NORMAL MODE ANALYSIS OF SECOND-ORDER PROJECTION METHODS FOR INCOMPRESSIBLE FLOWS. Jae-Hong Pyo and Jie Shen. (Communicated by Z.

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1 DISCRETE AND CONTINUOUS Webite: DYNAMICAL SYSTEMS SERIES B Volume 5 Number 3 Augut 005 pp NORMAL MODE ANALYSIS OF SECOND-ORDER PROJECTION METHODS FOR INCOMPRESSIBLE FLOWS Jae-Hong Pyo and Jie Shen Department of Mathematic Purdue Univerity Wet Lafayette IN USA Communicated by Z. Zhang Abtract. A rigorou normal mode error analyi i carried out for two econd-order projection type method. It i hown that although the two cheme provide econd-order accuracy for the velocity in L -norm their accuracie for the velocity in H 1 -norm and for the preure in L -norm are different and only the conitent plitting cheme introduced in [6] provide full econd-order accuracy for all variable in their natural norm. The advantage and diadvantage of the normal mode analyi v. the energy method are alo elaborated. 1. Introduction. We conider the movement of an incompreible fluid inide Ω whoe velocity u and preure p are governed by the Navier-Stoe equation: t u + u u + p ν u = f in Ω [0 T ] u = 0 in Ω [0 T ] u Γ = 0 and u t=0 = u 0 in Ω 1.1 where for the ae of implicity a homogeneou Dirichlet boundary condition i aumed for u on Γ = Ω. One of the main numerical difficultie in olving 1.1 i that the velocity and the preure are coupled together through the incompreibility condition u = 0. The original projection method wa introduced by Chorin [] and Temam [16] in the late 60 to decouple the computation of velocity from the preure it quicly gained popularity in the computational fluid dynamic community and over the year an enormou amount of effort have been devoted to develop more accurate and efficient projection type cheme we refer to [5] for a comprehenive and up-to-date review on thi ubject. The numerical analyi of projection type method are uually carried out by uing an energy method ee for intance [14 1 5] and the reference therein or a normal mode analyi e.g. [ ]: the energy method i capable of providing rigorou etimate for general etting but often overloo particular error tructure of projection error; on the other hand the normal mode analyi i only applicable to very pecial domain uch a a periodic channel or a quarter plane but often reveal more precie information on the error behavior. 000 Mathematic Subject Claification. 65N35 76D05. Key word and phrae. Naiver-Stoe equation projection method Fractional tep method Incompreibility Normal Mode Analyi Spectral approximation. Partially upported by NSF Grant DMS

2 818 JAE-HONG PYO AND JIE SHEN We conider in thi paper two econd-order projection type cheme namely the rotational preure-correction cheme [7] and the conitent plitting cheme [6]. The error analyi of the firt cheme wa performed in [1] for a emi-infinite periodic trip uing a normal mode analyi for one normal mode only and in [7] for general domain uing an energy method. It i hown in [1] that the velocity and preure error are of econd-order but ince only one normal mode wa conidered it i not clear the etimate hold in which functional pace. A it i a well nown fact by now that the convergence rate of projection type cheme may differ if meaured in different norm. In fact it i hown in [7] that the preure approximation of thi cheme although more accurate than the tandard preure-correction cheme i only 3/-order accurate in L 0 T ; L Ω for general domain. On the other hand there i no tability and error analyi available for the econd-order conitent plitting method. The purpoe of thi paper are to carry out a ytematic and rigorou normal mode error analyi for the two method to clarify their different error behavior and to exemplify the advantage and hortcoming of normal mode analyi. In particular We prove that the conitent plitting cheme provide fully econd-order accuracy. The gauge method propoed by E and Liu [4] i another cla of projection type method. It firt-order verion ha been analyzed in [ ]. The original gauge method involve boundary condition which are not uitable for finite element Nochetto and Pyo contructed the finite element gauge-uzawa method in [9 13] and proved the convergence of the bacward Euler time dicrete gauge method by variational approach under realitic regularity of given data in [10 13]. However a far a we now no rigorou tability/error analyi i available for the econdorder verion of the gauge or gauge-uzawa method. It turn out that in the pace continuou cae the econd-order gauge or gauge-uzawa method i equivalent to the conitent plitting cheme. Hence we have alo proved that in the pecial etting conidered in thi paper the econd-order gauge and gauge-uzawa method with properly choen boundary condition are alo fully econd-order accurate. The ret of the paper i organized a follow. In we decribe the rotational form of preure-correction projection method Algorithm 1 and the conitent plitting method Algorithm and we compute the normal mode reference olution of a econd-order coupled cheme. We compare the reference olution with the normal mode olution of Algorithm 1 in 3 and Algorithm in 4 to etimate their convergence rate. Several intereting iue are addreed in Section 5.. The two cheme and the reference olution. In thi ection we decribe the rotational preure-correction cheme the conitent plitting cheme and a econd-order coupled cheme a the reference olution. For the ae of implicity the time derivative in thee cheme will be approximated by the econd-order bacward difference formula BDF. Since the treatment of the nonlinear term doe not contribute in any eential way to the error behavior we do not chooe any pecific treatment for the nonlinear term intead we imply aume g n+1 i a certain econd-order approximation of f u u at t = n + 1τ where τ i the time tep..1. Decription of the two cheme. The tandard preure-correction projection cheme wa propoed by Van Kan in [18]. It i well nown that it uffer from an artificial preure Neumann boundary condition. Hence we conider it rotational form which wa firt introduced in [17]:

3 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 819 Algorithm 1 The Rotational Form of Preure-Correction Projection Method. Set initial value uing a uitable firt order projection method and repeat for n N = [ T τ 1]. Step 1: Find ũ n+1 a the olution of 3ũ n+1 4u n + u n 1 + p n ν ũ n+1 = g n+1 ũ n+1 Γ = 0. Step : Find ψ n+1 a the olution of ψ n+1 = 3 ũn+1 ν ψ n+1 Γ = 0 where ν i the unit outward normal vector. Step 3: Update u n+1 and p n+1 by: u n+1 = ũ n+1 3 ψn+1 p n+1 = ψ n+1 + p n ν ũ n In thi cheme the homogeneou Neumann boundary condition i impoed on a non-phyical variable ψ n+1 in.. Moreover by uing the identity ũ n+1 = u n+1 = u n+1 we find that u n+1 atifie 3u n+1 4u n + u n 1 + p n+1 ν u n+1 = g n+1 u n+1 = 0 u n+1 ν Γ = 0. Thu the preure p n+1 doe not uffer from the artificial boundary condition a in the tandard preure-correction cheme. A tated in the introduction the above cheme i till not fully econd-order accurate cf. [7]. Recently Guermond and Shen propoed the conitent plitting method in [6] and howed numerically that it econd-order verion i fully accurate. However no tability and error analyi i available. Below i a econd-order verion of the conitent plitting cheme baed on BDF: Algorithm The Conitent Splitting Method. Set initial value uing a uitable firt order projection method and repeat for n N = [ T τ 1]. Step 1: Find u n+1 a the olution of 3u n+1 4u n + u n 1 + p n p n 1 ν u n+1 = g n+1 u n+1 Γ = 0. Step : Find ψ n+1 a the olution of ψ n+1 = 3un+1 4u n + u n 1 ν ψ n+1 Γ = 0. Step 3: Update p n+1 by.4.5 p n+1 = ψ n+1 + p n p n 1 ν u n+1..6

4 80 JAE-HONG PYO AND JIE SHEN.. Normal Mode Analyi of the reference olution. Since we are concerned with normal mode analyi we hall retrict our attention to the following linearized Navier-Stoe equation on Ω = [ 1 1] [0 π]: t u + p = ν u in Ω 0 T ] u = 0 in Ω [0 T ] u±1 y t = 0 ux 0 t = ux π t x y t [ 1 1] [0 π [0 T ] u t=0 = u 0 in Ω..7 Although it i cutomary to chooe the olution of.7 a the reference olution ince we are only intereted in econd-order cheme we hall however ue the olution of the following econd-order coupled cheme a the reference olution: 3U n+1 4U n + U n 1 + P n+1 = ν U n+1 U n+1 = 0 U n+1 x=±1 = 0. It i obviou that.8 i a fully econd-order cheme. We write U n P n x y = Un P nxeiy and denote u := x x p u p := ip u := x u + iv.8 where u = u v. Then after a Fourier tranform in the y variable the problem.8 i reduced to a family indexed by of one-dimenional problem: 3U n+1 4U n + Un 1 U n+1 = 0 U n+1 x=±1 = 0. + P n+1 = ν U n+1.9 To implify the notation we hall drop the index when no confuion would occur and we aume that the olution of.9 tae the following normal mode form U n P n x = ρ n u px..10 Then.9 become an ODE ytem for u p: xu + 3ρ 4ρ + 1 νρ u = 1 ν p u = 0 Setting p = 0 u±1 = v±1 = 0. λ := + 3ρ 4ρ + 1 νρ and eliminating v and p from the firt equation in.11 we obtain.11 4 xu + λ xu + λu = 0 and u±1 = 0..1

5 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 81 It can be eaily verified that the neceary condition for.1 to have a non-trivial olution are 0 and λ < 0. Hence we et µ = λ = + 3ρ 4ρ + 1 νρ.13 o that the olution of.1 are linear combination of co µx coh x ymmetric and in µx inh x anti-ymmetric. A imple calculation how that the ymmetric olution of.11 are of the form 0: coh x ux = co µx co µ coh vx = µ i in µx + 1 x co µinh i coh.14 px = + µ inh x ν co µ coh. The remaining condition v±1 = 0 lead to µ tan µ + tanh = It can be eaily een that for any given 0 there exit a unique olution µ of.15 on each open interval I := 1π +1π where Z with 0. We denote it a µ I. Note however that no olution exit in I 0. For each µ I let ρ I be the olution of.13. We find from.13 that 1 ρ I 3 1 and µ I + 1 ν..16 Hence the olution µ I of.15 may not be compatible with.13. In particular for any given τ > 0 there exit K τ > 0 uch that for > K τ there i no olution for From.13 1 ρ I =. ± 1 νµ I + One of them with the poitive ign before the quare root converge to 1 3 a τ 0 and hence correpond to the trivial olution in view of.10. Therefore we conclude 1 ρ I =. 1 νµ I + We denote the ymmetric olution.14 a u I = u I x v I x and p I x for each 0 and admiible interval I. Similarly the antiymmetric olution are given by inh x ux = in µx in µ inh vx = µ i co µx + 1 x in µcoh i inh.17 px = + µ coh x ν in µ inh.

6 8 JAE-HONG PYO AND JIE SHEN Since v±1 = 0 we find µ cot µ coth = 0. For any given 0 there exit a unique olution µ on each open interval J = { π + 1π 1 1π π 1.18 where Z with 0. We denote it a µ J. Note however that no olution exit in J 0 = π π. In view of.13 we have 1 ρ J 3 1 and µ J + 1 ν.19 1 ρ J =. 1 νµ J + We denote the antiymmetric olution.17 a u J p J x for each 0 and admiible interval J. In ummary the olution of.8 can be written a = u J x v J x and U n P n x y = α ρ n I u I p I x + β ρ n J u J p J x expiy.0 where α and β are determined by the normal mode expanion of the initial velocity namely: U 0 x y = α u I x + β u J x expiy..1 In view of.16 and.19 the maximum value of µ I µ J have to be bounded by 1 τ. Hence.0 i a finite erie for any given τ 0. One of the difficultie in the normal mode analyi rigorou i that the normal mode {u I u J } are not mutually orthogonal a the Fourier mode are. Hence we are led to mae the following aumption on U 0 x y and it derivative: Aumption 1. We aume that for certain m to be pecified later there exit a poitive contant M m independent of τ uch that α D γ 1 x u I xd γ y expiy γ 1 +γ m + β D γ 1 x u J xd γ y expiy. M m x y Ω. The above aumption enure that the normal mode expanion of U 0 and all it derivative of order up to m are abolutely convergent. Since. hold for all x y Ω one can hown that it can be equivalently repreented by γ 1γ m { γ 1 + µ γ I α + γ 1 + µ γ J β } M m.

7 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS Error analyi for the Rotational Preure-Correction Method. In thi ection we compute normal mode olution of Algorithm 1 and to etimate the error by comparing with the reference olution computed in.. Similarly a for the reference olution we write the olution of Algorithm 1 a u n ũ n ψ n p n x y = un ũn ψn pn xeiy. After eliminating ũ n+1 from.1 by uing.3 and performing a Fourier tranform in y we obtain a family of one-dimenional problem for u n ũn ψn pn : 3u n+1 p n+1 = 0 4u n + un 1 u n+1 = 0 3ũ n+1 4u n + un 1 ν u n+1 = p n+1 ν ũ n+1 = p n ψ n+1 = 3 ũ n+1 ũ n+1 ±1 t = ṽ n+1 ±1 t = u n+1 ±1 t = x ψ n+1 ±1 t = We hall temporarily drop the index and aume that olution of the above ytem tae the following normal mode form: u n ũ n p n x = ρ n û ũ px. Then 3.1 become an ODE ytem for û ũ p: xû + 3 ρ 4 ρ + 1 ρ û = 1 ν ν p p = 0 û = 0 x ũ + 3 ũ = 4 ρ 1 ν ρ ν û + 1 ρν p ψ = 3 ũ ũ±1 t = ṽ±1 t = û±1 t = x ψ±1 t = The procedure for olving 3. i eentially the ame a for the reference olution. The neceary condition for 3. to have non-trivial olution are 0 and that there exit a real number µ atifying µ = + 3 ρ 4 ρ + 1 ρ. 3.3 ν Let u denote λ := + 3 ν. 3.4

8 84 JAE-HONG PYO AND JIE SHEN Then the ymmetric olution of the ODE ytem 3. are coh x ûx = co µx co µ coh vx = µ i in µx + 1 i co µinh x coh ũx = co µx co µ coh x coh µ ṽx = i in µx + 1 x co µinh i coh 1 3 ρ 1 ρ 1 i 3 ρ 3 co µ px = µ + ψx = inh x ν co µ coh 3 ρ 1 ρ 1 ρ 3 co µ + 3 ρ 1 ρ 1 3 ρ 3 co µ 1 inh x coh λ inh x coh 1 λ inh λx coh λ inh λx. coh λ coh x coh coh λx coh λ 3.5 A econd relation between µ and ρ can be obtained by the condition ṽ±1 = 0: µ tan µ + tanh = 3 ρ 1 ρ 1 3 ρ 3 tanh λ tanh λ. 3.6 For any given 0 the relation 3.3 and 3.6 determine a unique olution µ I in each open interval I 0. In view of 3.3 we have ρ I and µ I + 1 ν. 3.7 Similarly a for the reference olution we find that for each µ I 3.3 determine a unique ρ I given by 1 ρ I = ν µ I + We will denote the olution in 3.5 a û I = û I v I ũ I = ũ I ṽ I p I and ψ I.

9 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 85 Similarly the antiymmetric olution are given by inh x ûx = in µx in µ inh vx = µ i co µx + 1 i ũx = in µx in µ inh x inh in µcoh x inh µ ṽx = i co µx + 1 x in µcoh i inh 1 3 ρ 1 ρ 1 i 3 ρ 3 in µ px = µ + ψx = coh x ν in µ inh 3 ρ 1 ρ 1 ρ 3 in µ + 3 ρ 1 ρ 1 3 ρ 3 in µ 1 coh x inh λ coh x inh 1 λ coh λx inh λ coh λx. inh λ inh x inh inh λx inh λ 3.9 We determine from the condition ṽ±1 = 0 that 3 ρ 1 ρ 1 µ cot µ coth = 3 ρ 3 coth λ coth λ For any given 0 there i unique olution µ J of 3.10 in each open interval J 0. In view of 3.3 we have 1 ρ J 3 1 and µ J + 1 ν and 1 ρ J = ν µ J + We will denote the olution in 3.9 a û J = û J v J ũ J = ũ J ṽ J p J and ψ J. In ummary the normal mode olution of the Algorithm 1 can be written a u n ũ n p n = + α ρ n I û I ũ J p I x expiy β ρ n J û J ũ J p J x expiy. 3.1 Remar 3.1. We infer from 3.4 that λ Oντ 1. Hence ũ ṽ ψ in 3.5 and 3.9 exhibit a boundary layer of width Oντ 1 and we infer from 3.8 and 3.11 that it magnitude i of order Oντ. On the other hand it i clear that ũ ṽ p are free of any puriou term and the only error are related to µ µ which i hown below to be econd-order accurate. Remar 3. Initial Value. Since the pace panned by {u I u J } and {û I û J } are in general different it i impoible for Algorithm 1 to have the ame initial condition a the reference olution. Intead we hall aume that α and

10 86 JAE-HONG PYO AND JIE SHEN β are the ame a in the reference initial olution in.1 o the initial velocity for Algorithm 1 become u 0 = α û I x expiy + β û J x expiy. The lemma below indicate that the initial error will be of econd-order. Lemma 3.3. We have the following error bound: 8τ ν + 1 µ I µ I 3 1π 1 µ J µ J 8τ ν π 4 µ I + 4 µ J µ I µ I 8τ ν 3 µ J µ J 8τ ν µ I + 5 µ J and ρ I ρ I Cτ 3 ν 3 µ I ρ J ρ J Cτ 3 ν 3 µ J + 3. Proof. We hall only carry out the proof on the interval I = 1π +1π ince the etimate on J defined in.18 can be derived by the ame argument. Subtracting 3.6 from.15 we get µ I tan µ I µ I tan µ I = 3 ρ I 1 ρ I 1 3 ρ 3 tanh I λ I tanh λ We will firt bound the right hand ide of From 3.7 and 3.8 we obtain 1 ρ I ρ I = ν µ I + ν µ ν µ I + I On the other hand we infer from 3.4 that λ I which yield tanh λ I tanh λ I 1. Therefore in light of 3.3 and 3.17 we have 3 ρ I 1 ρ I 1 µ I tan µ I µ I tan µ I 3 ρ 3 I ντ 3 µ I + 1 ρ I ρ I 4τ ν µ I

11 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 87 Let fx = x tanx. It can be eaily verified that f x+inx cox x = co x x on I. Since µ I and µ I I the mean value theorem and 3.18 yield µ I µ I 1π µ I tan µ I µ I tan µ I 8τ ν 3 1π µ I where we note 0. Since µ I + µ I + 1π we infer from 3.19 that µ I µ I = µ I + µ I µ I µ I 3.0 8τ ν µ I +. Since µ I +1 1 µ I the above two etimate imply 3.13 and Next we etimate ρ I ρ I. Since ρ I and ρ I we find by the mean value theorem that ρ I ρ I ln ρ I ln ρ I 3 A i 3.1 i=1 with A 1 := ln ρ I 3 ρ I 4 ρ I + 1 ρ I A := ln ρ I 3ρ I 4ρ I + 1 ρ I 3ρ I A 3 := 4ρ I + 1 ρ 3 ρ I 4 ρ I + 1 I ρ I. Uing the Taylor expanion ln ρ I = 1 n+1 n=1 n ρ I 1 n we can write A 1 a A 1 = ρ 3 ρ I 1 1 n+1 I 1 ρ ρ I 1 n 1 I n n=1 = ρ ρ I 1 ρ I 1 1 n+1 I 1 ρ ρ I 1 n 1 I n n= = ρ I 1 ρ I 1 1 n+1 ρ ρ I 1 n I n n=3 ρ = ρ I 1 3 I + 3 ρ I 3 1 n+1 6 ρ ρ I 1 n 3. I n n=4 3. Therefore 3.17 lead to A 1 Cτ 3 ν 3 µ I + 3.

12 88 JAE-HONG PYO AND JIE SHEN By the ame argument we obtain A Cτ 3 ν 3 µ I + 3. Subtracting 3.3 from.13 yield A 3 = ντµ I µ I which along with 3.0 lead to A 3 8τ 3 ν µ I +. In conjunction with µ I +1 1 µ I inerting above etimate on A i i = 1 3 into 3.1 yield 3.15 and conclude the proof. Theorem 3.4. Let U n P n and u n ũ n p n be repectively the olution of.8 and of Algorithm 1 and Suppoe Aumption 1 hold with m = 8. Then we have U n u n L Ω CM 7ν 3 τ P n p n L Ω CM 8ν τ 1 n N ũ n L Ω CM 4τ Proof. Let u denote the error function at time t = 0 by: E I := u I ũ I E J := u J ũ J e I := p I p I e J := p J p J. 3.4 We firt compare.14 and 3.5 on a fixed interval I. Uing the velocity error can be bounded by E I L Ω 1 C + µ I µ I µ I Cτ ν µ I + 3. Similarly the preure error can be bounded by e I L Ω Cν µ I + µ µ I I µ I + µ I Cτ ν 3 µ I + 3. Since ρ J and ρ I the mean value theorem lead to ρ n I ρ n I n ρ I ρ I. So 3.15 lead to note nτ T : ρ n I ρ n I n ρ I ρ I Cτ ν 3 µ I + 3. The error E J ρ n J ρ n J and ej can be etimated in a imilar fahion. We now compute the error of Algorithm 1 by comparing 3.1 with.0 and uing the above three inequalitie. Since we have ρ I and u I L Ω C 1 + µ I we can derive U n u n L Ω α expiy ρ n I ρ n I u I + ρ n L I E I Ω + β expiy ρ n J ρ n J u J + ρ n L J E J Ω Cτ ν 3 α µ I + 3 µ I + β µ J + 3 µ J CM 7 ν 3 τ.

13 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 89 On the other hand uing p I C µ I + ν we can obtain P n p n L Ω α expiy ρ n I ρ n I p I + ρ n L I e I Ω + β expiy ρ n J ρ n J p J + ρ n L J e J Ω Cτ ν 4 α µ I β µ J + 4 CM 8 ν 4 τ. It remain to prove the lat inequality in 3.3. We compute ũ I from 3.5 and ue to get ũ L 3 ρ I 1 ρ I 1 λ I I Ω 3 ρ 3 I ν µ I + λ I λ I 3 τ 3 ν 3 µ I + 3 τ 3 ν 3 Similarly we can alo derive ũ L J 4 τ ν µ J Ω µ I 1 +. Then umming up the above two inequalitie for and yield 3.3. We note that 3.3 indicate that the divergence of ũ n i only 3 -order accurate. Conequently the error of ũ n in H 1 Ω i at mot 3 -order accurate. 4. Error analyi of the Conitent Splitting Method. Similarly a in the lat ection we compute normal mode olution of Algorithm and etimate the error by comparing with the reference olution in.. Writing the olution of Algorithm a u n ψ n p n x y = un ψn pn xeiy and performing a Fourier tranform in y we obtain a family of one-dimenional problem for u n ψn pn : 3u n+1 4u n + u n 1 ν u n+1 = p n p n 1 p n+1 = 0 ψ n+1 = 3un+1 4u n + u n 1 p n+1 = ψ n+1 + p n p n 1 ν u n+1 u n+1 ±1 t = v n+1 ±1 t = x ψ n+1 ±1 t = We hall temporarily drop the index and aume that olution of the above ytem tae the following normal mode form: u n p n ψ n x = ρ n û p ψx

14 830 JAE-HONG PYO AND JIE SHEN then 4.1 become xû + 3 ρ 4 ρ + 1 ρ ν û = ρ 1 ρ ν p p = 0 x ψ + 3 ρ 4 ρ ρ 1 ρ 13 ψ ρ = ν ρ 4 p ν û±1 t = v±1 t = x ψ±1 t = A before one can how that the neceary condition for 4. to have non-trivial olution are 0 and that there exit a poitive real number µ atifying The ymmetric olution of ODE ytem 4. are coh x ρ 1 ûx = co µx co µ coh ρ vx = µ i in µx + 1 i µ = + 3 ρ 4 ρ + 1 ρ. 4.3 ν co µinh x coh px = µ + inh x ν co µ coh ρ 1 µ ψx + = ρ ν µ The condition v±1 = 0 lead to which can be rearranged to Dividing both ide by + ρ 1 ρ in µx co µ ρ 1 µ tan µ + tanh = ρ co µx co µ coh x coh i µ in µx 1 x co µinh i coh inh x. coh µ tan µ tanh ρ µ + ρ 1 ρ µ µ tan µ = ρ 1 ρ tanh. ρ µ ρ µ + ρ 1 we obtain 4.4 µ tan µ + tanh = ρ 1 µ + ρ µ + ρ 1 tanh. 4.5 One can how that for any given 0 there i a unique olution µ I of 4.5 in each open interval I 0. In view of 4.3 we have 1 ρ I 3 1 and µ I + 1 ν. 4.6 We will denote the olution in 4.4 a û I = û I v I p I and ψ I.

15 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 831 Similarly the antiymmetric olution of 4. are inh x ρ 1 ûx = in µx in µ inh ρ vx = µ i co µx + 1 i px = µ + ρ 1 ψx = in µcoh x inh coh x ν in µ inh µ + ρ ν µ in µx in µ ρ 1 ρ co µx + in µ We determine from the condition v±1 = 0 that ρ 1 µ cot µ coth = ρ µ which can be rearranged to Dividing both ide by inh x inh i µ co µx + 1 x in µcoh i inh coh x. inh cot µ + coth ρ µ + ρ 1 ρ µ µ cot µ = ρ 1 ρ coth. ρ µ ρ µ + ρ 1 we obtain 4.7 µ cot µ coth = ρ 1 µ ρ µ + ρ 1 coth. 4.8 For any 0 there i a unique olution µ J of 4.8 in each open interval J 0. In view of 4.3 we have 1 ρ J 3 1 and µ J + 1 ν. We will denote the olution in 4.7 a û J = û J v J p J and ψ J. Hence the normal mode olution of Algorithm can be written a u n p n = α ρ n I û I p I x expiy + β ρ n J û J p J x expiy 4.9 where α and β are given contant in the initial value expanion.1 ee Remar 3.. Remar 4.1. We note that contrary to Algorithm 1 ee Remar 3.1 there i no puriou boundary layer term in the approximation olution û v p in 4.4 and 4.7. Lemma 4.. We have the following error bound: µ I µ I 8τ ν µ I + 3 1π 1 µ I µ J µ J 8τ ν µ J + 3 π µ J 4.10

16 83 JAE-HONG PYO AND JIE SHEN 9 µ I µ I + 1 8τ ν µ I µ I 9 µ J µ J + 1 8τ ν µ I + 3 µ I 4.11 and ρ I ρ I Cτ 3 ν 3 µ I ρ J ρ J Cτ 3 ν 3 µ J + 3. Proof. We hall only carry out the proof on the interval I = 1π +1π ince the etimate on J defined in.18 can be derived by the ame argument. Subtracting 4.5 from.15 we obtain: µ I tan µ I µ I tan µ I = ρ I 1 µ I + ρ I µ I + ρ I 1 tanh Note that 3.17 i alo valid here. Since tanh 1 the right hand ide of 4.13 can be bounded by µ I tan µ I µ I tan µ I ρ I 1 µ I + ρ I µ I τ ν µ I + 3 µ. I Let fx = x tanx. It can be eaily verified that f x+inx cox x = co x x on I. Since µ I and µ I I the mean value theorem and 4.14 yield µ I µ I 1π µ I tan µ I µ I tan µ I 8τ ν µ I + 3 1π µ. I Note 0. Since µ I + µ I + 1π 4.15 lead u µ I µ I = µi + µ I µ I µ I 8τ ν + 1 µ I µ I In light of µ I +1 1 µ I the above two etimate imply 4.10 and Similarly a in the proof of Lemma 3.3 ρ I ρ I can be plit a in 3.1 and the etimate 3. i till valid for A 1 and A namely: A 1 Cτ 3 ν 3 µ I + 3 and A Cτ 3 ν 3 µ I + 3. Subtracting 4.3 from.13 lead to A 3 = ντµ I µ I and 4.16 lead to A 3 8τ 3 ν µ I µ. I Inerting above A 1 A and A 3 into 3.1 yield 4.1 which conclude the proof.

17 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 833 Theorem 4.3. Let U n P n and u n ũ n p n be repectively the olution of.8 and of Algorithm and uppoe Aumption 1 hold with m = 8. Then we have U n u n L Ω CM 8ν 3 τ P n p n L Ω CM 8ν 3 τ 1 n N ũ n L Ω CM 6ν τ Proof. The proof i very imilar to that of Theorem 3.4. We ue the ame error function in 3.4 namely E I := u I ũ I E J := u J ũ J e I := p I p I e J := p J p J. We firt compare.14 and 4.4 on a fixed interval I. Uing the velocity error can be bounded by E I L Ω 1 C + µ I µ I µ I + C 1 + Cτ ν µ I + 4 µ I. µ I Similarly the preure error can be bounded by e I L Ω Cν µ I + µ µ I I µ I + µ I Cτ ν 3 µ I + 4 µ I. ρi 1 ρ I Since ρ J and ρ I which come from.19 and 4.6 the mean value theorem lead to ρ n I ρ n I n ρi ρ I. So 4.1 give u ρ n I ρ n I n ρi ρ I Cτ ν 3 µ I + 3. By the ame argument the above etimation hold for E J ρ n J ρ n J and e J. We now compute the error of Algorithm by comparing 4.9 with.0 and uing above three inequalitie. Since we have ρ I 1 C 1 + µ I 3 1 and u I L Ω U n u n L Ω α expiy ρ n I ρ n I u I + ρ n L I E I Ω + β expiy ρ n J ρ n J u J + ρ n L J E J Ω Cτ ν 3 α µ β I µ + 4 J CM 8 ν 3 τ.

18 834 JAE-HONG PYO AND JIE SHEN Uing p I C µ I + ν we can obtain P n p n L Ω α expiy ρ n I ρ n I p I + ρ n L I e I Ω + β expiy ρ n J ρ n J p J + ρ n L J e J Ω Cτ ν 3 α µ β I µ + 4 J CM 8 ν 3 τ. It remain to prove the lat inequality in We compute ũ I from 4.4 and then ue 4.3 and 3.17 to get ũ L ρ I 1 µ I I + Ω ρ I µ I τ ν µ I Similarly we can derive ũ 6 L + 1 J 4τ ν µ J Ω + 3. Summing up the above two etimate for and yield 4.17 which complete the proof. 5. Micellaneou iue Standard preure-correction cheme. We oberved that the normal mode analyi reveal explicit error tructure of Algorithm 1 and. It i alo intereting to tae a loo at a tandard preure-correction cheme the only difference of which with the rotational preure-correction cheme Algorithm 1 i that in the third tep.3 i replaced by u n+1 = ũ n+1 3 ψn+1 p n+1 = ψ n+1 + p n. 5.1 Similarly a for Algorithm 1 taing the Fourier tranform in the y variable in the Algorithm yield a family of one-dimenional problem indexed by Z; 3ũ n+1 4ũ n + ũ n 1 ν ũ n+1 = 1 3 7p n 5p n 1 + p n 3 ν p n+1 p n + p n+1 = 0 u n+1 = 0 u n+1 = ũ n+1 3 p n+1 p n ũ n+1 ±1 t = ṽ n+1 ±1 t = u n+1 ±1 t = x p n+1 ±1 t = 0. Auming the normal mode olution tae the form u n ũ n p n x = ρ n û ũ px

19 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 835 we find that x ũ + 3 ρ 4 ρ + 1 ρ ν 3 ρ 1 ρ ν p + p = 0 û = 0 ũ = 7 ρ 5 ρ ρ 3 p ν 5. û = ũ 3 ρ 1 ρ p ũ±1 t = ṽ±1 t = û±1 t = x p±1 t = 0. Let u denote and µ = + 3 ρ 4 ρ + 1 ρ ν λ = + 3 ρ ν ρ 1. The ymmetric olution of the ODE ytem 5. are coh x ûx = co µx co µ coh vx = µ i in µx + 1 i co µinh x coh ũx = co µx co µ coh x coh µ ṽx = i in µx + 1 x co µinh i coh 1 3 ρ 1 ρ 1 i 3 ρ 3 co µ px = µ + inh x ν co µ + 3 ρ 1 ρ 1 3 ρ 3 co µ inh x coh λ coh + µ + λ in λx co λ in λx ν co µ co λ. coh x coh Since ṽx ha 0 on boundary x = ±1 we have 3 ρ 1 ρ 1 µ tan µ + tanh = tanh 3 ρ 3 λ tan λ. co λx co λ We note that the normal mode olution for a cheme which i eentially the ame a Algorithm but with BDF replaced by Cran-Nicholon wa obtained in [3]. Remar 5.1. It i clear that ρ 1 = Oντ a for other cheme o we have λ = Oντ 1. Hence a pointed out in [3] ũ ṽ p contain a puriou high ocillatory part with magnitude of order Oντ for ũ ṽ and of order Oντ for p.

20 836 JAE-HONG PYO AND JIE SHEN 5.. The Convergence of The Conitent Splitting Method in H 1 Ω by energy method. We note that the main difference between the error behavior of the Algorithm 1 and i that there i no puriou term in the olution of Algorithm while there i a puriou boundary layer term of width Oντ 1 in the olution of Algorithm 1 which reulted in a lo of 1 -order in the divergence of ũ n and conequently ũ n in Algorithm 1 can be at bet 3 -order accurate. A hown below the error etimate in Theorem 4.3 are ufficient to etablih econd-order velocity error etimate in H 1 Ω via energy etimate without any additional regularity aumption. We denote by the norm in L Ω and by the inner product in L Ω. Let ut n pt n be the exact olution of 1.1 at time tep t n. If u n p n i the olution of the Algorithm then we denote the correponding error by From Theorem 4.3 we have E n := ut n u n and e n := pt n p n. E n L Ω + en L Ω Cτ. 5.3 By virtue of Taylor expanion of 1.1 we have 3ut n+1 4ut n + ut n 1 + pt n pt n 1 ut n+1 = R n+1 + g n where R n+1 := 1 t n+1 τ t n+1 t u t n 1 ttt tdt + t n+1 t n+1 t p t n 1 tt tdt i the truncation error. Subtracting.4 from 5.4 yield 3E n+1 4E n + E n 1 E n+1 + e n e n 1 = R n We tae the inner product of 5.5 with 4τE n+1 to get 3E n+1 4E n + E n 1 E n+1 + 4τ E n+1 =4τ R n+1 e n e n 1 E n+1 τ e n e n 1 t n+1 + Cτ 4 t n 1 Uing the following algebraic identity u ttt t + p tt t dt + E n+1. 3a n+1 4a n + a n 1 a n+1 = a n+1 a n + a n+1 a n + a n 1 and umming up n from 1 to m implie E m+1 + E m+1 E m + E 1 + E 1 E 0 + τ t m+1 + Cτ a n+1 a n a n a n 1 m E n+1 E n + E n 1 + n=1 m e n e n 1 n=1 u ttt t + p tt t dt. m E n+1 n=1

21 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 837 In conjunction with 5.3 the dicrete Gronwall lemma yield E m+1 + E m+1 E m m + E n+1 E n + E n 1 + Cτ 4 n=1 m E n+1 which implie that under the ame aumption of Theorem 4.3 we have additionally m τ ut n+1 u n+1 H 1 Ω Cτ 4 1 m N. n= The Gauge Method and The Gauge-Uzawa Method. The gauge formulation introduced by E and Liu in [4] conit of rewriting 1.1 in term of auxiliary vector field a and the gauge variable φ which atify u = a + φ. Upon replacing thi relation into the momentum equation in 1.1 we get a t + u u + φ t ν φ + p ν a = f. Setting p = φ t + ν φ we obtain the gauge formulation of 1.1: a t + u u ν a = f in Ω φ = a in Ω u = a + φ in Ω p = φ t + ν φ in Ω. To enforce the boundary condition u Γ = 0 a et of uitable boundary condition are ν φ Γ = 0 a ν Γ = 0 a τ Γ = τ φ where ν and τ are the unit vector in the normal and tangential direction repectively. We now introduce the BDF time dicretized gauge method [ ]. Algorithm 3 The Gauge Method. Set initial value uing a firt-order gauge method with φ 0 = 0 and repeat for n N = [ T τ 1]. Step 1: Find a n+1 a the olution of 3a n+1 4a n + a n 1 + ν a n+1 = g n+1 a n+1 ν Γ = 0 a n+1 τ Γ = τ φ n + τ φ n 1. Step : Find φ n+1 a the olution of φ n+1 = a n+1 Step 3: Update u n+1 by ν φ n+1 Γ = 0. n=1 5.6 u n+1 = a n+1 + φ n

22 838 JAE-HONG PYO AND JIE SHEN One may compute the preure p n+1 whenever neceary a p n+1 = 3φn+1 4φ n + φ n 1 + ν φ n Even though thi formulation i conitent with 1.1 the boundary condition a n+1 τ = τ φ n + τ φ n 1 i non-variational and thu difficult to implement within a finite element context epecially in three dimenional cae. To avoid the difficult aociated with the boundary differentiation the Gauge-Uzawa method wa deigned by Nochetto and Pyo in [9 13] by introducing an artificial velocity function ũ n+1 which vanihe on boundary: ũ n+1 = a n+1 + φ n φ n If we define ρ n+1 = φ n+1 φ n combining 5.7 and 5.9 yield We now inert 5.9 into 5.6 to obtain ũ n+1 = u n+1 ρ n+1 ρ n ũ n+1 4ũ n + ũ n 1 + p n p n 1 ν ũ n+1 = g n+1. Since u n+1 = 0 by taing the divergence of 5.10 we get ρ n+1 = ρ n + ũ n+1. In order to remove the variable φ n+1 we tae the difference between two conecutive tep of 5.8 to get p n+1 p n = 3ρn+1 4ρ n + ρ n 1 + ν ρ n+1. Hence a cheme equivalent to Algorithm i the following: Algorithm 4 The Gauge-Uzawa Method. Set initial value uing a firt-order Gauge-Uzawa method with ρ 0 = 0 and repeat for n N = [ T τ 1]. Step 1: Find ũ n+1 a the olution of 3ũ n+1 4ũ n + ũ n 1 + p n p n 1 ν ũ n+1 = g n+1 ũ n+1 Γ = 0. Step : Find ρ n+1 a the olution of { ρ n+1 = ρ n + ũ n+1 ν ρ n+1 Γ = 0. Step 3: Update u n+1 and p n+1 by u n+1 = ũ n+1 + ρ n+1 ρ n p n+1 = p n 3ρn+1 4ρ n + ρ n 1 + ν ρ n Note that it i not neceary to compute u n+1 in each time iteration for thi linearized problem. Since ũ n+1 0 while u n+1 = 0 it i adviable to compute u n+1 for the full Navier-Stoe equation to treat the convection term. Finally we how that Algorithm and 4 are equivalent.

23 NORMAL MODE ANALYSIS FOR PROJECTION TYPE METHODS 839 Remar 5. Equivalence of Algorithm and 4. Subtracting two conecutive preure equation in 5.11 and applying the Laplace operator we obtain p n+1 p n = p n p n 1 + 3ũn+1 4ũ n + ũ n 1 ν ũ n+1 which i exactly.5 and.6 in Algorithm. So we conclude that Algorithm and 4 are equivalent and conequently Algorithm 3 and 4 are alo fully econdorder accurate. 6. Concluding remar. We preented in thi paper a detailed and rigorou normal mode analyi for ome econd-order projection-type cheme. In particular it allowed u to etablih for the firt time but only in the pecial domain conidered here the tability and error analyi for the econd-order conitent plitting cheme. The main advantage of the normal mode analyi i that it reveal the particular error tructure uch a puriou boundary layer or puriou highly ocillatory term and their explicit dependence on the dynamic vicoity ν. It i important to note that the projection error decreae a ν decreae. Hence the projection-type cheme are particularly uitable for high Reynold number flow. The normal mode analyi for a ingle normal mode of a linear problem i relatively eay compared with the energy method. However a rigorou analyi which tae into account all normal mode which are not mutually orthogonal become tediouly complex even for linear problem. It alo require much higher regularity on the olution than the energy method doe. Although we have only conidered linear problem in thi paper but by combining the technique ued here and in [3] where the orthogonality of normal mode are implicitly aumed but in general doe not hold for nonlinear problem it can be hown that the error etimate etablihed in thi paper would hold for the fully nonlinear Navier-Stoe equation. Finally it i worthy repeating a already demontrated in [7] that the error etimate from a normal mode analyi on the pecial domain may not be valid in more general domain. REFERENCES [1] D.L. Brown R. Cortez and M.L. Minion Accurate projection method for the incompreible Navier-Stoe equation J. Comput. Phy no [] A.J. Chorin Numerical olution of the Navier-Stoe equation Math. Comp [3] W. E and J.-G. Liu Projection method. II. Godunov-Ryabeni analyi SIAM J. Numer. Anal no [4] W. E and J.-G. Liu Gauge method for vicou incompreible flow Int. J. Num. Meth [5] J.L. Guermond P. Minev and Jie Shen An overview of projection method for incompreible flow Inter. J. Numer. Method Eng. ubmitted. [6] J.L. Guermond and J. Shen A new cla of truly conitent plitting cheme for incompreible flow J. Comput. Phy [7] J.L. Guermond and J. Shen On the error etimate of rotational preure-correction projection method Math. Comput [8] G. E. Karniadai M. Iraeli and S. A. Orzag High-order plitting method for the incompreible Navier-Stoe equation J. Comput. Phy [9] R. Nochetto and J.-H. Pyo The Gauge-Uzawa finite element method part I : the Navier- Stoe equation SIAM J. Numer. Anal. to appear. [10] R. Nochetto and J.-H. Pyo Error etimate for emi-dicrete gauge method for the evolution Navier-Stoe equation Math. Comput

24 840 JAE-HONG PYO AND JIE SHEN [11] S. A. Orzag M. Iraeli and M. Deville Boundary condition for incompreible flow J. Sci. Comput [1] A. Prohl Projection and quai-compreibility method for olving the incompreible Navier- Stoe equation Advance in Numerical Mathematic. B. G. Teubner Stuttgart [13] J.-H. Pyo The Gauge-Uzawa and related projection finite element method for the evolution navier-toe equation Ph.D diertation at Univerity of Maryland 00. [14] Jie Shen On error etimate of the projection method for the Navier-Stoe equation: firt-order cheme SIAM J. Numer. Anal [15] J. C. Striwerda and Y. S. Lee The accuracy of the fractional tep method SIAM J. Numer. Anal no electronic. [16] R. Temam Sur l approximation de la olution de equation de Navier-Stoe par la methode de pa fractionnaire. II Arch. Rational Mech. Anal [17] L.J.P. Timmerman P.D. Minev and F.N. Van De Voe An approximate projection cheme for incompreible flow uing pectral element Int. J. Numer. Method Fluid [18] J. van Kan A econd-order accurate preure-correction cheme for vicou incompreible flow SIAM J. Sci. Statit. Comput no [19] C. Wang and J.-G. Liu Convergence of gauge method for incompreible flow Math. Comp no [0] B.R. Wetton Error analyi for Chorin original fully dicrete projection method and regularization in pace and time SIAM J. Numer. Anal Received September 004; revied January 005. URL: addre: pjh@math.purdue.edu URL: addre: hen@math.purdue.edu