PROJECTION METHOD III: SPATIAL DISCRETIZATION ON THE STAGGERED GRID

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1 MATHEMATICS OF COMPUTATION Volume 71, Number 237, Pages S (01) Article electronically published on May 14, 2001 PROJECTION METHOD III: SPATIAL DISCRETIZATION ON THE STAGGERED GRID WEINAN E AND JIAN-GUO LIU Abstract. In E & Liu (SIAM J Numer. Anal., 1995), we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous (we call this the semi-discrete case). In this paper, we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid. We prove that the numerical solution in velocity has full accuracy up to the boundary, despite the fact that there are numerical boundary layers present in the semi-discrete solutions. 1. Introduction In this paper, we continue our study of convergence and error structure for the projection method. In previous work [3], we studied semi-discrete situations when the temporal variable is discretized, but the spatial variable is kept continuous. We proved that as far as velocity is concerned, the method of Chorin and Temam [2, 12] is uniformly first order accurate up to the boundary, and the second order method of [6, 9] is uniformly second order accurate up to the boundary. However, due to the presence of numerical boundary layers, pseudo-pressure has only half order of accuracy near the boundary. We characterized explicitly the structure of the numerical boundary layer, and proved that full accuracy for pressure can be recovered if we subtract the numerical boundary layer profiles from the numerical solution. We also showed, using normal mode analysis, that the boundary layer modes turn into oscillatory modes for the pressure-increment formulation of [13, 1]. For a summary of these results, we refer to [4]. For related convergence results for the semi-discrete case, we refer to [10, 11]. The presence of singular modes in the semi-discrete solutions raises serious doubts on the convergence of the fully discrete method. Indeed, it has been known for some time that accuracy and even convergence can be lost if care is not exercised at the projection step. This is documented carefully in [5]. It has also been known for some time that full accuracy is kept if the spatial discretization is done on a staggered grid. This paper is devoted to a proof of this empirical fact. Our work still leaves open the very important issue of characterizing the minimum condition the spatial discretization has to satisfy in order to guarantee accuracy. The working assumption seems to be that as long as the projection step is truly a projection, i.e. the numerical projection operator P h satisfies: (1) P h is Received by the editor May 19, 1997 and, in revised form, March 1, Mathematics Subject Classification. Primary 65M06, 76M20. Key words and phrases. Viscous incompressible flows, projection method, numerical boundary layer, finite difference, convergence. 27 c 2001 American Mathematical Society

2 28 WEINAN E AND JIAN-GUO LIU self-adjoint; (2) P 2 h = P h (we will refer to this as the projection condition), accuracy in velocity will be kept. It remains to be seen whether this is true in general. We refer to [16] for some progress in this direction. On the other hand, there are many situations in which enforcing the projection condition is difficult. Therefore we are motivated to seek numerical methods which do not require the projection condition. The gauge method [5] has so far proved to be a very attractive alternative in this regard. This paper is organized as follows. In the next section, we review the projection method, with emphasis on spatial discretization on the staggered grid (also known as the MAC grid [7]). In Section 3, we summarize our main results. In Sections 4 and 5 we present the proof for the first and second order methods respectively. 2. Review of the projection methods In primitive variables, the Navier-Stokes equation (NSE) takes the form { t u +(u )u + p = u, (2.1) u =0. Here u =(u, v)isthevelocity,p is the pressure. For simplicity, we will only consider the case when the no-slip boundary condition (BC) is supplemented to (2.1): (2.2) u =0 on Ω, where Ω is an open domain in RI 2 with smooth or piecewise smooth boundary Time discretization. The first order projection method of [2, 12] proceeds in two steps. Step 1: Computing the intermediate velocity field u : u u n +(u n )u n = u, (2.3) u =0, on Ω. Step 2: Projecting to the space of divergence-free vector fields to obtain u n+1 : { u = u n+1 + p n+1, (2.4) u n+1 =0. The projection step enforces the boundary condition: u n+1 n p n+1 (2.5) =0, or =0, on Ω. n Second order schemes. We will concentrate on the formulation in [6, 9], and refer to [4] for a summary of other second order methods. We have u u n +(u n+1/2 )u n+1/2 = u + u n, 2 u + u n = p n 1/2, on Ω, (2.6) u = u n+1 + p n+1/2, u n+1 =0, u n+1 n =0, on Ω.

3 PROJECTION METHOD III 29 In this formulation, the homogeneous Neumann BC for pressure, p n+1/2 =0, on Ω, n is retained. An inhomogeneous BC for u is introduced so that the slip velocity of u n+1 at the boundary is of order () 2. Remark. The nonlinear convection term (u n+1/2 )un+1/2 can be treated in many 3 ways. In Theorems 1 and 2, we use the Adams-Bashforth formula: 2 (un )un 1 2 (un 1 )u n 1. This explicit treatment of the convection term requires a CFL stability constraint. In the convergence analysis, this is realized by an apriori estimate (see (4.58)). This stability restriction and the aprioriestimate are no longer needed in an implicit treatment of the convection term. It is readily seen that the projection step enforces p n+1 (2.7) n = pn p0 = = n n =0, on Ω, for the numerical solution. In general this is not satisfied by the exact solution of (2.1). Therefore we expect that p n / n has O(1) error at the boundary. This causes u and p n to have numerical boundary layers Spatial discretization. We will consider the case when the MAC scheme is used to discretize in x. An illustration of the MAC mesh near the boundary is given in Figure 1. Here pressure is evaluated at the square points (i, j), the u velocity at the triangle points (i ± 1/2,j), and the v velocity at the circle points (i, j ± 1/2). The discrete divergence is computed at the square points: ( u) i,j = u i+1/2,j u i 1/2,j x + v i,j+1/2 v i,j 1/2 y. Γ y Γ x Figure 1. The MAC mesh

4 30 WEINAN E AND JIAN-GUO LIU Other differential operators are discretized as follows: ( u) i+1/2,j = u i+3/2,j 2u i+1/2,j + u i 1/2,j x 2 + u i+1/2,j+1 2u i+1/2,j + u i+1/2,j 1 y 2, ( v) i,j+1/2 = v i+1,j+1/2 2u i,j+1/2 + u i 1,j+1/2 x 2 + v i,j+3/2 2v i,j+1/2 + v i,j 1/2 y 2, (p x ) i+1/2,j = p i+1,j p i,j x (p y ) i,j+1/2 = p i,j+1 p i,j y,, ū i,j+1/2 = 1 4 (u i+1/2,j + u i 1/2,j + u i+1/2,j+1 + u i 1/2,j+1 ), v i+1/2,j = 1 4 (v i+1,j+1/2 + v i+1,j 1/2 + v i,j+1/2 + v i,j 1/2 ), N h (u,a) i+1/2,j = u i+1/2,j a i+3/2,j a i 1/2,j 2 x a i+1/2,j+1 a i+1/2,j 1 + v i+1/2,j, 2 y N h (u,b) i,j+1/2 =ū i,j+1/2 b i+1,j+1/2 b i 1,j+1/2 2 x b i,j+3/2 b i,j 1/2 + v i,j+1/2, 2 y whereweusethenotationn to denote the nonlinear convection term. Clearly the truncation errors of these approximations are of second order. The boundary condition u = 0 is imposed at the vertical physical boundary, whereas v = 0 is imposed at the ghost circle points which are x 2 to the left or right of the physical boundary. The ghost points are eliminated using linear interpolation of the boundary conditions. More explicitly, v 1/2,j +v 1/2,j =0. Similarly the boundary condition v = 0 is imposed at the horizontal physical boundary, but u = 0 is imposed at the ghost triangle points a distance of y 2 away from the physical boundary. One shortcoming of the MAC scheme is the serious constraint on geometry. Although slightly more general situations can be studied, in the present paper we willconcentrateonthesituationwhenω=[ 1, 1] [0, 2π] with periodic boundary condition in the y direction and no-slip boundary condition in the x-direction: u(x, 0,t)=u(x, 2π, t), u( 1,y,t)=0, u(1,y,t) = 0. We will use Ωtodenote the part of the boundary at x = ±1. We will always assume that x y and h =min( x, y). Notation. We will use C to denote generic constants which may depend on the norms of the exact solutions.

5 PROJECTION METHOD III Summary of results The main results of this paper are the following (the constants are independent of and h). Theorem 1. Let (u,p) be a solution of the Navier-Stokes equation (2.1) with smooth initial data u 0 (x) satisfying the compatibility condition (3.1) u 0 (x) =0, y p(x, 0) = 2 xyp(x, 0) = 0, on Ω. Let (u h,p h ) be the numerical solution of the projection method (2.3)-(2.5) coupled with the MAC spatial discretization. Assume that <<h. Then we have (3.2) u u h L + 1/2 p p h L C( + h 2 ), (3.3) where (3.4) p p h p c L C( + h 2 ), p c (x,t) 1/2 β eα 1/2 e α 1 e α x 1 / D+p x h (x 1/2,y,t) eα + 1/2 β e α x+1 /1/2 e α D x 1 +p h (x + 1/2,y,t), α = 1/2 x 2 x arccosh(1 + 2 ), β = x 1/2 (1 e α x/ ) 1. 1/2 Remark 1. As will be seen more clearly in the proof (4.52), at the scaling = Ch 2 the boundary layer in pressure looks like t 1/2 A(y, t)κ j,wherea(y, t) issmooth, κ < 1, κ does not depend on h, andj is the horizontal grid index relative to the lower boundary. Theorem 2. Let (u,p) be a smooth solution of the Navier-Stokes equation (2.1) with smooth initial data u 0 (x) satisfying the compatibility condition (3.5) x α1 α2 y u0 (x) =0, on Ω, for α 1 + α 2 6. Let (u h,p h ) be the numerical solution of the projection method (2.6) coupled with the MAC spatial discretization. Assume that 2 << h. Then we have (3.6) u u h L + 3/2 p p h L + p p h L (0,T,L 2 ) C( 2 + h 2 ), (3.7) where (3.8) p p h p c L C( + h 2 ), p c 1/2 β eα e α 1 e α x 1 /1/2 D x + p h(x 1/2,y,t) eα + 1/2 β e α x+1 /1/2 e α D+ x 1 p h(x + 1/2,y,t), α = 1/2 x 2 x arccosh(1 + ), β = x 1/2 (1 e α x/ ) 1. 1/2

6 32 WEINAN E AND JIAN-GUO LIU Remark 2. The constraint on the size of is only technical and is different from the standard stability condition. The issue of the compatibility conditions on the initial data for the general domain are very complicated and are usually nonlocal [8]. In the case of periodic BC in y and no-slip BC in x (as in the present work), this reduces to the boundary condition (3.5). The correction term in (3.8) can also be given by 1 2 hp, as proposed in Kim & Moin s original paper [9]. The proofs of these results follow the general strategy outlined in Section 3 of [3]. Step 3 is made simpler since we can now use the inverse inequality. 4. First order schemes with spatial discretization We will concentrate on the following version of the first order projection method with the standard MAC spatial discretization: u u n + N h (u n, u n )= h u, u =0, on Ω, (4.1) u = u n+1 + h p n, h u n+1 =0, n u n+1 =0, on Ω. For a = (a, b), c = (c, d), u = (u, v), we define the following discrete inner products on the grid: (4.2) N 1 ((a, c)) = x y + x y N 1 ((u, h p)) = y + x i=1 j=1 N i=1 j=1 N N 1 (( h u,p)) = y + x N a i+1/2,j c i+1/2,j i=1 j=1 N b i,j+1/2 d i,j+1/2, N u i+1/2,j (p i+1,j p i,j ) i=1 j=1 i=1 j=1 N N v i,j+1/2 (p i,j+1 p i,j ), N (u i+1/2,j u i 1/2,j )p i,j i=1 j=1 N (v i,j+1/2 v i,j 1/2 )p i,j, and discrete norms (4.3) u =((u, u)) 1/2, u =max u i,j. i,j Denote h =min( x, y).

7 PROJECTION METHOD III 33 Lemma 4.1. We have the following. (i) Inverse inequality: (4.4) (4.5) (ii) f 1 h f. Poincaré inequality: if f x=±1 =0,then f C h f, where C depends only on the domain. (iii) Suppose n u x=±1 =0; then we have (4.6) (iv) (4.7) (v) (4.8) ((u, h p)) = (( h u,p)). Suppose u x=±1 =0; then we have 2((u, h u)) h u 2 h u 2. Suppose a x=±1 =0and c n x=±1 =0; then we have ((a, N h (u, c))) c h a u W 1,. Proof. The proof of (i iii) is standard. We show (iv). Summation by parts gives (4.9) ((u, h u)) = h u 2 + j [v 0,j+1/2 (v 1,j+1/2 v 0,j+1/2 ) v N,j+1/2 (v N+1,j+1/2 v N,j+1/2 )]. Since v x=±1 =0,wehave (4.10) v 1,j+1/2 = v 0,j+1/2, v N,j+1/2 = v N+1,j+1/2. Hence (4.11) ((u, h u)) = h u 2 +2 j (v 2 0,j+1/2 v 2 N,j+1/2 ). But (4.12) h u 2 h u 2 + j [(v 1,j+1/2 v 0,j+1/2 ) 2 +(v N+1,j+1/2 v N,j+1/2 ) 2 ] = h u 2 +4 j [(v 0,j+1/2 ) 2 +(v N+1,j+1/2 ) 2 ]. Combination of (4.11) and (4.12) gives (4.7). To show (v), denote I =((a, N h (u, c))). We have I = x y i,j a i+1/2,j (u i+1/2,j D x 0 c i+1/2,j + v i+1/2,j D y 0 c i+1/2,j) (4.13) + x y i,j b i,j+1/2 (ū i,j+1/2 D x 0 d i,j+1/2 + v i,j+1/2 D y 0 d i+1/2,j),

8 34 WEINAN E AND JIAN-GUO LIU whereweusedthenotationsa =(a, b) andc =(c, d). Summation by parts gives (4.14) I = x y c i+1/2,j [D0 x (u i+1/2,j a i+1/2,j )+D y 0 ( v i+1/2,ja i+1/2,j )] i,j x y i,j d i,j+1/2 [D x 0(ū i,j+1/2 b i,j+1/2 )+D y 0 (v i,j+1/2b i,j+1/2 )] x y j (ū N+1,j+1/2 d N,j+1/2 ū N,j+1/2 d N+1,j+1/2 )D x b N+1,j+1/2 1 4 x y j (ū 1,j+1/2 d 0,j+1/2 ū 0,j+1/2 d 1,j+1/2 )D x + b 0,j+1/2. Here we have used the fact that (4.15) b 1,j+1/2 = b 0,j+1/2, b N,j+1/2 = b N+1,j+1/2. Now, (4.8) follows directly. This completes the proof of the lemma. We set ε = 1/2, ξ =(x +1)/ε, x i = 1 +i x, ξ i = i ξ, ξ = x/ε, t n = n, t n 1/2 =(n 1/2), n =1, 2,. Clearly, we have (4.16) D ξ + a(ξ i,y j,t) = a(ξ i+1,y j,t) a(ξ i,y j,t) ξ = ε a(x i+1/ε, y j,t) a(x i /ε, y j,t) x = εd x + a(x i/ε, y j,t). This shows that D ξ + = εdx +. We will use the notation (4.17) and Dξ 2 = Dξ Dξ +, D2 y = Dy Dy +, (4.18) ξ =(D ξ +, 0), y =(0,D y + ). Denote the solutions of (4.1) as (u h, u h,p h). Motivated by the discussions in 4 of [3], we make the following ansatz, valid at t n = n, n =1, 2, : u h (x,t)=u 0(x,t)+ ε j [u j (x,t)+a j (ξ, y, t)], j=2 (4.19) u h (x,t)=u 0 (x,t)+ j=2 ε j u j (x,t), p h (x,t)=p 0 (x,t)+ j=1 ε j [p j (x,t)+ϕ j (ξ, y, t)]. Note that the functions involved are defined only on the numerical grid. So these formulas and the following ones should be understood as being valid on the grid. We have (4.20) h u h = hu 0 + ε j ( h u j + ε 2 Dξ 2 a j + D y 2 a j ), j=2

9 PROJECTION METHOD III 35 (4.21) h u h = h u 0 + j=2 ε j h u j, (4.22) h p h = h p 0 + ε 1 ξ ϕ 0 + y ϕ 0 + j=1 ε j ( h p j + ε 1 ξ ϕ j + y ϕ j ), u n+1 h (x) = u 0 (x,t n+1 )+ j=2 ε j u j (x,t n+1 ) (4.23) = k=0 1 k! ε2k u (k) 0 (x,tn )+ ε j 1 k! ε2k u (k) j (x,t n ). j=2 k=0 Next we substitute these relations into (4.1) in order to determine the coefficients of ε j in (4.19). We get hierarchies of equations by collecting equal powers of ε. The first equation in (4.1) gives (4.24) u 2 + a 2 u 2 + N h (u 0, u 0 )= hu Dξ 2 a 2. For j 1, j (4.25) u j+2 + a j+2 u j+2 + N h (u k, u j k )= h u j + D ξ 2 a j+2 + D y 2 a j. k=0 The second equation in (4.1) implies that (4.26) u 2 + a 2 = u 2 + t u 0 + h p 0 + ξ ϕ 1 + y ϕ 0. For j =2l 1, l 1, l (4.27) u j+2 + a 1 j+2 = u j+2 + t u j + h p j + ξ ϕ j+1 + y ϕ j + k! u(k) j 2k+2. k=2 For j =2l, l 1, u j+2 + a j+2 = u j+2 + t u j + h p j + ξ ϕ j+1 + y ϕ j (4.28) 1 l 1 + (l +1)! u(l+1) 0 + k! u(k) j 2k+2. From the third equation in (4.1), we obtain (4.29) h u j =0, j =0, 1,. The boundary conditions become (4.30) u j + a j =0, Dx + p j 1 + D ξ + ϕ j =0, at x = 1, ξ =0, for j>0. Next we go through all these equations, one by one, to see if they are solvable. Since this is very similar to 4.1 in [3], we will only give a summary of results. The coefficients in the expansions (4.19) can be obtained successively in the following order: (4.31) k=2 t u 0 + h p 0 + N h (u 0, u 0 )= h u 0, h u 0 =0, u 0 =0, at x = ±1, u 0 (, 0) = u 0 ( ).

10 36 WEINAN E AND JIAN-GUO LIU Using the following lemma, we know that (4.31) has a smooth solution in the sense that the divided differences of various orders are bounded. Lemma 4.2. Let (u,p) be a solution of the Navier-Stokes equation (2.1) with smooth initial data u 0 (x) satisfying some compatibility conditions. Let (u 0,p 0 ) be a solution of (4.31). Then(u 0,p 0 ) is smooth in the sense that its discrete derivatives are bounded. Moreover, we have (4.32) u u 0 L + p p 0 L Ch 2. The proof of this lemma, as well as Lemma 4.3, can be found in [15]. We next have (4.33) u 2 = u 2 + t u 0 + h p 0, (4.34) This gives (4.35) where (4.36) (4.37) (4.38) { ϕ1 = D 2 ξ ϕ 1, D ξ + ϕ 1 ξ=0 = D x + p 0 x= 1. ϕ 1 (ξ, y, t) =βd x + p 0( 1,y,t) e αξ, α = 1 ξ arccosh (1 + ξ 2 /2), β = ξ(1 e α ξ ) 1, a 2 = D ξ + ϕ 1, b 2 =0, u 3 = u 3, (4.39) (4.40) ϕ 2 =0, a 3 =0, b 3 = Dy + ϕ 1, t u 2 + h p 2 + N h (u 0, u 2 )+N h (u 2, u 0 ) = h u 2 + h ( t u 0 + h p 0 ) 1 2 t 2 u 0, h u 2 =0, u 2 x= 1 = h p 0 x= 1 ξ ϕ 1 ξ=0, on Ω. With suitable initial data, we know from the following lemma that (4.40) has a smooth solution. Lemma 4.3. Let (u,p) be a solution of the linear ODE t u + h p + N h (u 0, u)+n h (u, u 0 )= h u + f, h u =0, (4.41) u = g, at x = ±1, u(, 0) = u 0 ( ), where f, g and u 0 smooth and satisfy some compatibility conditions. Then (u,p) is smooth in the sense that its divided differences of various order are bounded. Continuing in this fashion, we get { ϕ3 = Dξ 2ϕ 3 + Dy 2 ϕ 1, (4.42) D+ϕ ξ 3 ξ=0 = D+p x 2 x= 1. The solution for (4.42) is (4.43) ϕ 3 (ξ, y, t) =βd+p x 2 e αξ + β 1 (ξ + γ)d+d x y 2 p 0 x= 1 e αξ,

11 PROJECTION METHOD III 37 where (4.44) (4.45) (4.46) (4.47) β 1 = 1 (1 e α ξ )(e α ξ e α ξ ), γ = ξ e α ξ 1 e α ξ, b 4 =0, a 4 = D ξ + ϕ 3. t u 3 + h p 3 + N h (u 0, u 3 )+N h (u 3, u 0 )= h u 3 + h h p 2, h u 3 =0, u 3 x= 1 = y ϕ 1 ξ=0, { ϕ4 = D 2 ξ ϕ 4, D ξ + ϕ 4 ξ=0 = D x + p 3 x= 1, (4.48) a 5 = D+ϕ ξ 4, b 5 = D+ϕ y 3. Obviously this procedure can be continued, and we obtain { ϕj = Dξ 2ϕ j + Dy 2ϕ j 2, (4.49) D ξ + ϕ j ξ=0 = D+ x p j 1 x= 1, (4.50) (4.51) Now if we let (4.52) ϕ j = [j/2] k=0 F j,k (y)ξ k e αξ, a j = Dξ + ϕ j 1, b j = Dy + ϕ j 2, 2N U = u 0 + ε j (u j + a j ), U n = u 0 + P n = p 0 + j=1 2N j=1 2N j=1 ε j u j, ε j (p j + ϕ j )+ε 2N+1 ϕ 2N+1, then we have (4.53) U U n + N h (U n,u n )= h U + α f, U =0, at x = ±1, U = U n+1 + h P n + α+1 g, h U n+1 =0, D x + P n = n U n+1 =0, at x = ±1, where α = N 1/2; f and g are bounded and smooth if (u 0,p 0 ) is sufficiently smooth. It is easy to see that (4.54) max U n ( ) W 1, C. 0 t T

12 38 WEINAN E AND JIAN-GUO LIU We point out that the constructed solution in (4.52) is different from the discretized solution. The constructed solution consists of two parts: the leading term is the exact solution of the NSE on MAC grid (4.31) and hence is bounded (see [15]), and the remaining terms are the solutions of the linearized (linearized on the exact solution) NSE (4.40), (4.46), and hence are bounded. As a direct consequence, the truncation terms f and g in (4.53) are also bounded. For the initial data, we have (4.55) U 0 (x) =u 0 (x)+w 0 (x), where w 0 is a bounded function. Furthermore, under the compatibility condition (3.5), we can construct a better approximate initial data (4.57) U 0 (x) =u 0 (x)+ 2 w 0 (x). The construction of the initial w 0 in (4.55) and (4.57) in general is very complicated. This is exactly where we need the compatibility conditions. In the case of periodic BC for y and no-slip boundary condition in x (the domain we considered in this paper), the construction of w 0 is more straightforward. The details of the construction are very similar to those used in the analysis of the gauge method in [14]. In that paper the details of the construction of w 0 were given. Proof of Theorem 1. Assume apriorithat (4.58) max 0 t n T un W 1, C. We will justify this aprioriassumption later (see (4.67)). In the following estimates, the constant will sometimes depend on C and C, and we will show that it is uniformly bounded. Later on we will estimate C. Let (4.59) e n = U n u n, e = U u, q n = P n p n. Subtracting (4.53) from (4.1), we get the following error equation: e e n + N h (e n,u n )+N h (u n, e n )= h e + α f n, e =0, at x = ±1, (4.60) e n+1 e + h q n = α g n, h e n+1 =0, D + x qn = e n+1 n =0, at x = ±1, e 0 = α w 0. Taking the scalar product of the first equation of (4.60) with 2e and integrating by parts, we obtain e 2 e n 2 + e e n 2 + h e 2 (4.61) 2α+1 f n 2 + e 2 2 ((e, N h (e n,u n ))) 2 ((e, N h (u n, e n ))) 2α+1 f n 2 + C( e 2 + e n 2 )+ 1 2 he 2.

13 PROJECTION METHOD III 39 Here we used Lemma 4.1. Taking the scalar product of the second equation of (4.60) with 2e n+1 yields (4.62) e n+1 2 e 2 + e n+1 e 2 e n α+1 g n 2. Combining (4.61) and (4.62), we get e n+1 2 e n 2 + e e n 2 + e n+1 e 2 + h e 2 (4.63) C ( e n e n 2 )+ 2α+1 ( f n 2 + g n 2 ). Applying the discrete Gronwall lemma to the last inequality, we arrive at (4.64) e n + e e n + e n+1 e + 1/2 h e C 1 α. Using the second equation of (4.60), we have (4.65) e n + h q n C 1 α. Now by the inverse inequality (4.4) we have e n L + h q n L + h e n α (4.66) W 1, C 1 h. If N =3and α << h 2,andwechoose small enough, we will always have (4.67) e n+1 W 1, 1. Therefore in (4.58) we can choose (4.68) C =1+ max U n ( ) W 1,, n [ T ]+1 which depends only on the exact solution (u,p). This proves that (4.69) u 0 u h L + p 0 p h L 2 + 1/2 p 0 p h L + p 0 p h p c L C. But we also have, from Lemma 4.2, (4.70) u u 0 L + p p 0 L Ch 2. Thus (4.71) u u h L + p p h L 2 + 1/2 p p h L + p p h p c L C( + h 2 ). This completes the proof of Theorem Second order schemes with spatial discretization In this section we carry out the same program as in 4 for the second order method (2.6) with the standard MAC spatial discretization: u u n u + u n = h, 2 u + u n = h p n 1/2, at x = ±1, (5.1) u = u n+1 + h p n+1/2, h u n+1 =0, n u n+1 =0, at x = ±1. Here we leave out the nonlinear term, since it does not affect the major steps but substantially complicates the presentation.

14 40 WEINAN E AND JIAN-GUO LIU We begin with the following ansatz: u (x) =u 0 (x,t n )+ ε j [u j (x,tn )+a j (ξ, y, tn )], j=2 (5.2) u n (x) =u 0 (x,t n )+ j=4 ε j u j (x,t n ), p n 1/2 (x) =p 0 (x,t n 1/2 )+εϕ 1 (ξ, y, t n 1/2 )+ε 3 ϕ 3 (ξ, y, t n 1/2 ) + j=4 ε j [p j (x,t n 1/2 )+ϕ j (ξ, y, t n 1/2 )]. Here again we set ε = 1/2, ξ =(x +1)/ε, t n = n, t n 1/2 =(n 1/2), n =1, 2,. The formulas are to be understood as being valid at the grid points. Substituting (5.2) into (5.1) and collecting equal powers of ε, we get the following equations: From the first equation in (5.1), we get (5.3) u 2 + a 2 u 2 = 1 2 ( hu 0 + D 2 ξ a 2 + h u 0 ). For j 1, (5.4) u j+2 + a j+2 u j+2 = 1 2 ( hu j + D 2 ξ a j+2 + D 2 y a j + hu j ). From the third equation in (5.1), we get (5.5) u 2 + a 2 = u 2 + t u 0 + h p 0 + ξ ϕ 1, (5.6) For j =2l, u 3 + a 3 = u 3 + t u 1 + h p 1 + ξ ϕ 2 + y ϕ 1. (5.7) u j+2 + a j+2 = u j+2 + t u j + h p j + ξ ϕ j+1 + y ϕ j 1 l 1 + (l +1)! u(l+1) 0 + k! u(k) j 2k l l! hp (l) + l k=1 k=2 k=2 1 2 k k! ( ξϕ (k) j 2k+1 + yϕ (k) j 2k ). l k=1 1 2 k k! hp (k) j 2k For j =2l +1, u j+2 + a j+2 = u j+2 + t u j + h p j + ξ ϕ j+1 + y ϕ j (5.8) l+1 1 l + k! u(k) j 2k k k! ( hp (k) j 2k + ξϕ (k) j 2k+1 + yϕ (k) j 2k ). k=1 From the incompressibility condition, we get (5.9) h u j =0, for j 0.

15 PROJECTION METHOD III 41 The boundary conditions imply that for x = 1, ξ =0, (5.10) (5.11) (5.12) for j =2l, l 1, u 0 =0, u 2 + u 2 + a 2 = hp 0 + ξ ϕ 1, u 3 + u 3 + a 3 = h p 1 + ξ ϕ 2 + y ϕ 1 ; (5.13) u j + u j + a j = h p j 2 + ξ ϕ j 1 + y ϕ j 2 + ( 1)l l 1 (l 1)! hp (l 1) 0 l 2 + k=1 for j =2l +1,l 1, ( 1) k l 1 2 k k! hp (k) j 2k 2 + k=1 ( 1) k 2 k k! ( ξϕ (k) j 2k 1 + yϕ (k) j 2k 2 ); u j + u j + a j = h p j 2 + ξ ϕ j 1 + y ϕ j 2 (5.14) l 1 + k=1 ( 1) k 2 k k! ( hp (k) j 2k 2 + ξϕ (k) j 2k 1 + yϕ (k) j 2k 2 ); and for j 0 (5.15) D x +p j + D ξ +ϕ j+1 =0. Next we go through all these equations, one by one, to see if they are solvable. It can be checked that the coefficients in the expansions (5.2) can be obtained successively in the following order: (5.16) t u 0 + h p 0 = h u 0, h u 0 =0, u 0 =0, at x = ±1, (5.17) (5.18) (5.19) where (5.20) (5.21) u 2 = u 2 + t u 0 + h p 0, { ϕ1 = 1 2 D 2 ξ ϕ 1, D ξ +ϕ 1 ξ=0 = D x +p 0 x= 1, ϕ 1 = βd x +p 0 x= 1 e αξ, α = 1 ξ arccosh (1 + ξ 2 ), β = ξ(1 e α ξ ) 1, a 2 = Dξ + ϕ 1, b 2 =0.

16 42 WEINAN E AND JIAN-GUO LIU We next have (5.22) (5.23) u 3 = u 3, ϕ 2 =0, a 3 =0, b 3 = D y +ϕ 1, (5.24) The solution for (5.24) is (5.25) where (5.26) β 1 = { ϕ3 = 1 2 (D ξ 2ϕ 3 + Dy 2ϕ 1), D ξ + ϕ 3 ξ=0 =0. ϕ 3 (y, ξ, t) =β 1 (ξ + γ)d x +D 2 y p 0 x= 1 e αξ, 1 2(1 e α ξ )(e α ξ e α ξ ), γ = ξ e α ξ 1 e α ξ, (5.27) a 4 = 1 2 Dξ + tϕ 1 + D ξ + ϕ 3, b 4 =0, (5.28) u 4 = u t u t h p 0, (5.29) (5.30) (5.31) ϕ 4 =0, a 5 =0, b 5 = 1 2 Dy + t ϕ 1 + D y +ϕ 3, u 5 = u 5, t u 4 + h p 4 = h u h( 2 t u 0 + t h p 0 ) t u t hp 0, h u 4 =0, u 4 x= 1 = 1 2 ( t h p t ξ ϕ 1 ) x= 1,ξ=0. Now if we let (5.32) 2N U = u 0 + ε j (u j + a j ), U n = u 0 + j=1 2N j=1 ε j u j, then we have (5.33) P n 1/2 = p 0 + 2N j=1 ε j (p j + ϕ j )+ε 2N+1 ϕ 2N+1, U U n U + U n = h + α f, 2 U + U n = h P n 1/2, at x = ±1, U = U n+1 + h P n+1/2 + α+1 g, h U n+1 =0, D x +P n+1/2 = n U n+1 =0, at x = ±1,

17 PROJECTION METHOD III 43 where α = N 1/2; f and g are bounded and smooth if (u 0,p 0 ) is sufficiently smooth. It is easy to see that (5.34) max U n ( ) W 1, C. 0 t T For the initial approximation, we have (5.35) U 0 (x) =u 0 (x)+ 2 w 0 (x) without the extra compatibility condition, and (5.36) U 0 (x) =u 0 (x)+ 4 w 0 (x) with the compatibility condition (3.17). Proof of Theorem 2. Assume apriorithat (5.37) max 0 t n T un W 1, C. As in the proof of Theorem 1, we let (5.38) where (5.39) e n = U n u n, e = Û û, q n = P n 1/2 p n 1/2, 2û = u + u n h p n 1/2, 2Û = U + U n h P n 1/2. From (5.1) and (5.34), we get (5.40) 2(e e n ( ) + h q n 1 ) 2 hq n = h e N h(e n 1,U n 1 ) N h(u n 1, e n 1 ) 3 2 N h(e n,u n ) 3 2 N h(u n, e n )+ α f n, e =0, at x = ±1, e n+1 + e n 2e + h (q n+1 q n )= α g n, h e n+1 =0, D +q x n+1/2 = e n+1 n =0, at x = ±1, e 0 = α w 0. Taking the scalar product of the first equation of (5.40) with e and integrating by parts, we get (5.41) e 2 e n 2 + e e n e e 2 e (q n 1 Ω 2 qn ) dx + C 2α+1 f n 2 +C ( e n 2 + e n e 2 )+ 1 2 e 2.

18 44 WEINAN E AND JIAN-GUO LIU Taking the scalar product of the second equation of (5.42) with e n+1,weobtain (5.42) e n+1 2 e 2 + e n+1 e ( en+1 2 e n 2 ) 1 2 en+1 e n 2 C 2α+1 g n 2 + C e n+1 2. Combining these two estimates, we get (5.43) e n+1 2 e n 2 + e n+1 + e n 2e 2 + h e 2 + h e 2 2 ((e, h (q n 1 2 hq n ))) + C ( e n 2 + e n e n+1 2 ) +C 2α+1 ( f n 2 + g n 2 ). To estimate the first term on the right hand side of (5.43), we let (5.44) I 2((e, h (q n 1 2 hq n ))) = 2((e, h q n )) 2 (( h e, h q n )) I 1 + I 2. Using the second equation and integrating by parts, we can write the first term as I 1 = 2((e, h q n )) = 2 (( h (q n+1 q n ), h q n )) α+2 ((g n, h q n )) (5.45) = ( h q n+1 2 h q n 2 ) h (q n+1 q n ) 2 α+2 ((g n, h q n )). Since (5.46) h (q n+1 q n ) 2 = 1 2 en+1 + e n 2e α+1 g n 2 + α+1 ((g n, e n+1 + e n 2e )), we have (5.47) I 1 = ( h q n+1 2 h q n 2 )+ 1 2 en+1 + e n 2e α+1 g n 2 + α+1 ((g n, e n+1 + e n 2e )) α+2 ((g n, h q n )).

19 PROJECTION METHOD III 45 Next we rewrite the second term as (5.48) I 2 = 2 (( h e, h q n )) = (( h (q n+1 q n ), h q n )) 1 2 α+3 (( h g n, h q n )) = ( h q n+1 2 h q n 2 ) h (q n+1 q n ) α+3 (( h g n, h q n )) = ( h q n+1 2 h q n 2 )+ h e α+3 h g n 2 α+2 (( h g n, h e )) 1 2 α+3 (( h g n, h q n )). Combining these two terms, we arrive at (5.49) I = ( h q n+1 2 h q n 2 ) ( h q n+1 2 h q n 2 ) en+1 + e n 2e 2 + h e 2 + α+1 ((g n, e n+1 + e n 2e )) α+2 ((g n, h q n )) α+2 (( h g n, h e )) 1 2 α+3 (( h g n, h q n )) α+3 h g n α+1 g n 2. This gives I ( h q n+1 2 h q n 2 ) ( h q n+1 2 h q n 2 ) (5.50) en+1 + e n 2e 2 + h e 2 + e n+1 + e n 2e h q n h q n α+1 ( g n 2 + g n H 2 ). 1 Going back to (5.45), we obtain (5.51) e n+1 2 e n en+1 + e n 2e 2 + h e ( h q n+1 2 h q n 2 ) ( h q n+1 2 h q n 2 ) 3 h q n h q n 2 + C ( e n 2 + e n e n+1 2 ) +C 2α+1 ( f n 2 + g n H 2 ). 1 Gronwall s lemma gives (5.52) e n + e + h q n + 3/2 h q n + 1/2 h e C 1 α. Now by the inverse inequality (4.4) we have e n L + h e n W 1, + h q n α (5.53) L C 1 h.

20 46 WEINAN E AND JIAN-GUO LIU If N =5and α << h 2,andwechoose small enough, we will always have (5.54) e n+1 L 1. Therefore in (5.39) we can choose (5.55) C =1+ max n [ ] U n ( ) W 1,, T which depends only on the exact solution (u,p). This proves that (5.56) u 0 u h L + p 0 p h L 2 + 1/2 p 0 p h L + p 0 p h p c L C 2. From Lemma 4.2, we have (5.57) u u h L + p p h L 2 + 1/2 p p h L + p p h p c L C( 2 + h 2 ). This completes the proof of Theorem Conclusions The main conclusion from the analysis presented in this paper is that the numerical error in the projection method has the same structure as in the semi-discrete case (analyzed in [4]) after it is spatially discretized on a staggered grid. This paper leaves open the very interesting question of what happens on a regular nonstaggered grid. Some progress in this direction has been made in [16]. Acknowledgments The research of the first author was supported in part by ONR grant N and NSF grant DMS The research of the second author was supported in part by ONR grant N and NSF grants DMS and DMS References 1. J.B. Bell, P. Colella, and H.M. Glaz, A second-order projection method for the incompressible Navier Stokes equations, J. Comput. Phys. 85 (1989), MR 90i: A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), MR 39: Weinan E and J.-G. Liu, Projection method I: convergence and numerical boundary layers, SIAM J. Numer. Anal. 32 (1995), ; Projection method II: Godunov-Ryabenki analysis, SIAM J. Numer. Anal. 33 (1996), MR 96e:65061; MR 97i: Weinan E and J.-G. Liu, Projection method for viscous incompressible flows, in Numerical Methods in Applied Sciences, edited by W. Cai and C.-W. Shu, Science Press New York, Ltd, New York, (1996), Weinan E and J.-G. Liu, Gauge method for viscous incompressible flows, preprint, (1996) 6. M. Fortin, R. Peyret, and R. Temam, Résolution numérique des équations de Navier-Stokes pour un fluide incompressible, J. Mecanique, 10 (1971) MR 54: F.H. Harlow and J.E. Welch, Numerical calculation of time-dependent viscous incompressible flowoffluidwithfreesurface,phys. Fluids, 8 (1965) J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier- Stokes problem, part III. Smoothing property and higher order error estimates for spatial discretization, SIAM J. Numer. Anal., 25, (1988) MR 89k: J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comp. Phys. 59 (1985), MR 87a: J. Shen, On error estimates of projection methods for Navier-Stokes equations: first order schemes, SIAM J. Numer. Anal. 29 (1992), MR 92m:35213

21 PROJECTION METHOD III J. Shen, On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations, Numer. Math. 62 (1992), MR 93a: R. Temam, Sur l approximation de la solution des equations de Navier-Stokes par la méthode des fractionnarires II, Arch. Rational Mech. Anal. 33 (1969), MR 39: J. van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Stat. Comp. 7 (1986), MR 87h: C. Wang and J.-G. Liu, Convergence of gauge method for incompressible flow, Math.Comp., 69 (2000), MR 20001a: B.R. Wetton, Ph.D Dissertation, New York University, B.R. Wetton, Error analysis for Chorin s original fully discrete projection method and regularizations in space and time, SIAM J. Numer. Anal. 34 (1997), MR 99c:65161 Courant Institute of Mathematical Sciences, New York, New York address: weinan@cims.nyu.edu Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland address: jliu@math.umd.edu