3/31/ = 0. φi φi. Use 10 Linear elements to solve the equation. dx x dx THE PETROV-GALERKIN METHOD

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1 THE PETROV-GAERKIN METHO Consder the Galern solton sng near elements of the modfed convecton-dffson eqaton α h d φ d φ + + = α s a parameter between and. If α =, we wll have the dscrete Galern form of the convectve-dffson eqaton. If α = we recover the Upwnd form. Becase we expect that there s a vale of α sch that the term / gves s the amont of nmercal dffson necessary to obtan the correct answer, we call ths term the " Balancng ffson" The Galern dscretzaton sng near elements of nform sze Δ x= h prodces the dfference eqatons α + ( α + ) φ + φ + + ( α ) φ+ + α ( + ) The solton s φ = A+ B + α ( ) It follows that the solton s non-oscllatory f α. We defne the " Crtcal Vale" αcr =. We defne the Trncaton Error TE as the dfference between the orgnal dfferental eqaton and the modfed dscrete form,.e. d d h h x= x φ φ α TE = + + ( α + ) φ + φ + + ( α ) φ + Keepng trac of dervatves p to order 8, we get 6 8 α () α h () h (6) h (8) TE φ h φ + φ + φ +! 6! 8! h (3) h (5) h (7) φ... φ φ h ! 5! 7! Now we se the dfferental eqaton wrtten as φ = φ h to wrte all hgher order dervatves n terms of the second dervatve. That s we get the recrsve relaton φ Sbstttng nto the trncaton error we get () () n ( n) () φ = h α α TE = φ! 6! 8! 3! 5! 7! The total trncaton error can be expressed n terms of a nmercal dffson that taes the form α () TE = + tanh snh( ) φ α Set f ( ) = tanh snh( ) + 3 () tanh( / ) snh( ) ) Set α =, and re-wrte f( ) as f( ) =. ( /) snh( ) tanh( / ) Becase > and < for all, f ( ) < for all. ( /) Therefore, the Galern method s always nderdffsed. ) Set TE = and solve for α, we get α=coth Ths vale of α s called the optmal vale αopt. It prodces the exact solton when the coeffcents are constant and sperconvergent soltons n the general case. αopt and αcr are shown below Use near elements to solve the eqaton d φ 6 + =, < x <, φ() =, φ() = x Solton sng near elements (h=.) Solton s exact to sgnfcant dgts, the average velocty was sed 5 n each element Evalatng α nvolves a coth, and as seen n the prevos fgre αopt and αcr are very close after > 5. The amont of addtonal nmercal dffson added sng αcr nstead of αopt at = 8 s less than %.The same s tre f we se α = when <.. So α s calclated from. f <. α = coth( / ) / f. 8. / f > 8. The concept of Balancng ffson, althogh effectve n smple statons s dffclt to se n more complex problems. We wll re-cast the deas nder the framewor of a PETROV-GAERKIN method that can be extended to all cases n -, -, and 3- dmensons. The name Petrov-Galern refers to a method n whch the weghtng fnctons are NOT the same as the shape fnctons n a Galern formlaton. 6

2 ets go bac to the weghted resdal form of the modfed convectve-dffson eqaton gnorng the bondary terms that are not mportant n ths dscsson. We have dw + + w = The Galern formlaton s (sng lnear shape fnctons) dn + + N =, where φ= Nφ+ Nφ Then re-wrte the ntegral n the form dn dn + N + = 7 dn Now we defne w = N +, then the wea form becomes dw + w = Ths s the basc Petrov-Galern method for the one-dmensonal Convecton ffson eqaton. Petrov-Galern methods sng qadratc elements have also been blt. In ths case two parameters α and β are needed de to the dfferent natre of the nternal node. Here we wll restrct orselves to the se of lnear elements. Introdce a non-zero sorce term n the eqaton. d + = S( x) dw The weghted resdal form s + w = ws 8 The followng example llstrates how the Petrov-Galern method Atomatcally extends consstently to treat other terms n the Eqaton. We set = and loo at the eqaton x x.75 = Sx ( ), < x<.5, φ() =, Sx ( ) = ( x ).75 x. x > Two nmercal soltons wth h =. are shown n the next slde. ) Usng added balancng dffson, whch does nothng to the rght hand sde. ) Petrov-Galern, whch consstently weghts the rght hand sde. 9 Note that n ths problem =, therefore fll pwndng ( α = ) s reqred. In page 39 n the text t s shown that the Petrov-Galern solton s exact f the sorce term s pecewse lnear and nodes are placed at the ponts of slope dscontnty. dt dt x, x, T() T() + = < < = = Reacton ffson Eqatons d φ + + aφ = S When the reacton term a φ domnates, these eqatons also experenece nmercal oscllatons. Stable soltons can be obtan by means of two stablzng parameters, bt t becomes more dffclt to determne them. References to the basc wor are gven n the text, page 33. THE PETROV GAERKIN METHO IN TWO IMENSIONS The basc eqaton, ( φ ) + ( ) + + v = x x y y x y wll be wrtten as φ + V φ = x where xy =, = and = V v y

3 We assme that the velocty vector s constant for convenence. Ths can be for example the average fld velocty n the element. Consder a rectanglar blnear element as shown n the fgre We ntrodce a local (element wse) rotaton of the operator to a new coordnate system s t sch that s s algned n the drecton of the velocty vector. In ths new coordnate system the eqaton becomes T st stφ + V = s 3 V s the magntde of the velocty and the gradent st = T xy cosθ snθ where T = s the rotaton matrx. snθ cosθ Now re-wrte the eqaton n the form φ φ φ + V = s s s t t Ths eqaton can be vewed as a one-dmensonal convecton- dffson eqaton n the s drecton, wth a sorce term. From ths pont of vew, we mst ntrodce abalancng dffson only n the s drecton. That s an ansotropc balancng dffson. Gven by α V h = = REMARKS: ) h s an average element length defned later. V h ) α s calclated as before sng = The modfed eqaton, after addng the balancng dffson s T ( + ) φ + V = s Rotatng bac to the x - y system we get v x x x x y V V y y αvh v v = y x y V V x y The wea Galern formlaton s wrtten as Ω N N + + v v x x V x y y y V x y + N + v dω N = x y n Γ 5 Fnally, after some more algebrac manplatons we can wrte N N N N + + N + + v + x x y y x y x Ω V N N vn + + v dω N = V x y y Γ n Ths last expresson sggests the Petrov Galern weghts N N w = N + + v V x y whch are a natral extenson of the one-dmensonal weghts, and do not ntrodce CROSS FOW NUMERICA IFFUSION Notce that the contrbton to the lne ntegral was omtted, the Fnctons act only n the nteror of the doman. It can be shown That ths s the consstent way to formlate the method. 6 Advecton of a cosne hll n a rotatng feld defned by = y v = x The otsde bondary s x = and y = where φ=. Along the nternal bondary OA φ s prescrbed as a cosne as shown n the fgre. We set = so ths s a prely convectve staton. Galern Petrov-Galern Upwnd 7 In practce the Petrov Galern weghts are sed only n the convectve term and the sorce term. In - and 3- dmensons t amonts to gnorng the cross dervatve constants arsng from the dffson term. However t can be shown that ths does not affect the accracy. The element length h n the drecton of flow s gven by h = ( h + h ) where h and h are gven by V h = a V,and h = b V as shown n the fgre. ax = ( x + x3 x x) ay = ( y + y3 y y) bx = ( x3 + x x x) by = ( y3 + y y y) 8 3

4 Algorthms for convectve flow stablzaton have also been developed for trangles (Tabata, Kch etc.) we wll not descrbe them here. A sgnfcant nmber of algorthms smlar to or Petrov Galern and varatons on t have also been proposed. Most notably the so Called dscontnty captrng methods. Most of these methods Introdce parameters for whch we do not have a clear crteron to Choose, or mae the problem non-lnear. The Petrov Galern method can be consdered d a partclar case Of a more general famly of algorthms nown as the Galern- east-sqares method. Whch we now explan n one dmenson. Wrte the resdal of the one-dmensonal convectve-dffson d eqaton R = + S = a least sqares solton reqres the mnmzaton of the fnctonal I Ω 9 over the space of tral fnctons. = RdΩ Approxmatng φ wth shape fnctons φ = N φ (where repeted ndces mply smmaton) the condtons for the mnmm are dn + φ S = and ths leads to the fnte element eqatons dn dn d dn + φ S= on the other hand, the Galern formlaton of the eqaton s dn N + φ S= The Galern east- Sqares formlaton conssts n satsfyng a lnear combnaton of the two eqatons above. That s dn dn d dn N + τ + φ S= Ths s clearly a Petrov-Galern formlaton sng the weght fncton d dn dn w = N + τ + Usng lnear elements and defnng τ = we recover the orgnal Petrov-Galern formlaton. Even more general formlatons, the Generalzed-Galern-east- Sqares have been proposed to deal wth tme dependent Convecton-ffson problems. Where two-parameters mst be determned. Methods have been proposed n whch p to for parameters are added n the formlaton. THREE IMENSIONS The extenson to three dmensons s obtaned by means of a ocal rotaton n the drecton of flow and the addton of an Ansotropc dffson n the same way as two dmensons. The Petrov - Galern weghtng fnctons obtaned are N N N w = N + v w, v + + V = V x y z w The parameters α and as n - and the length h s gven by h = ( h + h + h3 ) V h = a V h = b V h3 = c V a, b and c are gven by ax = ( x + x3 + x6 + x7 x x x5 x8) ay = y + y + y + y y y y y az = ( z + z3+ z6 + z7 z z z5 z8) bx = ( x5 + x6 + x7 + x8 x x x3 x) by = ( y5 + y6 + y7 + y8 y y y3 y) bz = ( z5 + z6 + z7 + z8 z z z3 z) cx = ( x+ x + x5 + x6 x3 x x7 x8) cy = ( y+ y + y5 + y6 y3 y y7 y8) cz = ( z+ z + z5 + z6 z3 z z7 z8) ( ) 3 Non-lnear Eqatons In smlatons of fld flow we mst solve the Brgers eqaton n One dmenson and the Naver-Stoes eqatons n two and three mensons, whch are all non-lnear n the convectve terms. d d Frst consder the Brgers eqaton ε + = ) It was establshed (H-Z 979) that sng the Petrov-Galern method n conncton wth a Newton-Raphson teraton leads to very slow convergence and sometmes nstablty wth strong convecton. ) Usng a drect teraton convergence was faster for the convecton domnated cases. An alternatve was also developed sng modfed qadratre formlae (H.-979) to resolve these problems. Example A x d d eε B ε + =, () =, () =, ( x) = A A x eε + B

5 A and B are ntegraton constants. We wll set ε =. and se lnear elements sng Galern and Petrov-Galern combned wth Newton-Raphson and a drect teraton method. Convergence s determned when the dfference at every node between two - consectve teratons s less than. Note that the Galern solton s non-oscllatory, even thogh The cell Peclet nmber Pe = n some elements. However, near the bondary layer at x =, Pe =.3. An mportant dfference wth lnear problems s that Pe s varable and depends on the solton. Therefore t s not possble to predct 5 the onset of oscllatons a pror. Notce that the Newton-Raphson-Petrov-Galern method becomes less effcent as convecton domnates. In smlatons of the Naver-Stoes eqatons eventally t fals to converge (H 979, H-Z 979). 6 For the Naver-Stoes eqatons, even thogh Newton-Raphson Is the method of choce to get steady-state soltons drectly, There are three problems that arse: ) For very hghly convectve flows the Petrov-Galern method mst be frther modfed to acheve convergence. There s no agreement as to what the best way s. ) For large Reynolds nmbers the solton mst be obtaned ncrementally startng from a low Re, then ncreasng the Re nmber and sng the prevos solton as ntal gess ntl the desred vale of Re s reached. 3) In many flow problems as Re ncreases there are bfrcatons ponts where the solton changes to a dfferent mode that s physcally more stable. rect soltons are rarely capable to Swtch to the new mode and contne along the branch that s Physcally nstable. For these reasons we wll prefer to se tme dependent algorthms To reach steady state. These always follow the physcally stable 7 Branches and allow s to se Petrov-Galern as n lnear systems. et s se drect sbsttton to solve the Brgers eqaton. + + n+ dw d d are obtaned from ε + w = Ths dffers from Galern n that the weghtng fnctons α h dn w = N + change at each teraton becase the h cell Peclet nmber s =, and α = coth ε We have chosen to apply the Petrov-Galern method to the non-lnear problems fndng α n the same way as n the lnear case. It s far from obvos that ths shold be the rght way to do t. However, t was proved (H-E 98) sng Taylor seres expansons that at least the frst two terms n the error expanson of the Brgers eqaton are dentcal to the lnear case. 8 For the Newton-Raphson teraton the Petrov-Galern teraton dw dδ dδ d taes the form ε + w + w Δ = Ths eqaton s of the Convecton-ffson-Reacton type, and That s the reason why the Petrov-Galern method as we have Appled s not sffcent to stablze the calclatons when t s Hghly convecton domnated. Varos modfcatons to resolve ths staton have been proposed (see e. g. Id 996). A smple way to mprove the stablty s to se The Petrov-Galern weghts n the mddle term only, and the shape Fnctons n the other terms. Ths s effectve to stablze the Calclatons bt t sacrfces some of the accracy. 9 5