SECOND-ORDER FULLY DISCRETIZED PROJECTION METHOD FOR INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

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1 Tenth MSU Conference on Differential Equations and Computational Simulations. Electronic Journal of Differential Equations, Conference 3 (06), pp ISSN: URL: or ftp ejde.math.txstate.edu SECOND-ORDER FULLY DISCRETIZED PROJECTION METHOD FOR INCOMPRESSIBLE NAVIER-STOKES EQUATIONS DANIEL X. GUO Abstract. A second-order fully discretized projection method for the incompressible Navier-Stokes equations is proposed. It is an explicit method for updating the pressure field. No extra conditions of immediate velocity fields are needed. The stability and convergence are investigated.. Introduction The projection method (or fractional step method) for solving the incompressible Navier-Stokes equations was originally introduced and studied independently by Chorin [, ] and Temam [9, 0], it has had multiple applications, see e.g., Guermond, Shen, and Yang [3], Guermond, Minev, and Shen [4], Kim and Moin [7], Temam [], Yanenko [], and the references therein, for the theoretical and numerical aspects. Despite many advantages and extensive uses in the past by numerous researchers, the projection method has a few major drawbacks for numerical computations. In general, the original method is only first-order accurate in time. It also needs the supplementary boundary conditions for the intermediate level velocity field and the pressure, which are not supplied in the original equations. A fully discretized projection method was studied in Guo [5, 6] on the staggered grid. The idea was originated in the block LU decomposition, see Perot [8]. This leaded to a whole class of methods (first-order, second-order and even higher order methods). It depends on how the Navier-Stokes equations are discretized, higher order methods can be possible. In this article, we investigate the stability and convergence of a second-order fully discretized projection method proposed in Guo [5, 6]. This article is organized as follows. In Section, we introduce the Navier-Stokes equations and the boundary conditions. The space discretization is listed in Section 3. The full discretization of the Navier-Stokes equations is shown in Section 4. In Section 5, we present fully discretized projection methods. The stability and convergence of second-order fully 000 Mathematics Subject Classification. 76D05, 74H5, 65C0, 65M. Key words and phrases. Fully discretized; projection method; Navier-Stokes equations; stability; convergence. c 06 Texas State University. Published March, 06. 9

2 0 D. X. GUO EJDE-05/CONF/3 discretized projection method is studied in Section 6. The conclusion is in the last section.. Incompressible Navier-Stokes equations We will consider the non-dimensionalized unsteady incompressible Navier-Stokes equations in space dimension two and three on a given regular domain Ω, namely with the boundary conditions on Ω: u t + (u )u u + p = f, Re u = 0, u t=0 = u 0, (.) ηu + ( η) u n = 0. Here Re is the Reynolds number; the parameter η has the limit values of 0 for the free-slip (no stress) condition (Neumann) and for the no-slip condition (Dirichlet). In general, we will not specify η, but keep in mind that 0 η. For the stability and convergence, what we need are the boundary conditions that guarantee the existence of the solution for the original equations. Actually, the boundary conditions are also important in discretizing the equations near the boundary. We may need to use different methods for the points near boundary. In this article, we suppose that all discretizations have the same order on all of grid points. 3. Space discretization Two families of finite-dimensional Hilbert spaces X h and V h are given, which depend on a parameter h R d + (d =, 3). For finite differences, h is the mesh, i.e. h = {h, h } = { x, y} in space dimension two and h = {h, h, h 3 } = { x, y, z} in space dimension three. Two scalar products ((, )) h and (, ) h with corresponding norms h and h are defined on each V h. Since V h is a finite-dimensional space the two norms h and h are equivalent. We assume that they are related as follows u h h c u h h, (3.) u h h S(h) u h h, u h V h. (3.) where c is independent of h and S(h) depends on h. We assume that S(h) as h 0. When convergence be studied, we will be interested in the passage to the limit h 0. The spaces V h with scalar product ((, )) h will approximate in some sense the space V, while the spaces V h with scalar product (, ) h will approximate the space H. The spaces V and H are defined as follows V = the closure of V in H 0 (Ω), V = {u D(Ω), div u = 0}. H = the closure of V in L (Ω), A trilinear operator b h is defined on V h V h V h as follows: b h (u h, v h, w h ) = ((u h )v h, w h ), u h, v h, w h V h,

3 EJDE-06/CONF/3 SECOND-ORDER FULLY DISCRETIZED PROJECTION METHOD and we have the properties: b h (u h, v h, w h ) c u h / h u h / h v h h w h / h w h / h, u h, v h, w h V h, (3.3) in space dimension two, and b h (u h, v h, w h ) c u h 4 h u h 3/4 h v h h w h 4 h w h 3/4 h, u h, v h, w h V h, (3.4) in space dimension three. The constant c in (3.3) and (3.4) is independent of h. We also assume the skewness property: which implies b h (u h, v h, v h ) = 0, u h, v h V h, (3.5) b h (u h, v h, w h ) = b h (u h, w h, v h ), u h, v h, w h V h. We also define two operators A h and B h that are the discrete analogs of A and B as follows: (A h u h, v h ) h = ((u h, v h )) h, u h, v h V h, (B h (u h, v h ), w h ) h = b h (u h, v h, w h ), u h, v h, w h V h. 4. Full discretization of Navier-Stokes equations Let T > 0 be fixed, and let the time step be denoted by = T/N where N is an integer. We construct recursively elements u n h V h, n = 0,,,..., N; u 0 h is an approximation of u 0 ; we assume that u h is constructed by a different scheme because our scheme has two levels in time. Hence starting from u h, we recursively define un h, n =,..., N, by applying the explicit Adams-Bashforth scheme of order two to the convective term and the Crank-Nicholson scheme of order two to the viscous terms, the pressure term and the right-hand side of equation (.), t(un+ h u n h) + 3 B h(u n h) B h(u n h ) (4.) + Re A h(u n+ h + u n h) + G h(p n+ h + p n h) = n+ (fh + fh n ), D h u n+ h = 0. (4.) where fh n is an approximation of f n in V h ; G h is the discrete gradient operator (grad) mapping X h into V h, and D h is the discrete divergence operator (div). In fact, they are related as follows, (u h, G h p h ) = (D h u h, p h ), for u h V h, p h X h. (4.3) One can easily show that (4.) and (4.) are a second-order scheme in time for the Navier-Stokes equations (.). We will assume that (4.) and (4.) are secondorder in space. It is possible to get higher order discretizations, but other procedure would be needed. Therefore, (4.) and (4.) have the overall second-order accuracy. They are a fully implicit coupled system. Generally speaking, it is hard to solve this system. Notice that if we consider the boundary conditions in equation (4.) and (4.), we may need to add one or more terms to the right-hand side of equations (4.) and (4.). For the sake of simplicity, we suppose that these terms are combined in the terms of the right-hand side. So, the idea is that during fully discretizing

4 D. X. GUO EJDE-05/CONF/3 procedure in time and spaces, we need all boundary conditions. And the final discretized equations are a complete linear system. For the rest of this article, we drop the subscript h. Define g n+ as follows, then we rewrite (4.) and (4.) as g n+ = (f n+ + f n ) 3 B(un ) + B(un ) (4.4) and (un+ u n ) + Re A(un+ + u n ) + G(pn+ + p n ) = g n+, (4.5) Du n+ = 0. (4.6) 5. Fully discretized projection method We then introduce the following method for the equations (4.5) and (4.6), t(un+ u n ) + Re A(un+ + u n ) = g n+, t(un+ u n+ ) + (I Re A)G(pn+ + p n ) = 0, Du n+ = 0. (5.) Observe that in (5.), A is a known operator, say a square matrix; G and D also are two known operators, but they may be two non-square matrices. However, they satisfy the equation below, (Du n, p n ) = (u n, Gp n ), for u n V h, p n X h. Let us check the difference between (4.5) and (5.). equations in (5.) and we obtain: First, we add the two i.e., where t(un+ u n ) + Re A(un+ + u n ) = g n+ G(pn+ + p n ) + R n+ ; t(un+ u n ) + 3 B(un ) B(un ) + Re A(un+ + u n ) + G(pn+ + p n ) = (f n+ + f n ) + R n+, (5.) R n+ = ( Re ) AAG(p n+ + p n ). Since R n+ is of order (), this method is a second-order method to the Navier- Stokes equations (.). However, to update p n+, we need to solve a linear system with the same size as A at each time step. In this paper, we study the stability and convergence of the equations (5.).

5 EJDE-06/CONF/3 SECOND-ORDER FULLY DISCRETIZED PROJECTION METHOD 3 Let 6. Stability analysis and convergence Φ n+ = (pn+ + p n ). Then R n+ = ( Re ) AAGΦ n+. Note that from the definition of G and D and the third equation of (5.), it holds, (GΦ n+, u n+ ) = (Φ n+, Du n+ ) = Stability analysis. It is assumed that n=n u 0 K, f n K, u + u / + Re { u + u / } K 3, where K, K and K 3 are independent of h and. Let δ n+ = (un+ u n ), δ n+ = (un+ u n ); then (un+ + u n ) = u n+ δ n+, (un+ + u n ) = u n+ δ n+. Now rewriting the first equation in (5.) with g n+ defined in (4.4), it reads (un+ u n ) + Re A(un+ + u n ) n=0 = (f n+ + f n ) 3 B(un ) + B(un ). (6.) (6.) Solve the second equation in (5.) for u n+ u n+ as follows = u n+ + (I Re A)GΦn+. Substituting u n+ in (6.), it results, (un+ u n ) + Re A(un+ δ n+ ) = (f n+ + f n ) 3 B(un ) + B(un ) GΦ n+ + R n+. Multiply equation (6.3) with u n+, we obtain or (u n+ u n, u n+ ) + Re ((un+ δ n+, u n+ )) = (f n+ + f n, u n+ ) b(u n, u n, u n+ ) [b(u n, u n, u n+ ) b(u n, u n, u n+ )] + (R n+, u n+ ), u n+ u n + 4 δ n+ + Re un+ = Re ((δn+, u n+ )) + (f n+ + f n, u n+ ) + 4b(u n, δ n+, u n+ ) [b(δ n, u n, u n+ ) + b(u n, δ n, u n+ )] + (R n+, u n+ ). (6.3) (6.4)

6 4 D. X. GUO EJDE-05/CONF/3 We now estimate each term in the right-hand side of (6.4) in space dimension two. The first one is majorized, thanks to the Schwarz and Young inequalities, and (3.) (note, S = S(h)): Re ((δn+, u n+ )) Re δn+ u n+ Re S δn+ u n+ 8 δn+ + 8 Re S u n+. For the second term we use the Schwarz and Young inequalities again and thanks to (3.): (f n+ + f n, u n+ ) ( f n+ + f n ) u n+ c ( f n+ + f n ) u n+ 4Re un+ + c Re( f n+ + f n ). For the third term we use (3.), (3.) and (3.3) and majorize it by 4b(u n, δ n+, u n+ ) 4c u n / u n / δ n+ / δ n+ / u n+ 4c S u n δ n+ u n+ 8 δn+ + 3c S u n u n+. (6.5) Thanks to (3.3) the fourth term is approximated by [b(δ n, u n, u n+ ) + b(u n, δ n, u n+ )] c δ n / δ n ( u n / u n / + u n u n / ) u n+ c S δ n ( u n + u n ) u n+ (6.6) 4 δn + 4c S ( u n + u n ) u n+. Finally, by the definition of R n+ we have (R n+, u n+ ) = ( Re ) (AAGΦ n+, u n+ ). Rewriting the second equation of (5.), it results if Re GΦ n+ = t(i Re A) (u n+ u n+ ) = (6.7) t[i + Re A + ( Re ) AA +... ](u n+ u n+ ), is small enough. Then, substituting GΦ n+ in R n+, we obtain (R n+, u n+ ) = ( Re ) (AA[I + Re A + ( Re ) AA +... ]u n+, u n+ ) + ( Re ) (AA[I + Re A + ( Re ) AA +... ]u n+, u n+ )

7 EJDE-06/CONF/3 SECOND-ORDER FULLY DISCRETIZED PROJECTION METHOD 5 = ( Re ) [(AAu n+, u n+ ) + Re (AAAun+, u n+ ) +... ] + ( Re ) [(AAu n+, u n+ ) + Re (AAAun+, u n+ ) +... ] ( Re ) (S u n+ + Re S4 u n ] + ( Re ) [S ( 4 un+ + u n+ ) + Re S4 ( 4 un+ + u n+ ) +... ] ( Re ) 5 4 (S + Re S ) u n+ + ( Re ) (S + Re S ) u n+ ( S Re ) (5 Re S 4 un+ + u n+ ) Re S ( 5 4 un+ + u n+ ), where we assume that Re S < /. Gathering all these estimates, we deduce from (6.4) that u n+ u n δn+ 4 δn Re un+ c Re( f n+ + f n ) + Re S u n Re S u n+ + c 3 S ( u n + u n ) u n+, where c 3 = 3c. We multiply equation (6.) by u n+ to obtain (u n+ u n, u n+ ) + Re ((un+ δ n+, u n+ )) = (f n+ + f n, u n+ ) b(u n, u n, u n+ ) [b(u n, u n, u n+ ) b(u n, u n, u n+ )]. Thanks to (3.5), the above equation can be rewritten as u n+ u n + 4 δ n+ + Re un+ = Re ((δn+, u n+ )) + (f n+ + f n, u n+ ) + 4b(u n, δ n+, u n+ ) [b(δ n, u n, u n+ ) + b(u n, δ n, u n+ )]. (6.8) (6.9) (6.0) Note that (6.0) is similar to (6.4) with u n+ and δ n+ replaced by u n+/ and δ n+/ respectively. So, we can repeat the estimation procedures as for (6.4). In space dimension two we have the following estimates: Re ((δn+, u n+ )) 8 δn+ + 8 Re S u n+. (f n+ + f n, u n+ ) 4Re un+ + c Re( f n+ + f n ) 4b(u n, δ n+, u n+ ) 8 δn+ + 3c S u n u n+ (6.)

8 6 D. X. GUO EJDE-05/CONF/3 [b(δ n, u n, u n+ ) + b(u n, δ n, u n+ )] 4 δn + 4c S ( u n + u n ) u n+. (6.) We gather these inequalities with (6.0), and obtain u n+ u n δn+ 4 δn Re un+ c Re( f n+ + f n ) + 8 Re S u n+ + c 3 S ( u n + u n ) u n+. Adding (6.3) to (6.8), we obtain u n+ + u n+ u n δn δn+ δn [ 4 Re 7Re S c 3 ReS ( u n + u n )]( u n+ + u n+ ) c Re( f n+ + f n ), where c 3 is the same as in (6.8). We now prove the following result. (6.3) (6.4) Theorem 6.. In dimension two we assume that (6.) holds. Then there exists K 4 independent of h and such that if < 6c Re, S(h) max{ Re 48, c ReK 4 } (6.5) where c is defined in (3.) and c = 56c 3, c 3 the constant is defined in (6.8). Then u n K 4, n = 0,..., N (6.6a) u n+ 4K 4, n = 0,..., N (6.6b) u n u n 3K 4, (6.6c) n= u n u n 6K 4, (6.6d) n= u n u n 3K 4, (6.6e) n= Re Re u n 7K 4, n= u n 7K 4, n= (6.6f) (6.6g) Proof. Taking the inner product of the equation (6.5) with u n+ and in light of (3.3), we obtain ([I + Re A + ( Re ) AA +... ](u n+ u n+ ), u n+ ) = 0;

9 EJDE-06/CONF/3 SECOND-ORDER FULLY DISCRETIZED PROJECTION METHOD 7 i.e., (u n+ u n+, u n+ ) = ([A + ( Re Re )AA +... ](un+ u n+ ), u n+ ). Thanks to (3.), we have (u n+ u n+, u n+ ) 5 4 ( Re ) ( un+ + u n+ ). Re S Since S Re/48 from (6.5), then Re S /6 and it results or ( un+ u n+ + u n+ u n+ ) 3 4 Re ( un+ + u n+ ) u n+ u n+ + u n+ u n+ 3 Re ( un+ + u n+ ). (6.7) We first show (6.6a) by induction. Assuming that (6.6a) holds for n = 0,..., m, we infer from (6.), (6.3) that u n+ + u n+ u n δn δn+ δn + 3 8Re { un+ + u n+ } c Re{ f n+ + f n }. Adding (6.8) to (6.7), we obtain (6.8) i.e., u n+ u n + u n+ u n δn δn+ δn + 8Re { un+ + u n+ } c Re{ f n+ + f n }; u n+ u n + un+ u n δn δn+ 4 δn + 6 { un+ + u n+ } c Re{ f n+ + f n }. Thanks to (3.), we obtain Therefore, if we define u n+ u n δn+ 4 δn + 6c Re un+ c Re{ f n+ + f n }. (6.9) (6.0) ξ n = u n + 4 δn, γ n = c Re{ f n+ + f n }, then, from (6.0) the following inequality holds ( + 6c Re)ξn+ ξ n + γ n.

10 8 D. X. GUO EJDE-05/CONF/3 By applying Lemma 6. below with β = 0 and α = (from (6.5), α < ), 6c Re we find, for n =,,..., m, that ξ n+ ( + n 6c {ξ + c Re ( f n+ + f n )}; Re) n i.e., We choose j= u n+ + 6 un+ u n u + 6 u u 0 + c Re 9 8 u + 8 u0 + c Re 9 8 K K + c ReK. K 4 = 3 K 3 + K + c ReK, f j, j= f j, and conclude that (6.6a) is proved for n = 0,,..., m +, hence for all n. Then (6.7) is valid for all n and summing these relations for n =,,..., N, we find Hence u N u j u j + δ j j= { u j + u j } j= u + 4 δ + c Re j= f j j= u + 6 u u 0 + c Re u j u j K 4, j= u j u j K 4, j= Thanks to (6.), we know that Re j= δ j j= f j K 4. j= u j u j K 4, j= { u j + u j } 6K 4. j= u u / u + u / 4K 3 4K 4, u u 0 u + u 0 K 3 + K K 4, u / u 0 u / + u 0 K 3 + K K 4, Re ( u + u / ) K 3. Then (6.6c) (6.6g) are proved.

11 EJDE-06/CONF/3 SECOND-ORDER FULLY DISCRETIZED PROJECTION METHOD 9 Finally, from the inequality u n+ u n+ + u n+ u n+, then (6.6b) holds following (6.6a) and (6.6d). We state the Lemma used in the proof of Theorem 6.. Lemma 6.. Consider two sequences of numbers ξ n, γ n 0 such that ( α)ξ n ( + β)ξ n + γ n, for all n and for some α < and β >. Then for all n: ξ n ( + β α )n ξ 0 ( + β)n n + ( α) n γ j. (6.) If γ j γ, for all j, we also have Proof. For m = 0,..., n, we write j= ξ n ( + β α )n (ξ 0 + γ ). (6.) β + α ξ n m + β α ξn m + α γ n m. We multiply this relation by ((+β)/( α)) m and add the corresponding inequalities for m = 0,..., n ; (6.) and (6.) follow easily. 6.. Convergence. We introduce the approximate functions u k, u k, ũ k (k = ) defined as follows: u k = u n+ for t [n, (n + )), n = 0,..., N, u k = u n+ for t [n, (n + )), n = 0,..., N, ũ k is continuous from [0, T ] to H, linear on each interval ((n ), n) and equal to u n at n, n = 0,..., N. Then (6.4) yields the following result. Theorem 6.3. As k = 0, the functions u k, u k and ũ k remain bounded in L (0, T ; L (Ω) d ); u k and u k remain bounded in L (0, T ; H (Ω) d ), and the same is true for ũ k if u 0 V. Furthermore u k u k and u k ũ k converge to 0 in L (0, T ; L (Ω) d ) as k = 0, their norm being bounded. The above theorem follows directly from Theorem 6.. Because of Theorem 6.3 there exists a subsequence k 0 such that u k u k ũ k u in L (0, T ; H) weak-star and L (0, T ; H (Ω) d ) weakly, u in L (0, T ; H) weak-star and L (0, T ; H (Ω) d ) weakly, u in L (0, T ; H) weak-star and L (0, T ; H (Ω) d ) weakly. Theorem 6.3 also implies that u = u = u and thus u L (0, T ; H) L (0, T ; V ).

12 0 D. X. GUO EJDE-05/CONF/3 Conclusions. In this article, we proved the stability and convergence of a second order fully discretized projection method for the incompressible Navier-Stokes equations. We did not specify the boundary conditions, but all necessary boundary conditions should be supplied in order to have a solution. It is possible to construct higher order methods, but the numerical analysis is much more complicated. References [] A. J. Chorin: Numerical solution of the Navier-Stokes equations, Math. Comp., (968) [] A. J. Chorin: Numerical solution of incompressible flow problems, Studies in Numer. Anal., (968) [3] J. L. Guermond, Jie Shen,, X. Yang: Error analysis of fully discrete velocity-correction methods for incompressible flows, Mathematics of Computation, 77 (008) [4] J. L. Guermond, P. Minev, Jie Shen: An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 95 (006) [5] D. Guo: A Second Order Scheme for the Navier-Stokes Equations: Stability and Convergence, Numer. Funct. anal. and Optimiz., 0 (999), [6] D. Guo: A Second Order Scheme for the Navier-Stokes Equations: Application to the drivencavity problem, Applied Numerical Mathematics, 35 (000) [7] J. Kim, P. Moin: Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (985) [8] J. B. Perot: An Analysis of the Fractional Step Method, J. Comput. Phys., 08 (993) [9] R. Temam: Sur l approximation de la solution d equations d evolution par la methode des pas fractionnaries, C. R. Acad.Sci. Paris, Serie A, 63 (966) 4 44, 65 67, [0] R. Temam: Sur l approximation de la solution des equations de Navier-Stokes par la methodd des pas fractionnaries (II), Arch. Ration. Mech. Anal., 33 (969) [] R. Temam: Navier-Stokes Equations: theory and numerical analysis, revised ed., North- Holland-Amsterdam.New York.Oxford, 984. [] N. N. Yanenko: The Method of Fractional Steps, Springer-Verlag, New York., 97. Daniel X. Guo Department of Mathematics and Statistics, University of North Carolina at Wilmington, NC 8403, USA address: guod@uncw.edu