2.3. Quantitative Properties of Finite Difference Schemes. Reading: Tannehill et al. Sections and

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1 .3. Quattatve Propertes of Fte Dfferece Schemes.3.. Cosstecy, Covergece ad Stablty of F.D. schemes Readg: Taehll et al. Sectos ad Three mportat propertes of F.D. schemes: Cosstecy A F.D. represetato of a PDE s cosstet f the dfferece betwee PDE ad FDE,.e., the trucato error, vashes as the grd terval ad tme step sze approach zero,.e., whe lm(pde FDE) = 0. Δ 0 Commet: Cosstecy deals wth how well the FDE approxmates the PDE. Stablty For a stable umercal scheme, the errors from ay source wll ot grow uboudedly wth tme. Commets: A cocept that s applcable oly to marchg (tmetegrato) problems. Geerally we are much more cocered wth stablty tha cosstecy. Some hard wor s ofte eeded to establsh aalytcally the stablty of a scheme. Covergece It meas that the soluto to a FDE approaches the true soluto to the PDE as both grd terval ad tme step sze are reduced.
2 Lax's Equvalece Theorem For a wellposed, lear tal value problem, the ecessary ad suffcet codto for covergece s that the FDE s stable ad cosstet. The theorem has bee proved for tal value problems govered by lear PDE's (Rchtmyer ad Morto 967). We wll dscuss the three cocepts oe by oe..3.. Cosstecy Cosstecy meas PDE FDE 0 whe Δx 0 ad Δt 0. Clearly cosstecy s the ecessary codto for covergece. Example: Cosder a D dffuso equato: u t u = K ( K > 0 ad costat) x We use the forwardtme ad ceteredspace (FTCS) scheme:
3 u u u u + u Δt Δx + + = K To show cosstecy, we eed to determe the trucato error τ. Usg Taylor seres expaso method, u ( Δt) u ( Δt) u u = u +Δ t t! t 3! t 3 3 u ( Δx) u ( Δx) u u± = u ±Δ x + ± +... x 3! x 3! x Substtutg to the FDE, we have u Δt u u + + O( Δ t ) = K + O( Δ x ) +... t t x therefore Δt u τ = + Δ +Δ t O( t x ) τ 0 whe Δx 0 ad Δt 0 => the scheme s cosstet.. A couter example: The DufortFrael method for the same dffuso equatos: 3
4 + + u u ( u+ + u ) ( u + u ) = K. Δt Δx It's a ceteredtme scheme that s dorder accurate both space ad tme. We ca fd aga the trucato error (do t yourself!) 4 3 K( Δx) u Δt u ( Δt) u τ = K HRT x Δx t 6 t We ca see that lm τ 0 except for the secod term. Δx, Δt 0 If Δ t lm = 0 Δx Δx, Δt 0 the the scheme s cosstet therefore Δt must approaches zero faster the Δx. If they approaches zero at the same rate, the lm Δx, Δt 0 Δ t = β, the Δx lm Δx, Δt 0 τ = K u β, t our equato becomes t t x u u u + K β = K 4
5 Thus, we are solvg the wrog equato. I fact ths equato s hyperbolc stead of parabolc. Δt Note that f ~ Δx you mght see spurous waves your soluto, due to the hyperbolc ature of the "ew" PDE Covergece Geeral Dscusso Defto s gve earler. Symbolcally, t s lm u = u( x, t). Δx, Δt 0 Covergece s geerally hard to prove, especally for olear problems. The Lax's Theorem we preseted earler s very helpful uderstadg the covergece for lear systems, ad s ofte exteded to olear systems. We wll also dscuss umercal covergece ad methods for measurg soluto accuracy later. We wll frst show a covergece proof for a dffuso problem. Certa cocept troduced wll be useful later. Covergece proof for a D dffuso problem Cosder 5
6 u t u = K ( K > 0 ad costat) x for 0 x L whch has a tal codto: π x uxt (, = 0) = a s( ) = f( x). L = The B.C. s u(0, t) = u( L, t) = 0. Ths s a wellposed, lear tal value problem (otce the I.C. satsfes the B.C. as well). Frst, let's fd the aalytcal soluto to the PDE. Because the problem lear, we eed oly to exame the soluto for a sgle waveumber, ad ca assume a soluto of the form: π x u( x, t) = A( t)s( ) L Here A s the ampltude ad s gves the spatal structure ad the fal soluto should be the sum of all wave compoets. Note that ths soluto satsfes the boudary codtos. Substtutg the soluto to the PDE, we obta a ODE for the ampltude A : 6
7 da () t dt π = L KA dl( A ) π = K dt L A() t = A(0)exp K π / L t. ( ) It says that the ampltude of the solutos for all wave umbers decreases wth tme. From the I.C., A (0) = a, so we have (, ) exp ( / ) π x u x t = a K π L t s( ) L ad uxt (, ) = u ( xt, ) whch s the aalytcal soluto to the orgal dffuso equato. A umercal approxmato to the dffuso equato should coverge to ths soluto as Δx, Δt 0. Cosder the forwardtme ceteredspace (FTCS) scheme we derved earler. 7
8 Goal: Show that u t u(x,t) as Δx, Δt 0. Frst, fd the umercal soluto. Ths tme, we use the FDE ad substtute a dscrete Fourer seres to the equato. Let the I.C. be gve by π x f x u a for = 0,,,.., J J 0 ( ) = = s( ) = 0 L where J+ = total umber of grd pots used to represet the tal codto. The coeffcet a s gve by a dscrete Fourer trasform: J π x a = f( x)s( ) for =0,,,., J. J L = 0 Note : L = JΔx. As Δx 0, J, the dscrete Fourer seres becomes cotuous ad a a. Note : The umber of harmocs or Fourer wave compoets that ca be represeted s a fucto of the umber of grd pots (J+), whch s the umber of degrees of freedom. Spectral methods represet felds terms of spectral compoets, whose ampltudes are solved for. The waveumber for the wave compoets s π the above equatos. L Recall that wavelegth 8
9 π π L λ = = = w.. ( π / L) where s the umber (dex) of wave compoets. Logest wavelegth =, correspodg to waveumber zero (=0). Next logest wave = L ( = )... Shortest wave = L/J = JΔx/J = Δx. Commets: A Δx wave s the shortest wave that ca be resolved o ay grd ad t taes at least 3 pots to represet a wave. 9
10 Δx waves ofte have some specal propertes. They are also represeted most poorly by umercal methods recall that smooth felds are more accurately represeted by a fte umber of grd pots. As the cotuous case, we exame oly oe waveumber (the soluto s the sum of all waves), so for our dscrete problem, assume a soluto of the form u π x = A( )s L (It satsfes B.C.) ad s the tme level. We also have from I.C. A (0) = a. Substtutg ths to the FDE ad lettg u u u u + u Δt Δx + + = K, 0
11 S π x = s L, we have A( + ) S A( ) S KA( ) = [ S + S + S ]. Δt ( Δx) Sce x = Δx, x + = (+) Δx therefore S + π ( x+δx) = s L. Usg stadard trgoometrc dettes, we ca wrte the above the form of a recurso relato: π Δx A( + ) = A( ) 4μ s L KΔt where μ =. ( Δ x) π Δx If we let M() 4μ s L, the we have A ( + ) = M( ) A ( ).
12 Wrte t out for =0,,,., : A () = M( ) A (0) = M( ) a () = ( ) () = [ ( )] A M A M a... A ( ) = M( ) A ( ) = [ M( )] a we therefore have the soluto of u for wave mode : π x ( u ) = A( ) S = a [ M( )] s. L Defto: M() s ow as the amplfcato factor, ad f M(), the soluto wll ot grow tme as. Ths had better to be the case because the ampltude of the aalytcal soluto s supposed to always decrease wth tme. For our problem we ca see that M() meas π Δx L 4μ s. If we tae the maxmum possble value of s ( ) =, 4μ
13 μ /. Ths codto eeds to be met for all to prevet soluto growth. Based o the defto of μ, the codto becomes ( Δx) Δt. K Ths mposes a upper boud o the Δt that ca be used for a gve value of Δx, ad such a codto s uow as the Stablty Costrat. Now we have our soluto, let's chec covergece for sgle mode. By defto of covergece, we tae 4 π Δ lm ( u ) lm a t s s Δx, Δt 0 Δx, Δt 0 K x πx = Δ ( Δx) L L 4K π Δx or f we let f(δx) s ( Δx) L, π x lm ( u ) = lm a [ Δt f( Δx) ] s. L Δx, Δt 0 Δx, Δt 0 3
14 It ca be show that f f(x) s a complexvalued fucto of a real argumet, say Δx, such that lm f ( Δ x) = a, the, Δ x 0 lm [ ±Δtf( Δ x)] = e ± at (we wll show ths later). Δx, Δt 0 s( y) We ow thatlm =, therefore y 0 y π Δx s K( π) L π lm f ( Δ x) = lm = K a L π x = Δ L L Δ x 0 Δ x 0 Also lm a Δ x 0 = a, therefore Δx, Δt 0 π π x j lm ( u ) = aexp K t s. L L Iterestgly, ths soluto s detcal to our aalytcal soluto derved earler! Therefore the umercal soluto coverges to the PDE soluto whe Δx, Δt 0. 4
15 Fally, we show here (otg that Δt = t) ( ) ( )( )! 3! 3 a a 3 = ± at + ( Δ t) ± ( Δ t) +...! 3! lm( ± fδ t) = ± fδ t+ ( fδ t) + 3 ( fδ t) +... Δ x 0 3 ( at) ( at) = ± at + ± +...! 3! ± at = e.3.4. Numercal Covergece Readg: Fletcher (hadout), Sectos 4.., 4.., Numercal Covergece Covergece s ofte hard to demostrate theoretcally. True aalytcal soluto s hard or eve mpossble to fd s oe of the reasos. We ca, however, fd out the covergece of a gve scheme umercally. We compute solutos at successvely hgher resolutos ad see how the error chages wth the resoluto. Does τ 0 whe Δx 0? 5
16 Ad how fast τ decreases? The procedure ca be very expesve (remember the cost factor crease as Δ doubles). A typcal measure of error s the L orm or RMS error: L [ u Δ u ] = x where u s a true soluto or a 'coverged' umercal soluto whe exact soluto s ot avalable. Example: D dffuso equato usg FTCS scheme: 4 Δt u ( Δx) u τ = K t x Mag use of 4 u u u u u u = K = K K K = =. 4 t x t t x x t x we have 4 K( Δx) u τ = s + O Δ x +Δt 4 6 x 4 ( ). K t where s = Δ ( Δ x) 6
17 Table 4. (from Fletcher) shows the error reducto wth Δx for two values of s. The above fgure shows plots of log 0 (err) as a fucto of log 0 (Δx). 7
18 Recall that τ = A (Δx) log τ = log A + log Δx. Ths s a straght le loglog dagram wth a slope of ad tercept A. Thus the slope of the le gves the rate of covergece. I the above fgure, we see that whe s = /6, the error le has a steeper slope ad the error s smaller for all Δx. Ths s because for ths value of s, the frst term τ drops out ad the scheme because 4thorder accurate. The scheme s secodorder accurate for all other values of s. Note that you ca choose Δx ad Δt such that s=/6 oly whe K s costat the etre doma. I cases where o exact solutos s avalable, a socalled 'grdcovergece' or referece soluto s ofte sought ad ths soluto ca be used the place of true soluto the estmatg the soluto error. 8
19 A example from Straa et al (993). It shows a referece soluto obtaed at Δx=5 m, for a desty curret resultg from a droppg cold bubble. 9
20 Fgure (from Straa et al 993). Graph of θ' L orms ( C) from selfcovergece tests wth the compressble referece model (REFC). The bold sold le labeled wth 'selfcovergece solutos' represets the L orms for spatal trucato errors of solutos made wth Δt = costat ad varyg grd spacgs. The L orms were computed agast a 5.0 m referece soluto. The bold dashed les labeled wth, for example, '00.0 m solutos' represet L orms for temporal trucato errors of solutos made wth Δx =costat (e.g m) ad varyg tme steps. The referece solutos for these computatos were made usg a tme step cosstet wth Δt =.5 s tmes a costat each of the cases. The sold fes labeled O(l) ad O() represet frst ad secodorder covergece, respectvely. 0
21 Rchardso Extrapolato As the grd becomes very fe, the error behaves much le that predcted from the leadg terms τ. Further refemets are expesve, so we use aother techque to mprove the soluto Rchardso Extrapolato. Cosder two umercal solutos obtaed at Δx a ad Δx b. Wth FTCS scheme (assumg s /6), τ a 4 K( Δxa ) u 4 = s + O Δ x +Δt 6 x 4 ( a ) 4 K( Δxb ) u 4 τ b = s + O Δ x +Δt 6 x 4 ( b ) Fd a lear combato of the two solutos, u a ad u b u c = a u a + b u b where a + b = ad a ad b are chose so that the leadg terms τ a ad τ b cacel ad the scheme becomes 4thorder accurate (Of course ths assumes that u x s the same both case, whch s reasoably assumpto oly 4 4 at relatvely hgh resolutos whe the soluto s well resolved). If Δx b = Δx a /, the 4a + b = 0 wth a + b = a = /3, b = 4/3 ad u c = /3 u a + 4/3 u b.
22 .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 ts 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 tmematchg problems oly The eergy method Read Durra Secto. (hadout, see web l)
23 Ths method s used much less ofte tha the vo Neuma method. It's attractve because t wors for olear problem ad problems wthout perodc 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 upstreamforward scheme for wave (advecto) equato + c = 0 : t x u u u u + = Δt Δx + c 0 Let μ = cδt/δx u = ( μ) u + μu + Squarg both sdes ad summg over all grd pots: ( u ) = [( μ) ( u ) + ( μ) μu u + μ ( u ) ] () + Assumg perodc B.C. ( u ) = ( u ) ad usg the Schwarz equalty (whch says that for two vectors U ad V, U V U V ), 3
24 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 Δt/Δx 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 Δt 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 u + = [ M(t) ] + u 0 4
25 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 5
26 If ω I >0, the soluto wll grow expoetally tme. Example : D dffuso equato wth FTCS scheme. u u = μ( u u + u ) (3) + + KΔt where μ =. ( Δ x) Let's exame a sgle wave : u = U e ( x ωt) + ( x ωt) ωδt = = ( x ωt) ± Δx ± u U e e u U e e Substtute the above to (3) Ue ( e ) = μue [ e + e ] ( x ωt) ωδt ( x ωt) Δx Δx ( x ωt ) ω t Ue [ e Δ μ(cosδx )] = 0 ( x ωt) ω t Ue e μ x [ Δ + 4 s ( Δ /)] = 0 For otrval soluto, we requre 6
27 e ω Δ t + 4μ s ( Δ x/) = 0 e ω Δ t = 4μ s ( Δ x/) Here, t e ω Δ s actually the amplfcato factor, the same as the M dscussed earler. λ t e ω Δ = u + /u  the amplfcato factor. For stablty we requre λ 4μ s ( Δx/) μ / as before! I practce, whe μ=/, the soluto (amplfcato factor) swtches betwee ad + for Δx waves ( s ( Δ 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! Read Pele (984) secto 0.. (hadout, see web l). 7
28 Shorthad otatos for dscrete/fte dfferece operators ad dscretzato dettes Notatos: Idettes: A + A x A = Aj A δxx = Δx Aj+ Aj δ + xx = Δx Aj Aj δ xx = Δx j+ / j / + / j / δ A δ A δ A x x x x x x Aδ B δ ( A B) Bδ A x x x x x A δxa δx( A /) 8
29 .3.6. 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: δ+ = αδ + α δ (4) + tu K[ xxu ( ) xxu ] 9
30 (Note: shorthad otatos for F.D. are used. See Appedx. e.g., δ u t ( u + u )/ t + = Δ ). Whe α = 0, the scheme s explct, ad s the FTCS scheme dscussed earler. Whe α = /, t s mplct ad s called CraNcolso scheme. For other values of α, t s a geeral mplct scheme. We ca show that 4 u ( Δx) τ = K KΔt α O x t 4 + Δ +Δ x (show f for yourself!). 4 ( ) We ca see that whe α = /, t s dorder accurate tme ad space. Otherwse, t's frstorder tme whch s expected for ucetered tmedfferecg 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 secodorder accurate). Whe [ ] = 0, the scheme becomes fourthorder space. Let's perform stablty aalyss o (4) usg vo Neuma method. u = U e = U e e U λ e (5) ( x ωt) ωδt jδx jδx j Hereλ t e ω Δ. Note that we are ow usg j as the grd pot dex. 30
31 Substtute (5) to (4) jδ x + jδ x + Δx Δx Ue ( λ λ ) = μue αλ + ( α) λ ( e + e ) Dvdg Ue jδx λ o both sdes, ad rearragg 4 s ( x/ )[ ( )] λ = μ Δ αλ + α λ = 4( αμ ) s ( Δx/) + 4αμ s ( Δx/) Loo at several cases: Case I: α = 0, λ = 4μs ( Δx/) μ / as before. The scheme s codtoally stable. Case II: α = / (CraNcolso) μ s ( Δx/) + μ s ( Δx/) λ = 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. 3
32 λ =, aga for all values of μ, + 4μ s ( Δx/) therefore the scheme s also absolutely stable. However, ths scheme s oly frstorder accurate tme, as dscussed earler (cosstet wth the tme dfferece scheme beg ucetered). 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 forwardbased 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 olocal 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 multtme 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 Δ, ad the logest s L, where L s the doma wdth. 3
33 Trdagoal Solver 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 u u + u Δt Δx Δx = αk + ( α) K (6) It ca be rearraged to ΔtK ΔtK u α ( u u + u ) = ( α) ( u u + u ) + u Δx Δx ΔtK ΔtK ΔtK α u + ( + α u ) α u = d Δx Δx Δx u u + u+ where d =Δt( α) K + u Δx αδtk αδtk Let A = C =, B ( ) Δ = +, we the have x Δ x Au + Bu + Cu = d, (7) 33
34 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) 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 (79) 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 = d for =,, N, 34
35 D = d Au +, 0 N N N N D = d C u +. 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 ozero ad the ozero 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) 35
36 whch the subdagoal coeffcets A are elmated ad the dagoal coeffcets are ormalzed. For the frst equato C D C' =, D' =. (4a) B B For the geeral equatos: C D AD' C' =, D' = B AC B AC ' ' for =,, N. (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) 36
37 Note both (4) ad (5) volve reducto, the algorthm s heretly oparallelzable. 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. 37
38 .3.7. 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 (6) 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( Δ t, Δ 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. 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 vector V),.e., we ca obta from (7) 38
39 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 MTT U + Let V = T U, we have V NV. + = Therefore + = where λ are the egevalues. ( v) λ( v) 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 Taehll et al. 39
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