Key words. Navier-Stokes, Unconditional stability, IMEX methods, second order convergence, Crank-Nicolson, BDF2

Transcription

1 AN OPTIMALLY ACCURATE DISCRETE REULARIZATION FOR SECOND ORDER TIMESTEPPIN METHODS FOR NAVIER-STOKES EQUATIONS NAN JIAN, MUHAMMAD MOHEBUJJAMAN, LEO. REBHOLZ, AND CATALIN TRENCHEA Abstract. We propose a new, optimally accurate numerical regularization/stabilization for (a family of) second order timestepping metods for te Navier-Stokes equations (NSE). Te metod combines a linear treatment of te advection term, togeter wit a stabilization terms tat are proportional to discrete curvature of te solutions in bot velocity and pressure. We rigorously prove tat te entire new family of metods are unconditionally stable and O( t ) accurate. Te idea of curvature stabilization is new to CFD and is intended as an improvement over te commonly used speed stabilization, wic is only first order accurate in time and can ave an adverse affect on important flow quantities suc as drag coefficients. Numerical examples verify te predicted convergence rate and sow te stabilization term clearly improves te stability and accuracy of te tested flows. Key words. Navier-Stokes, Unconditional stability, IMEX metods, second order convergence, Crank-Nicolson, BDF AMS subject classifications. 76D05, 65L0, 65M. Introduction. We consider optimally accurate stabilizations for second order time-stepping metods for te Navier-Stokes equations (NSE) on a bounded domain Ω R d, d= or 3: u t + u u u + p = f, for x Ω, 0 < t T, (.) u = 0, for x Ω, 0 < t T, u = 0, on Ω, for 0 < t T, u(x, 0) = u 0 (x), for x Ω, wit te pressure satisfying te usual zero-mean normalization. Developing efficient, accurate, and robust numerical metods for solving te NSE remains a great callenge in Computational Fluid Dynamics (CFD). For time-stepping metods, common approaces combine linearizations at eac timestep wit stabilizations/regularizations tat damp oscillations and unstable modes. An important linearization metod is CNLE (Crank-Nicolson wit linear extrapolation), proposed by Baker [], wic is comparable in stability and accuracy wit te more expensive, fully implicit Crank-Nicolson (CN) metod, [5,, ]. However, wile a nonlinear solver for CN requires several linear solves at eac time step, CNLE requires just one. A similar linearization exists for BDF timestepping (called BDFLE). Herein, we will consider a new, optimally accurate stabilization to be used wit CNLE, BDFLE, Department of Matematics, 30 Tackeray Hall, University of Pittsburg, Pittsburg, PA 560, naj@pitt.edu. Partially supported by Air Force grant FA Department of Matematical Sciences, Clemson University, Clemson SC 963, mmoebu@clemson.edu. Partially supported by NSF rant DMS593. Department of Matematical Sciences, Clemson University, Clemson SC 963, rebolz@clemson.edu. Partially supported by NSF rant DMS593 and US Army rant 659- MA Department of Matematics, 30 Tackeray Hall, University of Pittsburg, Pittsburg, PA 560, trencea@pitt.edu. Partially supported by Air Force grant FA

2 Jiang, Moebujjaman, Rebolz and Trencea and te family of metods in between tem. Recall tis family of metods (witout stabilization) is given by (θ + )u n+ θu n + (θ )u n θ u n+ ( θ) u n (.) + ((θ + )u n θu n ) (θu n+ + ( θ)u n ) + θ p n+ + ( θ) p n = f n+θ, u n+ = 0, (.3) were θ [, ]. If θ =, BDFLE is recovered, and if θ =, ten CNLE is recovered. For any oter θ (, 0), a second order metod is still recovered. Since CNLE exactly conserves energy, and BDFLE numerically dissipates it, te parameter θ can be used to control te dissipation. A successful stabilization metod to be used wit (.) must be able to damp te instabilities tat frequently arise in NSE simulations, but witout over-smooting or removing important flow structures, i.e. witout urting accuracy. A common approac for tese timestepping metods is to add α u n+ to te left and side, and α u n to te rigt and side, were α is a tuning parameter generally taken to be on te order of te meswidt. Suc a stabilization as been used in metods for Navier-Stokes (see [0, 7] and references terein), and is related in principle to tecniques used in turbulence modeling [6, ], ocean modeling [5], and also to te discretization of te Voigt term in a turbulence model recently studied by Titi and oters, e.g. [6, 8]. As sown in tese works, tis stabilization can be effective for several different types of flows, and also can improve conditioning of linear systems by increasing te coefficient of te stiffness matrix, e.g. in te case of BDFLE, from to + α. However, as sown in te analysis of [0], tis tecnique is O(α t) accurate, and tus can potentially be a dominant error source in second order timestepping metods if te usual coice of α = O() is made. If one instead takes α = O( t), tis creates a need for a careful retuning of α eac time te time step size is canged, wic could make its use wit adaptive time-stepping very difficult. Te purpose of tis paper is to introduce and analyze a new stabilization for time-stepping metods of te form (.), tat can sufficiently stabilize but is O( t ) (i.e. optimally) accurate. Te design of te stabilization is inspired by te idea of stabilizing curvature (u n+ u n + u n ), instead of stabilizing speed (u n+ u n ), wic is done by te stabilization discussed above. Not only is curvature stabilization more accurate tan speed stabilization (wit respect to t), but in CFD it does not directly alter important flow quantities suc as drag coefficients, as speed penalization does (see section 5.3). To our knowledge, te idea of curvature stabilization is brand new to CFD, and was first introduced for any timestepping metod last year, in a paper of te autors[6], for a timestepping metod for a particular class of ODEs. Te new family of second-order, unconditionally stable, IMEX time-stepping

3 metods we propose are given by: Second order unconditionally stable IMEX sceme 3 (θ + )u n+ θu n + (θ )u n (.) θ( + ɛ) u n+ ( θ( + ɛ)) u n θɛ u n ( + ((θ + )u n θu n ) θ + ɛ u n+ + ( θ + ɛ )u n + θ ɛ ) u n + θ + ɛ p n+ + ( θ + ɛ ) p n + θ ɛ p n = f n+θ, u n+ = 0, were θ [, ], ɛ 0. Our analysis will sow tere is optimal convergence for any constant ɛ, owever te natural norms of te problem dictate tat ɛ = O() is most appropriate for simulations. Altoug tis appears to be quite small compared to stabilization constants in oter common metods, our numerical experiments demonstrate tat suc ɛ do provide sufficient stabilization. Te family of metods (.) are stabilizations of te family of metods (.), and if ɛ = 0, ten (.) is recovered. Tus, for θ = in (.), we obtain a regularized BDFLE sceme wic we will call BDFLEReg. Similarly, te sceme (.) wit θ = will be called CNLEReg. Our stability and convergence analysis will be general, and apply to te entire family (.); we prove bot unconditional stability and O( t ) accuracy. For our numerical examples, we perform numerical tests wit bot CNLEReg and BDFLEReg. As is typical wit stabilization metods, te positive effect is seen more troug numerical tests tan troug analysis, and we provide several examples wic sow convincing evidence of te effectiveness of te proposed stabilization. Te analysis we perform erein to prove unconditional stability and optimal convergence can be applied to oter advection-diffusion models, e.g., Lions yperviscosity model [], te modified NSE of Ladyzenskaya [] and nonlinear spectral eddyviscosity models of turbulence [9]. In particular, we note our proposed metod(s) are related to one discussed by H. Jonston and J.. Liu in [7], and our analysis can be extended to sow tat (3.7) in [7] is stable (it is stated as an open problem in te paper). We note also tat our analysis also applies in a straigtforward manner to te implicit version of (.). Tis paper is arranged as follows. In section, we give notation and preliminaries to make for a smooter analysis to follow. Section 3 proves unconditional stability of te proposed family of metods, and section proves optimal convergence for tem. In section 5, we provide numerical tests tat sow te metods are very effective on several bencmark problems. Finally, conclusions are drawn in section 6.. Notations and Preliminaries. Let Ω be an open, regular domain in R d, d = or 3. We denote te usual L (Ω) norm and te inner product by and (, ). Te L p (Ω) norms and te Sobolev Wp k (Ω) norms are denoted by L p and W k p respectively. In particular, H k (Ω) is used to represent te Sobolev space W k (Ω). k and k denote te norm and te seminorm in H k (Ω). For functions v(x, t) defined on te time interval (0, T ), we define ( m < ) ( T /m. v,k := EssSup [0,T ] v(t, ) k and v m,k := v(t, ) m k dt) 0

4 Jiang, Moebujjaman, Rebolz and Trencea We consider te finite element metod (FEM) for te spatial discretization. Let X be te velocity space and Q be te pressure space: X := (H0 (Ω)) d = {v : v (H (Ω)) d, v = 0 on Ω}, Q := L 0(Ω) = {q : q L (Ω), q dx = 0}. A weak formulation of (.) is: Find u : [0, T ] X, p : [0, T ] Q for a.e. t (0, T ] satisfying (u t, v) + (u u, v) + ( u, v) (p, v) = (f, v), v X u(x, 0) = u 0 (x) in X and ( u, q) = 0, q Q. Te space of divergence free functions is given by Ω V := {v X : ( v, q) = 0, q Q}. Let V denote te dual space of V. Te norm on V is defined as f : = sup 0 v V (f, v) v. Let X X, Q Q denote conforming velocity, pressure finite element spaces based on an edge to edge triangulations of Ω wit maximum triangle diameter. Te velocity-pressure FEM spaces (X, Q ) are assumed to satisfy te usual discrete inf-sup / LBB condition, see unzburger s book [0], for stability of te discrete pressure: inf q Q (q, v ) sup v X q v β > 0, were β is independent of. Taylor-Hood elements, [0], are one commonly used coice of suc velocity-pressure finite element spaces. Oter coices can also be found in [0, 8]. We also assume te following approximation properties typical of piecewise polynomials of degree (k, k ), see [], old for (X, Q ): inf u v C k+ u k+, u H k+ (Ω), v X inf (u v ) C k u k+, u H k+ (Ω), v X inf p q C k p k, p H k (Ω). q Q Te discretely divergence free subspace of X is V := {v X : ( v, q ) = 0, q Q }. Define te usual explicitly skew symmetrized trilinear form b (u, v, w) satisfies te bound, see [8], b (u, v, w) := (u v, w) (u w, v). b (u, v, w) C(Ω) u v w, for any u, v, w X,

5 Second order unconditionally stable IMEX sceme 5 and, if v, v L (Ω) (see e.g., Lemma. in [9]), b (u, v, w) C(Ω)( v L (Ω) + v L (Ω)) u w. To simplify te analysis, we denote te interpolation at t = t n+θ by J ɛ n+θ(a) := θ + ɛ and extrapolation at t = t n+θ by a n+ + ( θ + ɛ )a n + θ ɛ a n, H n+θ (a) := (θ + )a n θa n. Te fully discrete approximation of (.) we study is: iven u n, u n, p n, p n, find u n+ X, p n+ Q satisfying ( (θ + )u n+ θu n + (θ )u n, v ) + (J ɛ n+θ( u ), v ) (.) + b (H n+θ (u ), J ɛ n+θ(u ), v ) (J ɛ n+θ(p ), v ) = (f n+θ, v ), v X, ( u n+, q ) = 0, q Q. 3. Stability Analysis. We prove unconditional, long time stability of (.). To analyze te stability, we first define two matrices. Define symmetric positive matrix F R n n by F = θ(θ )I + θ ɛ I, and te symmetric matrix R n n as follows, see [, 6, 7] for reference. = ( θ(θ+3) ( (θ+)(θ ) +ɛ +ɛ I θ(θ+) ɛ I I + ( θ)(θ+) ɛ I) (θ+)(θ ) ( For any u, v R n, define F norm of te n vector u: u F = (u, F u), wic is non-negative, and norm of te n vector ] u [ = v θ(θ ) ([ [ u u,, v] v]) wose value could be negative. Lemma 3.. For any vector u, v R n, we ave ([ u [ u, v] v]) = θ + u + θ + v + θ + [ u v] : +ɛ +ɛ (θ + )(θ ) u θ v, I + ( θ)(θ+) I + θ( θ+3) ɛ I ɛ I) ). u v + θ ɛ u v

6 6 Jiang, Moebujjaman, Rebolz and Trencea and ([ ] [ u u, v v ( θ + ] ) θ + u + + (θ + )(θ ) + θɛ (θ + )(θ ) u v + θ ɛ u v ) ( u + (θ + )(θ ) + θɛ ) v. Proof. Te inequality follows from direct algebraic manipulations. Teorem 3.. Te algoritm (.) is unconditionally stable and N, te following energy estimate olds u N + N u n+ u n + u θ + n F + θ + n= ( ) N θ u θ N [ ] u θ + u 0 N n= J ɛ n+θ( u ) + N N f n+θ (θ + ). Remark 3.. Setting θ = in te stability result above and expanding te term J ɛ n+θ ( u ) gives te BDFLEReg stability result, u N + 3 ( 3 N n= u n+ u n + u n F + 3 [ ] u + N 3 ) N u 0 + N 3 u 0 N n= N n= n= u n+ + ɛ ( u n+ u n + u ) n f n+. We observe tat te viscous term as an additional curvature term (u n+ u n + u n ) compared to standard NSE stability estimates. It is tese terms wic act to penalize curvature in te BDFLEReg metod. Moreover, te viscous term suggests tat ɛ O() is most appropriate, oterwise te curvature terms could dominate te viscous dissipation. Our numerical experiment sow tat ɛ = O() works very well. Proof. Set in (.) v = J ɛ n+θ (u ). Ten we ave [ ] u n+ u [ ] u n n u n = (f n+θ, Jn+θ(u ɛ )). + u n+ u n + u n F + J ɛ n+θ( u ) Applying Caucy-Scwarz inequality to te rigt and side and ten taking sum from n = to n = N, we ave te following inequality [ ] u N u + N [ ] u u 0 N n= u n+ u n + u n F + + N n= f n+θ. N n= J ɛ n+θ( u )

7 Second order unconditionally stable IMEX sceme 7 By Lemma 3., we ave te energy estimate as follows u N + N u n+ u n + u θ + n F + θ + n= θ θ + u N + [ ] u θ + By induction, we obtain te following energy estimate u N + N u n+ u n + u θ + n F + θ + n= u 0 ( θ ) N u θ N [ ] u θ + u 0 N n= J ɛ n+θ( u ) + N f n+θ (θ + ). n= N n= J ɛ n+θ( u ) + N N f n+θ (θ + ). n=. Error Analysis. In tis section we present a detailed error analysis of te proposed metods. We prove ere tat te entire family of stabilized metods are optimally accurate. Te use of te -norms, and te extrapolation and interpolations operators, are critical for a smoot analysis of te entire family of metods. Let t n = n t, n = 0,,,..., N T, and T := N T t. Denote u n = u(t n ). We introduce te following discrete norms: N T v m,k := ( v n m k t) /m, v,k = max v n k. 0 n N T n=0 In order to establis te optimal asymptotic error estimates for te approximation we assume tat te true solution satisfies te following regularity u L (0, T ; H (Ω)) H (0, T ; H k+ (Ω)) H 3 (0, T ; L (Ω)) H (0, T ; H (Ω)), p L (0, T ; H s+ (Ω)) H (0, T ; L (Ω)), and f L (0, T ; L (Ω)). Denote te error between te true solution and te approximation by e n = u n u n, ten we ave te following error estimates. Teorem.. Consider te metods (.). For any 0 < N N T, tere exists a positive constant C independent of te mes widt and timestep t suc tat e N + N e n+ e n + e n F + N θ + θ + Jn+θ( e) ɛ (.) n= exp ( C T ) [ ( θ ) N ( θ ) N ) ( e e0 + C( + e 0 ) θ + θ + ( ( θ ) N ) ( + C t p tt,0 + s+ p,s+ + t u ttt,0 θ + + t u tt,0 + t u tt,0 + t u,0 u tt,0 + t u,0 u tt,0 + k+ u t,k+ + k u,k+ + k u,0 u,k+) ]. n=

8 8 Jiang, Moebujjaman, Rebolz and Trencea Corollary.. Consider Taylor-Hood approximation elements for te spatial discretization, i.e. (X, Q ) = (P, P ). Assuming e 0, e to be 0, we ave optimal accuracy in space and time: e N + N e n+ e n +e n F + N θ+ θ+ Jn+θ( e) ɛ C( t + ). n= Proof. (of Teorem.) At time t n+θ = (n + θ), te true solution of te NSE satisfies: were Let ( (θ + )u n+ θu n + (θ )u n, v ) + ( J ɛ n+θ( u), v ) (.) + b ( H n+θ (u), J ɛ n+θ(u), v ) (p n+θ, v ) = (f n+θ, v ) + Intp(u n+θ ; v ), ( (θ + Intp(u n+θ ; v ) = )u n+ θu n + (θ )u n u t (t n+θ ), v ) (.3) + ( ( Jn+θ(u) ɛ ) u n+θ, v ) + b ( H n+θ (u) u n+θ, Jn+θ(u), ɛ v ) + b ( u n+θ, J ɛ n+θ(u) u n+θ, v ). n= e n = u n u n = (u n I u n ) + (I u n u n) = η n + ξ n, were I u n V is an interpolant of u n in V. Subtracting (.) from (.) gives ( (θ+ )e n+ θe n + (θ )e n, v ) + ( J ɛ n+θ( e), v ) + b ( H n+θ (u), Jn+θ(e), ɛ v ) = Intp(u n+θ ; v ) ( Jn+θ(p) ɛ p n+θ, v ) + ( J ɛ n+θ(p) q, v ) + b (H n+θ (e), J ɛ n+θ(u ), v ), wic can be rewritten as ( (θ + )ξ n+ θξn + (θ )ξ n, v ) + ( J ɛ n+θ( ξ ), v ) (.) + b ( H n+θ (u), Jn+θ(ξ ɛ ), v ) = Intp(u n+θ ; v ) ( Jn+θ(p) ɛ p n+θ, v ) + ( Jn+θ(p) ɛ q, v ) ( (θ + )η n+ θη n + (θ )η n, v ) ( J ɛ n+θ( η), v ) b (H n+θ (u),j ɛ n+θ(η), v )+b (H n+θ (ξ ),J ɛ n+θ(u ), v )+b (H n+θ (η),j ɛ n+θ(u ), v ). Set v = J ɛ n+θ (ξ ), ten (.) becomes [ ξ ] n+ ξn [ ξ ] n ξn + ξ n+ ξn +ξn F + Jn+θ( ξ ɛ ) (.5) = Intp ( u n+θ ; Jn+θ(ξ ɛ ) ) ( Jn+θ(p) ɛ p n+θ, Jn+θ(ξ ɛ ) )

9 Second order unconditionally stable IMEX sceme 9 + ( Jn+θ(p) ɛ q, (Jn+θ(ξ ɛ ) ) ( (θ + )η n+ θη n +(θ )η n ( Jn+θ( η), ɛ Jn+θ( ξ ɛ ) ) b ( H n+θ (u), Jn+θ(η), ɛ Jn+θ(ξ ɛ ) ) + b ( H n+θ (ξ ), J ɛ n+θ(u ), J ɛ n+θ(ξ ) ) + b ( H n+θ (η), J ɛ n+θ(u ), J ɛ n+θ(ξ ) ). Te pressure terms on te rigt and side can be bounded as follows. ( J ɛ n+θ (p) p n+θ, J ɛ n+θ(ξ ) ) 6 J ɛ n+θ( ξ ) + C J ɛ n+θ(p) p n+θ ), Jn+θ(ξ ɛ ) 6 J ɛ n+θ( ξ ) +C θp n+ +( θ)p n p n+θ +C 3 ɛ θ p n+ p n +p n 6 J ɛ n+θ(ξ ) +C θ ( θ) t 3 tn+ and t n p tt dt+c 3 ɛ θ t 3 tn+ t n p tt dt, (J ɛ n+θ(p) q, J ɛ n+θ(ξ ) ) 6 J ɛ n+θ( ξ ) + C J ɛ n+θ(p) q 6 J ɛ n+θ( ξ ) + C ( J ɛ n+θ(p) p n+θ + p n+θ q ) tn+ 6 J n+θ( ξ ɛ ) + C θ ( θ) t 3 p tt dt + C 3 ɛ θ t 3 tn+ Next, we bound Intp ( u n+θ ; J ɛ n+θ (ξ ) ). First, t n t n p tt dt + C s+ p n+θ s+. ( (θ + )u n+ θu n + (θ )u ) n u t (t n+θ ), J ɛ n+θ(ξ ) 6 J ɛ n+θ( ξ ) + C (θ + )u n+ θu n + (θ )u n tn+ 6 J n+θ( ξ ɛ ) + C θ 6 t 3 u ttt dt. Secondly, t n u t (t n+θ ) ( ( J ɛ n+θ(u) u n+θ ), J ɛ n+θ (ξ ) ) 6 J ɛ n+θ( ξ ) + C ( J ɛ n+θ(u) u n+θ ) 6 J ɛ n+θ( ξ ) +C (θu n+ +( θ)u n u n+θ ) +C ɛ θ (u n+ u n +u n ) 6 J ɛ n+θ(ξ ) +Cθ ( θ) t 3 tn+ Tirdly, b (H n+θ (u) u n+θ, J ɛ n+θ(u), J ɛ n+θ(ξ )) t n u tt dt+c ɛ θ t 3 C (H n+θ (u) u n+θ ) J ɛ n+θ( u) J ɛ n+θ( ξ ) tn+ 6 J ɛ n+θ( ξ ) + C (H n+θ (u) u n+θ ) J ɛ n+θ( u) t n u tt dt. tn+ 6 J n+θ( ξ ɛ ) + C θ ( + θ) t 3 Jn+θ( u) ɛ u tt dt. t n

10 0 Jiang, Moebujjaman, Rebolz and Trencea Finally, b ( u n+θ, J ɛ n+θ(u) u n+θ, J ɛ n+θ(ξ ) ) C u n+θ (J ɛ n+θ(u) u n+θ ) J ɛ n+θ( ξ ) 6 J ɛ n+θ( ξ ) + C u n+θ ( J ɛ n+θ(u) u n+θ ) tn+ 6 J n+θ( ξ ɛ ) + C θ ( θ) t 3 u n+θ u tt dt tn+ + C 3 ɛ θ t 3 u n+θ u tt dt. t n We ten bound te oter terms on te rigt and side of (.5) ( (θ + )η n+ θη n + (θ )η n ), Jn+θ(ξ ɛ ) t n (.6) (θ + )η n+ θη n + (θ )η n J ɛ n+θ(ξ ) C (η n+ η n ) + θ(η n+ η n ) θ(η n η n ) J ɛ n+θ( ξ ) C tn+ tn+ t n ( η t dt + θ η t dt θ η t dt) J n+θ( ξ ɛ ) t n t n t n ( tn+ C η t dt + tn+ t n θ η t dt + t n t n θ η t dt ) Jn+θ( ξ ɛ ) t n C (θ + ) C(θ + ) Also, tn+ tn+ t n η t dt + 6 J ɛ n+θ( ξ ) t n η t dt + 6 J ɛ n+θ( ξ ). ( Jn+θ( η), ɛ Jn+θ( ξ ɛ ) ) Jn+θ( η) J ɛ n+θ( ξ ɛ ) (( C θ+ ɛθ ) ηn+ + ( θ ɛθ ) ηn + ɛ θ η n ) + 6 J n+θ( ξ ɛ ). We estimate te nonlinear terms as follows b ( H n+θ (u), J ɛ n+θ(η), J ɛ n+θ(ξ ) ) and C H n+θ ( u) Jn+θ( η) J ɛ n+θ( ξ ɛ ) C ( (θ + ) u n + θ u n )( ( ɛθ ) ηn+ θ + + ( θ ɛθ ) ηn + ɛ θ η n ) + 6 J n+θ( ξ ɛ ), b ( H n+θ (η), J ɛ n+θ(u ), J ɛ n+θ(ξ ) ) C H n+θ ( η) J ɛ n+θ( u ) J ɛ n+θ( ξ )

11 Second order unconditionally stable IMEX sceme C Jn+θ( u ɛ ) H n+θ ( η) + 6 J n+θ( ξ ɛ ) C ( ( ɛθ ) u θ + n+ + ( θ ɛθ ) u n + ɛ θ u n ) ( (θ + ) η n + θ η n ) + 6 J n+θ( ξ ɛ ). Te last trilinear term is bounded as b ( H n+θ (ξ ), J ɛ n+θ(u ), J ɛ n+θ(ξ ) ) (.7) C J ɛ n+θ( u ) H n+θ (ξ ) J ɛ n+θ( ξ ) + C J ɛ n+θ(u ) H n+θ (ξ ) J ɛ n+θ( ξ ) C ( J ɛ n+θ( u ) + J ɛ n+θ(u ) )( (θ+) ξ n +θ ξ n ) + 6 J ɛ n+θ( ξ ). After bounding all te terms on te rigt and side of (.5), we now take te sum from n = to n = N [ ] ξ N ξn + [ ] ξ + C ξ 0 N n= N ξn+ ξn + ξn F + N n= [ θ ( θ) t 3 tn+ + s+ p n+θ s+ + θ 6 t 3 tn+ + C ɛ θ t 3 tn+ n= J ɛ n+θ( ξ ) (.8) t n p tt dt + 3 ɛ θ t 3 t n u ttt dt + Cθ ( θ) t 3 t n u tt dt + C θ (+θ) t 3 J ɛ n+θ( u) tn+ t n p tt dt tn+ t n tn+ u tt dt t n u tt dt tn+ tn+ +C θ ( θ) t 3 u n+θ u tt dt+c 3 ɛ θ t 3 u n+θ u tt dt t n t n + (θ +) tn+ ( (θ+ η t ɛθ ) ηn+ dt+ + ( θ ɛθ ) ηn + ɛ θ t n η n ) + ( (θ + ) u n + θ u n )( ( ɛθ ) ηn+ θ + + ( θ ɛθ ) ηn + ɛ θ η n ) + ( ( ɛθ ) u θ+ n+ + ( θ ɛθ ) u n + ɛ θ u n ) ( (θ+) η n +θ η n ) + ( J ɛ n+θ( u ) + J ɛ n+θ(u ) )( (θ+) ξ n +θ ξ n )]. Using Lemma 3. and absorbing constants into C we obtain ξn + N ξn+ ξn + ξ θ + n F + N θ + Jn+θ( ξ ɛ ) n= n= ( θ ) N ξ ( θ ) N ) θ ( [ [ ] ξ ( θ + ξ0 + C t p tt,0 + s+ p,s+ + t u ttt,0 + t u tt,0 + t u tt,0

12 Jiang, Moebujjaman, Rebolz and Trencea + t u,0 u tt,0 + t u,0 u tt,0 + k+ u t,k+ ) N + k u,k+ + k u,0 u,k+ + C ξn ]. Notice tat 0 ( ) N θ <, for any N 0. θ + Ten by Lemma. in [8, p. 76], we obtain ξn + N ξn+ ξn + ξ θ + n F + N θ + Jn+θ( ξ ɛ ) n= n= exp ( C T ) [ ( θ ) N ( ξ θ ) N ) θ ( [ ] ξ θ + ξ0 ( ( θ ) N ) ( + C t p tt,0 + s+ p,s+ + t u ttt,0 θ + + t u tt,0 + t u tt,0 + t u,0 u tt,0 + t u,0 u tt,0 + k+ u t,k+ + k u,k+ + k u,0 u,k+) ]. n=0 Recall e n = η n + ξn. inequality (.). By triangle inequality and Lemma 3., we obtain te error 5. Numerical Experiments. Tis section presents numerical experiments to test te family of proposed metods. Two typical metods, i.e., CNLEReg (θ = ) and BDFLEReg (θ = ), are tested on several bencmark problems, including a problem wit known analytical solution in order to verify predicted convergence rates, Poiseuille flow at low viscosity over a long time interval, flow past a normal flat plate, and cannel flow wit a contraction and two outlets. In all cases, te proposed metods perform very well. For flow past a normal flat plate, we compare solutions for BDFLEReg to anoter common stabilization metod used wit BDFLE, wic is discussed in te introduction as a speed penalization. We will call it BDFLEStab, and it is given by 3u n+ u n + u n u n+ α (u n+ u n ) + (u n u n ) u n+ + p n+ = f n+, u n+ = 0. (BDFLEStab) It will use a standard finite element spatial discretization in our simulations. 5.. Convergence rate verification. Our first experiment tests te predicted convergence rates for te metod. To do tis, we selected te analytical solution ( ) cos(y)e t u = sin(x)e t, p = (x y)( + t),

13 Second order unconditionally stable IMEX sceme 3 took te domain to be te unit square, set te viscosity =, and ten calculated te forcing f from tis data and te NSE. Errors and convergence rates were ten calculated by varying te mes widt and timestep t, using (P, P ) Taylor-Hood elements on uniform meses, and te CNLEReg algoritm. Full Diriclet boundary conditions were enforced nodally on te boundary. For te spatial convergence rates, we fixed t = T/6 and te end time T = 0., and for te temporal convergence rates, we fixed te mes widt to be = /8 and set T =. Denoting u u, := ( t N n= ( ) / u(t n ) u ) n, te results are sown in tables (for spatial error and temporal error, respectively), for different coices of te stabilization parameter ɛ. Te spatial convergence rate appears to be for eac coice of ɛ, wic is optimal based on our element coice. Te temporal convergence rate also appears to be for eac coice ɛ, wic is also optimal, altoug te actual error can be observed to increase as ɛ increases. Hence, tese tables are consistent wit te convergence rates predicted by te teory above, and tus provide some verification tat te teory (and te code) are correct. ɛ = 0 ɛ = 0 ɛ =.0 u u, Rate u u, Rate u u, Rate 7.58e- 7.58e- 7.58e e e e e e e e e e e e e e e e-7.98 Table 5.: Tis table gives errors and convergence rates for analytical test problem wit =.0, T = 0., t = T/6, using CNLEReg and varying mes widts. ɛ = 0 ɛ = 0 ɛ =.0 t u u, Rate u u, Rate u u, Rate T.936e-.837e-.7906e- T e e e-.7 T e e e T e e e T 3.533e e e Table 5.: Tis table gives errors and convergence rates for analytical test problem wit =.0, T =.0, = /8, using CNLEReg and varying time step. 5.. Poiseuille flow wit small viscosity. Our second experiment tests te long-time stability of a Poiseuille cannel flow simulation wit viscosity = 0. Altoug tis type of flow is a smoot NSE solution, it is well known to ave numerical instabilities wen viscosity values are small.

14 Jiang, Moebujjaman, Rebolz and Trencea Te computations use CNLEReg on te [0,] [0,] rectangular domain, t =.0, f = 0, (P, P ) Taylor-Hood elements on a uniform coarse mes wit = 0., and an end time of T =, 500. We begin wit an initial condition of u 0 = ( y( y) 0 and enforce boundary conditions to be no-slip on te top and bottom of te cannel, and parabolic (same as te initial condition) at te inlet and outlet. Solutions were ten computed using ɛ = 0 (no stabilization) and wit ɛ = 0 5, 0. Plots of energy un versus time are sown in figure 5. for te tree solutions, and reveals tat wile te stabilized solutions maintain te same (correct) energy trougout te simulation, at around T = 800 te unstabilized solution undergoes a transition away from te correct solution. After T = 800, te energy in te unstabilized system becomes incorrect and ten oscillates in time. Plots of te velocity field speed contours for te ɛ = 0 and 0 solutions at T=,500 are sown in figure 5., and from te plots we observe tat te stabilized solution as maintained its smoot parabolic profile, wile te unstabilized solution as clearly diverged from te correct solution it started wit. ), Fig. 5.: u vs. Time plots wit = 0, T = 500, = 0., t = for different values of ɛ. ɛ = 0 ɛ =e- Fig. 5.: CNLEReg computed velocity field speed contours for Poiseuille flow at T = 500, wit ɛ = 0 (left) and ɛ = (rigt) Flow past a normal flat plate. Next, we consider simulating flow past a normal flat plate wit te proposed sceme BDFReg, following te experiments

15 Second order unconditionally stable IMEX sceme 5 from [3, ]. We use te [ 7, 0] [ 0, 0] rectangular cannel as te domain, wit a 0.5 flat plate vertically centered and placed 7 units into te cannel from te left. Te inflow velocity is set wit u in =, 0 T, no forcing is applied (f = 0), and te kinematic viscosity is taken to be = 0.0, wic gives a Reynolds number of Re = 00 (based on te plate eigt). No-slip conditions are enforced on te walls and plate, and for te outflow, te zero-traction boundary condition is enforced weakly via te do-noting condition. Te quantities of interest are te Stroual number St (te frequency of vortex sedding) and te time averaged drag coefficient, bot of wic are computed from te final 0 periods of te flow after te flow reaces a statistically steady (periodic-in-time) state (wic occurs at around T=30). Te Stroual number is calculated as in [3, ], using te fast Fourier transform (FFT) of te transverse velocity at (.0,0.0) (note since te maximum inlet speed is and fluid density is assumed to be, we get directly tat St is te dominant frequency of te FFT). Te drag coefficients are computed at eac timestep and ten averaged, using a global integral formula (see [5] for details). Te computations used (P, P ) Taylor-Hood elements for velocity and pressure, on a Delaunay-generated triangulation wic provided 6,03 total degrees of freedom (dof). For tese tests we used θ = (i.e. BDFLEReg), varied t =0.05, 0., and 0.5, and set te regularization parameter ɛ=0.00. For comparison, we also computed using no stabilization wit te same discretizations, and also no stabilization on a muc finer discretization wit tat used 59K total dof and t = 0.0. We also compare our results wit data from direct numerical simulations of [3, ]. Comparisons between te solutions are given in table 5.3 and figure 5.3. Table 5.3 sows te time averaged drag coefficients and Stroual numbers for te simulations, as well as for DNS data. We first observe tat our solution on a fine mes wit a small timestep matces te DNS of Saa in bot te Stroual number and te time averaged drag coefficient. On te coarser discretizations, te stabilized solutions are significantly more accurate tan te unstabilized solutions, particularly as te timestep size increases. In particular, te Stroual numbers for te stabilized solutions matc te DNS data (up to te spacing of discrete frequencies from te FFT), and te drag coefficients are reasonably accurate, even for t = 0.5. Note tat te curvature stabilization of BDFLEReg as a decreasing affect on te drag coefficient, wic is in sarp contrast to te results below for BDFLEStab (were speed is stabilized). In figure 5.3, we plot te speed contours (zoomed in near te plate) of te fine mes solution, and stabilized and unstabilized coarse mes solutions wit t = 0.5, at a snapsot in time, T = 00. We observe tat te stabilized coarse mes solution qualitatively matces te fine mes solution muc better tan te unstabilized solution. Tat is, te stabilized solution as te same flow structures in nearly te same locations as te fine mes solution, wile te unstabilized solution appears quite different. Flow structures of te unstabilized solution not matcing te fine mes solution are consistent wit te significant error in Stroual number of te unstabilized solution. We also compare results for BDFLEReg to tose of BDFLEStab. Hence on te same coarse spatial discretization, we compute solutions wit BDFLEStab wit t =0. and 0.5, and varying α. Time averaged drag coefficients and Stroual numbers for eac simulation are given in table 5.. We observe tat te BDFLEReg solutions provides significantly better drag coefficients compared to BDFLEStab (for any parameter α). It is not unexpected tat BDFLEStab does a poor job wit drag coefficient prediction, since it stabilizes speed; in fact it is clear in te table tat

16 6 Jiang, Moebujjaman, Rebolz and Trencea Fine mes Coarse mes Coarse mes t=0.0, ɛ=0 t=0.5, ɛ=0 t=0.5, ɛ= Fig. 5.3: Sown above is a (zoomed) contour plot of te speed at time T=00 for a resolved solution (left), and unstabilized (center) and stabilized (rigt) coarse discretization solutions. Metod mes t ɛ C d Stroual number DNS of Saa [3, ] fine BDFLE (no stab) fine BDFLE (no stab) coarse * BDFLEReg coarse * BDFLE (no stab) coarse BDFLEReg coarse * BDFLE (no stab) coarse BDFLEReg coarse * Table 5.3: Sown above are long-time average drag coefficients and Stroual numbers for te simulations wit and witout stabilization, along wit DNS data from finer discretizations. We note tat te Stroual numbers wit asterisks are te closest discrete frequency values in te FFT to increasing α increases te drag coefficient. For t = 0., BDFLEStab does accurately predict te Stroual number, even toug C d prediction is inaccurate and gets worse as te stabilization parameter is increased. For t = 0.5, owever, te BDFLEStab solutions do not do a good job wit Stroual number prediction. Only for α = 0. does it accurately predict te Stroual number, but tis particular solution as a terrible prediction of C d. 5.. Cannel flow wit two outlets and a contraction. For our final numerical test, we apply te proposed metod to a D bencmark contracted cannel flow problem, first studied by Turek et. al. in [3], wit one inlet on te left and side, one vertical outlet in te middle and one outlet on te rigt and side, as sown in Figure 5.. We enforce a parabolic inflow profile u = y( y), 0 T wit maximum

17 Second order unconditionally stable IMEX sceme 7 Metod mes t α C d Stroual number BDFLE (no stab) coarse BDFLEStab coarse * BDFLEStab coarse * BDFLEStab coarse * BDFLEStab coarse * BDFLE (no stab) coarse BDFLEStab coarse BDFLEStab coarse BDFLEStab coarse BDFLEStab coarse * Table 5.: Sown above are long-time average drag coefficients and Stroual numbers for te simulations wit BDFLEStab, wit varying α and t, using te same 8,8 dof mes as BDFReg uses above. We note tat te Stroual numbers wit asterisks are te closest discrete frequency values in te FFT to Fig. 5.: above. Te pysical model of te domain for contracted cannel flow wit two outlets is sown orizontal velocity component u max inlet = at te center, no slip boundary conditions on its sides, and at te outflows, zero traction is imposed (wit te do-noting condition). We impose no external forcing (f = 0), set te kinematic viscosity = 0.00, start te flow from rest, and te simulation is run to T =. A resolved solution of te NSE was found using a timestep of t = and 60,378 total spatial dof in [3], and its T= speed contours are sown in figure 5.5. We computed using (P, P ) Taylor-Hood elements on a coarse mes tat provided 8,8 total dof, using time step size t = 0.0 wit bot BDF witout stabilization (i.e. BDFReg wit ɛ = 0), and BDFReg wit stabilization parameter ɛ = Speed contours of tese solutions at T= are sown in figure 5.5, and we observe tat te unstabilized solution is destroyed by oscillations. Te stabilized solution, owever, remains stable and is in general qualitative agreement wit te DNS, altoug tere are some differences near te end of te cannel; owever on suc a coarse discretization, it is not expected to get exact agreement. 6. Conclusions. A family of efficient, stabilized, second order, unconditionally stable timestepping metods for NSE are analyzed and numerically tested. Tese metods require te solution of only one linear system per time step, and ave en-

18 8 Jiang, Moebujjaman, Rebolz and Trencea Resolved DNS BDF (no stab), coarse discretization, = BDFReg, coarse discretization, = Fig. 5.5: at T=. Sown above is contour plot of te speed of te resolved Navier-Stokes velocity solution anced stability due to a penalization of te velocity curvature, and acieve second order temporal accuracy because te penalization is itself second order consistent. Rigorous proofs of unconditional stability and convergence are given for te entire family of metods. In addition to te teoretical results, several numerical experiments are performed wic sow te effectiveness of te metods. In addition to verifying te teoreticallypredicted convergence rates, a significant positive effect of te metod is seen on te stability and accuracy of solutions of tree bencmark flow problems, wen compared to solutions of unstabilized scemes and common, related, stabilized scemes. In particular, te metod BDFLEReg is sown to provide significantly more accurate predictions of te drag coefficient for flow across a flat plate, compared to BDFLEStab. Te metods can be easily applied to many oter models wit minor modifications, and tis report is an initial attempt to study tese linearized metods and new stabilization in a CFD context. How te stabilization terms affect te stability separately still requires more investigation. REFERENCES [] F. Armero and J. C. Simo, Long-term dissipativity of time-stepping algoritms for an abstract evolution equation wit applications to te incompressible MHD and Navier-Stokes equations, Comput. Metods Appl. Mec. Engrg., 3 (996), pp. 90. []. Baker, alerkin approximations for te Navier-Stokes equations, tec. rep., Harvard University, 976. [3] A. Bowers and L. Rebolz, Numerical study of a regularization model for incompressible flow wit deconvolution-based adaptive nonlinear filtering, Computer Metods in Applied Mecanics and Engineering, 58 (03), pp.. 7

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