, ad herefore ceered differeces are more accurae ha uceered differeces. See schemaic Fig. 3.a ad he leapfrog scheme, based o ceered differeces i space ad i ime, laer i his secio. Fig. 3.a: Schemaic of ceered (" u / " ) # ( u+ u ) /( $ ), forward ( u / ) " ( u+ # u) / $ ad backward (" u / " ) # ( u u ) / $ fiie differeces esimaig he ime =. The hree esimaes are as derivaive u / a ime cosise wih u / sice hey all coverge o u / " 0. However, he slope calculaed from ceered differeces is much closer o he exac derivaive because is rucaio errors are secod order. u() forward ceered exac backward - + Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 5

6 The secod quesio, covergece, i.e., wheher ( x, ) " u( x, ) whe x " x, ", x " 0, " 0 is of evide pracical imporace, bu ca oly be aswered afer cosiderig aoher problem, ha of compuaioal sabiliy. u Cosider agai he advecio equaio = " c u x, which has he soluio u( x, ) = u( x c,0), show schemaically i Fig. 3.b (he iiial shape of u raslaes wih velociy c). Fig. 3.b: Schemaic of he soluio of he wave equaio movig alog he characerisic Δx=cΔ Δ =0 x Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 6

7 The upsream FDE + c + " " x = 0 ca be wrie as + = ( µ ) + µ (.4) where c µ = x is he Coura umber. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 7

8 Assume ha 0 < µ = c x <, as i Fig. 3.3a. + x - x + = ( µ ) + µ shows ha he FDE The soluio a he ew ime level + is ierpolaed bewee he values ad - (where he sars are). I his case he advecio scheme works he way i should, because we kow he rue soluio is i bewee hose values (movig alog he characerisic). However, if his codiio is o saisfied, ad µ = c x > (as i Fig. 3.3b) or if + x - x Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 8

9 c µ = < 0 (as i Fig. 3.3c), x + x - x he he value of + is exrapolaed from he values ad -. The problem wih exrapolaio is ha he maximum absolue value of he soluio icreases wih each ime sep. Takig absolue values of = ( µ ) + µ + ad leig = max, we ge + " µ + µ, so ha + { " µ + µ } The + if ad oly if 0 µ. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 9

10 If he codiio 0 µ is o saisfied, he he soluio is o bouded ad i grows wih. If we le, x " 0 wih µ=cos, i oly makes higs worse, because he ". I pracice, if he codiio 0 µ is o saisfied, he FDE blows up i a few ime seps, faser for oliear problems. We defie ow compuaioal sabiliy: we say ha a FDE is compuaioally sable if he soluio of he FDE a a fixed ime =Δ remais bouded as Δ 0. We ca ow sae he fudameal Lax-Richmyer heorem: Give a properly posed liear iiial value problem, ad a fiie differece scheme ha saisfies he cosisecy codiio, he he sabiliy of he FDE is he ecessary ad sufficie codiio for covergece. The heorem is useful because i allows us o esablish covergece by examiig separaely he easier quesios of cosisecy ad sabiliy. We are ieresed i covergece o because we wa o le, x " 0, bu because we wa o make sure ha if, x are small, he he errors u( x, ) " (accumulaed or global rucaio errors a a fiie ime) are accepably small. To deermie he ecessary codiio for sabiliy of he FDE () we used he crierio of he maximum mehod. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 0

11 We ca also use he crierio of he maximum o sudy he sabiliy codiio of he followig FDE which approximaes he parabolic diffusio equaio u " u = x " " " = # # x (.5) The verificaio of cosisecy is immediae. Noe ha, because he differeces are ceered i space bu forward i ime, he rucaio error is firs order i space ad secod order i ime We ca wrie (.5) as O( ) O( x) +. + = µ + + ( µ) + µ µ = " where " x. If we ake absolue values, ad le = max, we ge + { µ + " µ + µ } (.6) If all he erms iside he absolue values are posiive, we ca drop hem, ad ge +. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM

12 So we obai a codiio 0 µ / o isure ha he soluio remais bouded as, i.e., as he ecessary codiio for sabiliy of he FDE. Exercise : The codiio o he wave equaio 0 µ for he upsream FDE is ierpreed as he ime sep should be chose so ha a sigal cao ravel more ha oe grid size i oe ime sep. Give a physical ierpreaio of he sabiliy codiio ad he equivale Coura umber # µ = " / # x for he diffusio equaio. foruaely, he crierio of he maximum, which is iuiively very clear, ca oly be applied i very few cases. I mos FDE some coefficies of he equaios aalogous o (.6) are egaive, ad he crierio cao be applied. Aoher sabiliy crierio ha has much wider applicaio is he vo Neuma sabiliy crierio: Assume ha he boudary codiios allow expasio of he soluio of he FDE i a appropriae se of eigefucios. For simpliciy we will assume a expasio io Fourier series (e.g., periodic BC): i ( x, ) = Z e k i x k k (.7) The space variable, x, ad he wave umber k ca be mulidimesioal, e.g., x = ( x, x, x3), k = ( k, k, k3). The Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM

13 depede variable ca also be a vecor for a sysem of equaios. Le x = x (or = ( x, x, 3 x3) x ). We defie p as he discree wave umber for he fiie differeces space: p = kx = "x L. p is o-dimesioal ad varies bewee zero (for ifiiely log waves) ad (for he shores wave L = x ) For mulidimesioal problems, p = (k x,k x,k 3 x 3 ). Le =. The he Fourier expasio is Z ip = pe (.8) p (where for muliple dimesios pi = p + p + p3 3 ). Whe we subsiue his Fourier expasio io a liear FDE, we obai a sysem of equaios for each waveumber p Z + p = G Z p p G p is a amplificaio marix ha, whe applied o he p Fourier compoe of he soluio a ime Δ advaces i o he ime (+)Δ. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 3

14 G p depeds o p, Δ ad Δx. If we kow he iiial codiios ip = Z pe (.9), 0 0 p he he soluio of he FDE is Z p = (G p ) Z p 0 (.0). sice we ca plug i back io = Z p e ip ad compue he soluio. p Therefore, sabiliy, i.e., boudedess of he soluio for ay permissible iiial codiio a ay fixed ime, is guaraeed if he marix G p ( ) is bouded for all p whe Δ 0 ad. So, for some orm, we mus have. ( G p ) < M for all p, as If G p ( ) = max i " i is he specral radius of G, i.e., he maximum eigevalue of G, he i ca be show ha for ay orm, Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 4

15 ( ) " # G p $ " # % & Gp $ % & G p (.) The a ecessary codiio for sabiliy of a FDE, ad herefore a ecessary codiio for covergece, is ha lim [ ( G)] = fiie = " # 0, " # e CONST (.) The (G) " # $ (G) CONST / % & " e CONST ' = e ( + CONST ' Or ( G) " + O( # ) (.3), So, he vo Neuma ecessary codiio for compuaioal sabiliy saes ha he maximum eigevalue of he amplificaio marix should be max " + O(#). Noe: The erm O(Δ) allows bouded growh wih ime if his growh is legiimae, i.e., if i arises from a physical isabiliy prese i he PDE. If he exac soluio grows wih Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 5

16 ime, he he FDE cao boh saisfy (G) " ad be cosise wih he PDE. However, i pracice we us check ha max ". Noe: Sufficie codiios are very complicaed, ad are kow oly for special cases. I pracice i is geerally observed ha elimiaig he equal sig i (.3) is eough o esure compuaioal sabiliy. Noe: I priciple his mehod ca also be used o sudy he sabiliy of he boudary codiios, if hey are appropriaely icluded i he amplificaio marix. I pracice his is complicaed, ad compuaioal sabiliy of he BC is usually obaied by esurig well-posedess, ad esig he sabiliy experimeally. For simple caoical equaios he vo Neuma crierio ca be simplified by assumig soluios wih a amplificaio facor p raher ha a marix. The soluio for he amplificaio facor p he coicides wih he eigevalues of he amplificaio marix, ad he vo Neuma sabiliy crierio is ". Example of vo Neuma s sabiliy crierio: u PDE: + c u x = 0 FDE: + c + " " x = 0 (upsream scheme) Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 6

17 We sudied already he cosisecy, ad used he crierio of he maximum o ge a sufficie codiio for sabiliy. Le s apply ow he vo Neuma crierio: Assume = Z p e ip = A " p p ( ) e ip We replace i he FDE, cacel ou he commo facor Ae ip ad obai p ( ) + p " ( ) p # ( + c ) (" e "ip ) p #x = 0, for all p so ha a every sep he soluio ges amplified by a facor p = " µ(" e "ip ) #$ % & The amplificaio facor p is he x amplificaio marix G, so p = "(G), ad he sabiliy codiio is p " for all wave umbers p. We eed o esimae he maximum value of he specral radius (or amplificaio facor i his case): p = " µ(" e "ip ) = " µ(" cos p + isi p) or p = (" µ(" cos p)) + µ si p Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 7

18 We make use of he rigoomerical relaioships p p p p cos p = cos si, si p = si cos ad obai p = " 4µ( " µ)si p Recall ha he o-dimesioal waveumber p varies bewee zero (for ifiiely log waves) ad (for he shores wave L = x ). The maximum value of si p = is for p = ( L = x ). µ ( µ ), is a parabola whose maximum value is 0.5 whe µ=0.5. So he amplificaio facor squared will remai less or equal o as log as 0 µ. This coicides wih he codiio we obaied from he crierio of he maximum (ad also wih he oio ha we should o exrapolae bu ierpolae he ew values a ime level =(+)Δ, cf. schemaic Fig. 3.3). I is impora o oe ha he amplificaio facor p idicaes how much he ampliude of each waveumber p will decrease or icrease wih each ime sep. The upsream scheme decreases he ampliude of all Fourier wave compoes of he soluio, sice, if 0<µ<, p <. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 8

19 This is herefore a very dissipaive FDE: i has srog umerical diffusio. Fig. 3.4 shows he decrease i ampliude whe usig he upsream scheme afer oe ime sep ad afer 00 ime seps for each waveumber p, usig a Coura umber µ=0.5, a reasoable pracical choice. Sice is rucaio errors are large (of firs order), he upsream is i geeral o recommeded excep for special siuaios (e.g., for ouflow boudary codiios, or whe modified i such a way ha he dissipaio rae becomes lower). A aleraive, less dampig scheme kow as Masuo or Euler-backward, frequely used i combiaio wih he Leap-Frog scheme is also show. (Noe ha a dowsream scheme (Fig. 3.3c) is usable). Masuo scheme (a.k.a. Euler-backward), a predicorcorrecor scheme: * " + c + "x = 0 predicor sep (forward-ceered) + " + c * + * "x = 0 correcor sep Exercise: calculae he amplificaio facor p Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 9

20 Fig. 3.4: Amplificaio facor of wave compoes of he wave equaio usig eiher he upsream FDE, ad he Masuo or Euler-backward schemes wih µ=0.5. L is he wavelegh i uis of x. Maciosh HD:sers:ekalay:Documes:AOSC64- DOCS:PPTClasses:ch3 FiieDiffereces.docCreaed o Sepember 6, 007 9:05 AM 0