Direct Iteration Method. Numerical Solution of Equations. Introduction

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1 School of Mechaical Aerospace ad Civil Egieerig Direct Iteratio Method This is a fairly simple method, which requires the problem to be writte i the form = f() (3) Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 for some fuctio f() We start with a iitial guess to the solutio,, ad the calculate a ew estimate as = f( ) (4) f() Cotets: Itroductio to CFD Numerical solutio of equatios Fiite differece methods Fiite volume methods Pressure-velocity couplig Solvig sets of liear equatios Usteady problems Turbulece ad other physical modellig Body-fitted coordiate systems Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg, W Malalasekara, A Itroductio to Computatioal Fluid Dyamics: The Fiite Volume Method SV Patakar, Numerical Heat Trasfer ad Fluid Flow Notes: - People - T Craft - Olie Teachig Material This process is cotiued, at each step geeratig a ew approimatio + = f( ) (5) The iteratios are stopped whe the differece betwee successive estimates becomes less tha some prescribed covergece criterio ǫ ie whe + < ǫ If the process is coverget, the takig a smaller value for ǫ results i a more accurate solutio, although more iteratios will eed to be performed - p 3 Itroductio The Navier-Stokes equatios are a set of coupled, o-liear, partial differetial equatios As idicated earlier, solvig these umerically cosists of two steps: Approimatio of the differetial equatios by algebraic oes Solutio of the system of algebraic equatios Before cosiderig these steps for differetial equatios we first review briefly how we use umerical methods for approimate solutios of sigle o-liear algebraic equatios Sice most such methods are iterative, this itroduces a umber of cocepts ad geeric treatmets that will also be met later whe dealig with iterative solutio methods for large sets of coupled equatios The geeral problem to be cosidered is that of solvig the equatio f() = 0 () where f is some arbitrary (o-liear) fuctio For some methods we eed to reformulate this as = F() () for some fuctio F As a eample, cosider solvig the equatio (which has the eact solutio = 4) = 0 (6) We first eed to write the equatio i the form = f(), ad there is more tha oe way of doig this Oe way is to write the equatio as = ( ) /3 so f() = ( ) /3 (7) If we start with a iitial guess of = 5, the above iterative method the gives: Iteratio f() The scheme coverges, although ot particularly rapidly Some methods will probably have bee met i earlier courses, but it is useful to review these, to uderstad their behaviour ad eamie some of their advatages ad weakesses - p - p 4

2 If, however, we rewrite equatio (6) i the form The graphs of the two particular forms of f() eamied i the eample earlier are show below = ( 3 3 4)/3 so f() = ( 3 3 4)/3 (8) ad agai take = 5, the the iteratio process leads to: Iteratio f() = ( ) /3 coverges I this case the process is clearly divergig I fact, eve if we had started from a iitial guess much closer to the real solutio, with this formulatio the iterative process would still have diverged + = ( 3 3 4)/3 diverges As ca be see from this eample, the direct method as described is ot always guarateed to coverge, ad the particular formulatio chose for f() ca be highly ifluetial i determiig the covergece behaviour To uderstad why this should be, we ote that the differece betwee successive iteratios will be + = f( ) f( ) (9) This highlights a importat aspect of umerical solutios: oe usually eeds a good uderstadig of the problem to be solved ad the solutio methods i order to select the most appropriate scheme - p 5 - p 7 Assumig that we are close to the true solutio, r, we ca approimate f( ) ad f( ) by Taylor series epasios about r: where f deotes the derivative df/d f( ) f( r ) + ( r )f ( r ) + (0) f( ) f( r) + ( r)f ( r) + () Sice f( r) = r, substitutig these epressios for f( ) ad f( + ) ito equatio (9) gives + f ( r ) () Thus, if the gradiet of f is such that f ( r) <, the differece betwee successive approimatios decreases, ad the scheme will coverge If f ( r) > the differece betwee successive approimatios icreases ad the method will ot coverge f() f() The Bisectio Method This is desiged to solve a problem formulated as f() = 0 for some fuctio f I this method we start off with two poits ad, chose to lie o opposite sides of the solutio Hece f( ) ad f( ) have opposite sigs We the bisect this iterval, so take 3 = 05( + ), ad evaluate f( 3 ) i order to fid its sig For the et iteratio we retai 3 ad whichever of or gave the opposite sig of f to f( 3 ) We thus kow that the solutio lies betwee the two poits we retai We cotiue bisectig the iterval as above util it becomes sufficietly small, ie < ǫ for some small covergece criteria ǫ We the call the process coverged f() This behaviour ca also be iferred graphically, as show Clearly, as we reduce the covergece criteria ǫ we get a more accurate approimatio to the solutio, but have to perform more iteratios (ie more bisectios of the iterval) Scheme coverges Scheme diverges - p 6 - p 8

3 Solvig the same eample as earlier we write f() = = 0 (3) Applyig the bisectio method iteratio, with iitial poits = 5 ad = 5 ow gives: Iteratio ( ) (, ) ( 3, ) ( 3, 4 ) ( 3, 5 ) ( 6, 5 ) ( 7, 5 ) f() ( 7, 8 ) ( 7, 9 ) ( 0, 9 ) With the earlier eample, solvig f() = = 0, ad with the same startig poits, we would get: Iteratio f() The process shows a fairly rapid covergece The method will always coverge, sice the iterval size always decreases The method ca be rather slow, sice the iterval size is oly halved i each iteratio - p 9 - p The Secat Method However, if we istead start with the two poits = 0 ad = 5, we get the sequece: This method solves the system f() = 0 It also requires two startig poits, ad, but they eed ot be o opposite sides of the eact solutio f() Iteratio f() We ow fit a straight lie through the two poits (,f( )) ad (,f( )), ad take the et estimate as the poit at which this lie cuts the ais I this case the process does ot appear to be covergig This iteratio method ca be formulated mathematically as «+ = f( ) f( ) f( ) (4) By plottig the sequece, we ca see that the iterative process oscillates across the local maimum of f aroud = 04 The process is agai repeated util the covergece criteria is reached - p 0 - p

4 Uder-Relaatio Uder-relaatio is commoly used i umerical methods to aid i obtaiig stable solutios Essetially it slows dow the rate of advace of the solutio process by liearly iterpolatig betwee the curret iteratio value, ad the value that would otherwise be take at the et iteratio level I the preset eample of usig the Secat method, equatio (4) could be modified to read «+ = ωf( ) f( ) f( ) where the uder-relaatio factor ω is take betwee 0 ad If, for eample, we take ω = 05, the Secat method applied to the previous eample gives: Iteratio f() By reducig the jump betwee successive values of i the early iteratios the solutio estimates stay to the right side of the local miimum i f, ad the process ow coverges Note that as the solutio gets closer to covergig, the uder-relaatio factor could be icreased to speed up covergece (5) - p 3 This also solves the equatio f() = 0 The Newto-Raphso Method I this method we start with a iitial guess We the draw the taget to the curve of f at, ad take our et estimate to be the poit where this taget cuts the ais Mathematically, we ca write this iteratio process as where f deotes the derivative df/d + = f() f ( ) 4 3 We agai cotiue the process util we satisfy some suitable covergece criteria f() (6) - p 5 Covergece Criteria I all the eamples preseted here we have tested for covergece by checkig o the differece betwee successive solutio estimates We thus take the solutio as coverged whe + < ǫ for some predefied covergece criteria ǫ However, this coditio should ot always be applied i a purely blid fashio For eample, if usig the secat method i a regio where the gradiet f is very steep the chages i betwee iteratios could be very small, particularly if heavy uder-relaatio is also beig used However, as show i the figure, the estimates could still be some way from the true solutio f() Usig the same eample as before, = 0, with a startig poit of = 5, we obtai the followig sequece: Iteratio f() f () Note the large jump i the first step, which occurs sice we chose a value for which f ( ) is small After this the covergece is fairly rapid A small value of + could simply be idicatig a very slow covergece rate 3 A safer way of checkig for covergece is to also test whether the actual fuctio value f( ) is sufficietly small - p 4 - p 6

5 If, however, we had started the iteratio process from a iitial value of = 0, we would have got: Iteratio f() f () Systems of Algebraic Equatios Most of the methods outlied for solvig a sigle equatio ca be eteded to apply to a system of equatios of the form: f (,,, ) = 0 f (,,, ) = 0 f (,,, ) = 0 Because of the shape of the fuctio f, the method first speds a umber of iteratios oscillatig aroud the local maimum at 04 As i the case of sigle equatios, to solve the system efficietly (ad sometimes to solve it at all) requires a uderstadig of the method adopted, ad of the ature of the equatios to be solved - p 7 - p 9 I this case covergece ca agai be achieved by itroducig uder-relaatio If we uder-rela the solutio with a factor of 09, takig we the get the iteratio sequece: + = 09f( )/f ( ) (7) It f() f () The Direct Iteratio Method For the equivalet of the direct iteratio method, we first have to rewrite the system of equatios ito the form = f (,,, ) = f (,,, ) = f (,,, ) This ca be writte more compactly i vector form: = f() The uder-relaatio here results i the solutio rapidly reachig a poit ear the maimum of f where f is small eough ad positive that the et value of lies to the right of the root (ie > 4) Note that i this case covergece could agai have bee improved further by revertig to usig o uder-relaatio after iteratio 4 Oe the proceeds, with a startig guess 0, costructig the sequece + = f( ) (8) stoppig the iteratio whe + < ǫ for a suitably small covergece criterio ǫ The orm + is simply a scalar measure of the size of the vector + For eample, it could be the L orm L = () / = X i / (9) - p 8 - p 0

6 As i the sigle equatio case the method is ot always guarateed to coverge I additio to covergece behaviour beig highly depedet o the precise formulatio used for f i i the equatio i = f i (,,, ) (0) the choice of which equatio is used for updatig which variable is ow also very importat It is possible to devise epressios for the coditios ecessary to esure covergece, i the same way as was doe i the sigle equatio case However, the result is rather comple ad will ot be cosidered here I geeral, a useful rule of thumb is to try ad write the equatios i a form where the right had sides cotai the variables raised to powers less tha oe Uder-relaatio is ofte also used, to aid covergece by avoidig large chages i the estimates for from iteratio to iteratio With this icluded, the iteratio sequece ca be writte as + = ( ω) + ωf( ) () where ω is the uder-relaatio factor, with 0 < ω < Rearragig, ad takig (r) as the estimate to the solutio at iteratio level i + leads to a system of liear equatios for the elemets of the vector (i+) (i) : 0 0 (i+) (i) 0 f (i) (i+) (i) f (i) = C B C B C A (i+) (i) f (i) which ca be solved usig a variety of methods, some of which will be discussed later Uder-relaatio ca also be applied i this method Whe the scheme coverges it geerally does so more rapidly tha the simpler direct iteratio method However, more work has to be doe per iteratio, sice the derivatives f i / j have to be computed, ad the matri system of equatio (3) solved (3) - p - p 3 The Newto-Raphso Method Other schemes ca also be adapted to obtai solutios to a system of equatios The Newto-Raphso method for a sigle equatio ca be devised by retaiig the first term i a Taylor series epasio of f about the poit If r is the actual root, the oe ca write 0 = f( r) f( ) + ( r )f ( ) + () Neglectig the higher order terms ad rearragig leads to r f( )/f ( ) I the case of a system of equatios we could write 0 = f (r) f (i) + ( (r) (i) ) f(i) + ( (r) (i) ) f(i) + + ( (r) 0 = f (r) f (i) + ( (r) (i) ) f(i) + ( (r) (i) ) f(i) + + ( (r) 0 = f (r) f (i) + ( (r) (i) ) f(i) + ( (r) (i) ) f(i) + + ( (r) (i) ) f(i) (i) ) f(i) (i) ) f(i) Summary We have eamied a umber of umerical iterative methods for solvig sigle o-liear equatios of the form f() = 0 ad systems of the form f() = 0 Some schemes, such as the bisectio method, will always coverge, but may do so rather slowly Others coverge more rapidly for some cases, but may ot be guarateed to always coverge Uder-relaatio ca be applied i some cases to aid covergece by reducig the chage i solutio estimates betwee iteratios The uder-relaatio factor ca ofte be icreased as the true solutio is approached To select ad apply the most appropriate method for a particular problem requires a good uderstadig of the characteristics of the method ad of the problem beig solved where f (i) deotes the value f ( i ) - p - p 4