Stability For a stable numerical scheme, the errors in the initial condition will not grow unboundedly with time.

Transcription

1 .3.5. Stablty Aalyss Readg: Taehll et al. Secto 3.6. Stablty For a stable umercal scheme, the errors the tal codto wll ot grow uboudedly wth tme. I ths secto, we dscuss the methods for determg the stablty of F.D. schemes. Ths s very mportat whe desgg a F.D. scheme ad for uderstadg t behavor. There are several methods: Eergy method vo Neuma method Matrx method (for systems of equatos) Dscrete perturbato method (wll ot dscuss) Note: Stablty refers to the F.D. Does ot volve B.C. or I.C. Refers to tme-matchg problems oly The eergy method Read Durra Secto. (hadout, see web l)

2 Ths method s used much less ofte tha the vo Neuma method. It's attractve because t wors for olear problem ad problems wthout perod B.C. The ey s to show that a postve defte quatty le ( u ) s bouded for all. u u We llustrate ths method usg the upstream-forward scheme for wave (advecto) equato + c = 0 : t x u u u u + c = t x + 0 Let µ = c t/ x u = ( µ ) u + µ u + Squarg both sdes ad summg over all grd pots: ( u ) = [( µ ) ( u ) + ( µ ) µ u u + µ ( u ) ] () + Assumg perodc B.C. ( u ) = ( u )

3 ad usg the Schwarz equalty (whch says that for two vectors U ad V, U V U V ), uu ( u) ( u ) = ( u). If µ ( µ ) 0, all coeffcets of RHS terms () are postve, ad we have ( u ) [( µ ) + ( µ ) µ + µ ] ( u ) = ( u ), +.e., the L orm at + s o greater tha that at, therefore the scheme s stable! The codto µ ( µ ) 0 gves µ = c t/ x whch s the stablty codto for ths scheme. As we dscussed earler, t says that waves caot propagate more tha oe grd terval durg oe t order to mata stablty. vo Neuma method Read Taehll et al, Secto I a sese, we have already used ths method to fd stablty of the -D dffuso equato whe we were provg the covergece of the soluto usg FTCS scheme. We foud the 3

4 u + = [ M(t) ] + u 0 where M(t) = amplfcato factor. If M(t) by some measure (M ca be a matrx or a complex umber), the u + u ad the soluto caot grow tme the scheme s stable. Essetally, vo Neuma method expads the F.D.E. a Fourer seres, fds the amplfcato factor ad determes uder what codto the factor s less tha or equal to for stablty. Assumptos:. The equato has to be Lear wth costat coeffcets.. It s assumed that the soluto s perodc. Wth ths method, the depedet varable s decomposed to a complex (or a real) Fourer seres: uxyzt (,,, ) = Ulm (,, )exp[ x ( + ly+ mz ωt)] () lm,, where U s the complex ampltude ad ω = ω R + ω Ι s the complex frequecy. I fact ω R gves the wave propagato speed ad ω Ι gves the growth ad decayg rate. e = e = e e ωt [ ω + ω ] t ω t ω t R I R I Rt e ω I e ω t - phase fucto of Fourer compoets - growth or decay rate 4

5 If ω I >0, the soluto wll grow expoetally tme. Example : -D dffuso equato wth FTCS scheme. u u = µ ( u u + u ) (3) + + K t where µ =. ( x) Let's exame a sgle wave : u = U e ( x ωt) + ( x ωt) ω t = = ( x ωt) ± x ± u U e e u U e e Substtute the above to (3) Ue ( e ) = µ Ue [ e + e ] ( x ωt) ω t ( x ωt) x x ( x ωt ) ω t Ue [ e µ (cos x )] = 0 [ + 4µ s ( /)] = 0 ( x ωt) ω t Ue e x 5

6 For o-trval soluto, we requre e µ ω t + 4 s ( x/) = 0 e µ ω t = 4 s ( x/) Here, t e ω s actually the amplfcato factor, the same as the M dscussed earler. λ t e ω = u + /u - the amplfcato factor. For stablty we requre λ µ 4 s ( x/) µ / as before! I practce, whe µ=/, the soluto (amplfcato factor) swtches betwee ad + for x waves ( s ( x/ ) = ) very other step, whch s urealstc. The stadard requremet s therefore µ /4. Therefore, do ot avely th dffuso terms a umercal model does ot cause umercal stablty! Whe tegrated stably, the dffuso term CFD models teds to stablze the soluto by llg off/dampg small scale waves, but whe stablty codto s ot met, t tem tself wll cause problem! 6

7 Read Pele (984) secto 0.. (hadout, see web l) Implct Methods Read Taehll et al, secod part of secto So far, we have dealt wth oly explct schemes whch have the form of + u = f( u, u,...). Wth these schemes, the future state at each grd pot s oly depedet o the curret ad past tme levels, therefore the soluto ca be obtaed drectly or explctly. c.f., explct fuctos such as y = x. Implct scheme volves varables of the future tme level at more tha oe grd pot (ofte resultg from fte dfferece of varable(s) at the future tme level). Mathematcally t ca be expressed as + + u = f( u, u, u,...). Ths s aalogous to mplct fuctos such as x = s(x). As oe ca mage, mplct schemes are more dffcult to solve. Usually matrx verso s volved. We wll frst loo at the stablty property of a mplct scheme. Example. Cosder the -D dffuso equato u t = K u xx aga. It s approxmated by the followg F.D. scheme: 7

8 + + tu = K[ xxu + ( ) xxu ] (4) δ αδ α δ (Note: shorthad otatos for F.D. are used. See Appedx. e.g., δ + ( )/ u u u t + t = ). Whe α = 0, the scheme s explct, ad s the FTCS scheme dscussed earler. Whe α = /, t s mplct ad s called Cra-Ncolso scheme. For other values of α, t s a geeral mplct scheme. We ca show that 4 u ( x) τ = K K t α O x t x (show f for yourself!). 4 ( ) We ca see that whe α = /, t s d-order accurate tme ad space. Otherwse, t's frst-order tme whch s expected for u-cetered tme-dfferecg scheme (whe α=/, the rght had sde s a averaged betwee the curret ad future tme levels vald at +/. Relatve to ths RHS, the LHS tme dfferece becomes cetered tme. We ow that the smplest cetered dfferece scheme s secod-order accurate). Whe [ ] = 0, the scheme becomes fourth-order space. Let's perform stablty aalyss o (4) usg vo Neuma method. u = U e = U e e U λ e (5) ( x ωt) ω t j x j x j 8

9 Hereλ t e ω Substtute (5) to (4). Note that we are ow usg j as the grd pot dex. j x + j x + x x Ue ( λ λ ) = µ Ue αλ + ( α) λ ( e + e ) Dvdg Ue j x λ o both sdes, ad rearragg λ = µ x αλ + α 4 s ( / )[ ( )] λ = 4( αµ ) s ( x/) 4 s ( x/) + αµ Loo at several cases: Case I: α = 0, λ µ = 4 s ( x/) µ / as before. The scheme s codtoally stable. Case II: α = / (Cra-Ncolso) x λ = µ s ( /) + µ s ( x/) for all values of µ, therefore the scheme s absolutely or ucodtoally stable. Case III: α =, the tme dfferece s bacward, relatve to the RHS terms. 9

10 λ =, aga for all values of µ, + µ 4 s ( x/) therefore the scheme s also absolutely stable. However, ths scheme s oly frst-order accurate tme, as dscussed earler (cosstet wth the tme dfferece scheme beg u-cetered). I geeral, whe 0 α < /, t s requred that µ /( - 4α ), therefore the scheme s codtoally stable. Whe / α, the scheme s ucodtoally stable (t s sometmes referred to as the forward-based scheme). I the ARPS, the mplct dffuso scheme s a opto for treatg the vertcal turbulet mxg terms. Ths treatmet s ecessary order to remove the severe stablty costrat from these terms whe vertcal mxg s strog sde the plaetary boudary layer (PBL). The latter occurs whe the PBL s covectvely ustably ad the o-local PBL mxg s voed wth the Su ad Chag (986) parameterzato. Parameter alfcoef arps.put correspods to -α here (see hadout). Fally, we ote that for mult-tme level schemes, there s usually multple solutos for the amplfcato factor λ. some of them mght represet spurous computatoal modes due to the use of extra (artfcal) tal codtos. The expresso of λ ca be too complcated so that a graphc plottg s eeded to uderstad ts depedecy o wave umber. λ has to be o greater tha for all possble waves. The shortest wave resolvable o a grd has a wavelegth of, ad the logest s L, where L s the doma wdth. Trdagoal Solver 0

11 -D mplct method ofte leads to trdagoal systems of lear algebrac equatos. (I the ARPS, ths appears twce oce whe soud waves are treated mplctly the vertcal drecto ad oce whe the vertcal turbulece mxg s treated mplctly). For example, Eq. (4) ca be rewrtte as u u u u + u+ = α K + t x f (6) where f cotas other ow tmes at tme level +. It ca be rearraged to + tk ( ) u = α u u + u+ + tf + u x tk + tk + tk + α u + ( + α u ) α u + ) = tf + u x x x. Let A = C = αµ, B = ( + αµ ), ad d = tf + u, where tk + µ = u, we have x Au + Bu + Cu = d (7) for =,,, N-, assumg the boudares are at =0 ad N. If we have Drchlet boudary codtos,.e., u at =0 ad N are ow, the for =, the equato becomes B u + Cu = d Au (8)

12 ad for = N, the equato s A u + B u = d C u. (9) N N N N N N N For =, 3,., N-, the equato remas of the form Eq.(7). If we wrte the equatos (7-9) a matrx form, we have B C u D A B C u D..... A B C u = D..... AN BN CN un DN A B u D N N N N (0) where D = t f u Au +, 0 0 N N N N N D = t f u C u +, D = tf u for =,, N-.

13 If we have Neuma boudary codtos,.e., we ow the gradet of u at the boudares whch dscretzed form are u u0 = L ad un un = R. Plug these relatos to Eq.(7) for = ad =N-, we obta equatos smlar to (8) ad (9): ( A + B) u + Cu = d + AL () N N ( N N ) N N N A u + B + C u = d C R. () I ths case, the fal coeffcets (0) are dfferet for the frst ad last equato. Sce each except for the frst ad last row of the coeffcet matrx, oly three elemets are o-zero ad the o-zero elemets of the matrx are alged alog the dagoal axs, ths system s called trdagoal system of equatos. It ca be solved effcetly usg Thomas Algorthm. The procedure cossts of two parts. Frst, Eq.(7) s mapulated to the followg form: C ' u D' C ' u D'..... C ' u = D'..... C ' N un D' N u D' N N (3) whch the subdagoal coeffcets A are elmated ad the dagoal coeffcets are ormalzed. For the frst equato 3

14 C D C' =, D' =. (4a) B B For the geeral equatos: C D AD' C' =, D' = B AC B AC ' '. (4b) Equatos (4) represet a forward sweep step (see fgure below). It s followed by a bacward substtuto step that fds soluto u from (3). The soluto s: u = D' N N u = D' u C' for from N to. + (5) 4

15 Note both (4) ad (5) volve reducto, the algorthm s heretly o-parallelzable. Fortuately, for multdmesoal problems, multple systems of equatos ofte eed to be solved, ad oe ca explot parallelsm alog other dmesos (e.g., j stead of drecto). Read Secto of Taehll et al ad Appedx A Stablty Aalyss for Systems of Equatos Whe we are dealg wth a system of equatos, we ca also apply the vo Neuma method to fd the stablty property of a gve F.D. scheme. As wth sgle equatos, vo Neuma ca oly be used for lear systems of equatos. For olear systems, learzato has to be performed frst. Wthout gog to detals, we pot out that a system of lear equatos ca be expressed a matrx form le u u + [ A] = 0 t x The equato s frst dscretzed usg certa F.D. scheme, u ca be wrtte terms of a dscrete Fourer seres ad the wave compoet s the substtuted to the dscrete equato to obta somethg le: U + = [ M( t, x)] U (7) where U s the ampltude vector for wave at tme level, ad [M] s called the amplfcato matrx. The scheme s stable whe the maxmum absolute egevalue of [M] s o greater tha. (6) 5

16 Why the maxmum absolute egevalue? Because as you saw earler ( Chapter ) that a system of equato le (7) ca be trasformed to a system of decoupled equatos, ad the egevalues of [M] become the amplfcato factors for each of the ew depedet varables, v (the elemet of vectorv ),.e., we ca obta from (7) V [ N] V, + = where [N] s a dagoal matrx wth the egevalues of [M] as ts dagoal elemets. {[T] - [M] [T] = [N] [ T] U + = [ T] [ M][ T][ T] U V + = [ N] V where + V = [ N] U. Therefore ( v ) + = λ ( v ) where v s a elemet of vectorv.} Sce the system s stable oly whe all depedet varables rema bouded, the absolute value of the maxmum egevalue has to be o greater tha. Read secto 3.6. of Taehl et al. 6

17 Appedx Shorthad otatos for dscrete/fte dfferece operators ad dscretzato dettes Notatos: A A x j+ x+ Aj x x = δ x = + x A A δ x = A δ x = x Idettes: A j+ / j / j+ j x A x A x j j δ A δ A δ A x x x x x Aδ B δ ( A B) Bδ A x x x x δx δx( /) A A A x x