Modified Pressure-Correction Projection Methods

Modified Pressure-Correction Projection Methods: Open Boundary and Variable Time Stepping Andrea Bonito, Jean-Luc Guermond, and Sanghyun Lee

Abstract In this paper, we design and study two modifications of the first order standard pressure increment projection scheme for the Stokes system. The first scheme improves the existing schemes in the case of open boundary condition by modifying the pressure increment boundary condition, thereby minimizing the pressure boundary layer and recovering the optimal first order decay. The second scheme allows for variable time stepping. It turns out that the straightforward modification to variable time stepping leads to unstable schemes. The proposed scheme is not only stable but also exhibits the optimal first order decay. Numerical computations illustrating the theoretical estimates are provided for both new schemes.

1 Introduction We consider the time-dependent Stokes system on a bounded domain ˝  Rd , d D 2; 3, with Lipschitz boundary @˝ and over a finite time interval Œ0; T . For a given force f W ˝  Œ0; T  ! Rd , the velocity u W ˝  Œ0; T  ! Rd and the pressure p W ˝  Œ0; T  ! R are related via the following system   @t u  2div r S u C rp D f

and div.u/ D 0

in ˝  Œ0; T ;

(1)

where  and  are the fluid density and viscosity of the fluid assumed to be  constant (and positive) and r S WD 12 r C r T denotes the symmetric part of the gradient. Relations (1) is supplemented by a boundary condition either prescribing the velocity or the force at the boundary. In order to simplify the presentation, we consider homogeneous cases uD0

on @˝  Œ0; T 

(2)

A. Bonito () • J.-L. Guermond • S. Lee Texas A&M University, College Station, TX 77843, USA e-mail: [email protected]; [email protected]; [email protected] © Springer International Publishing Switzerland 2015 A. Abdulle et al. (eds.), Numerical Mathematics and Advanced Applications - ENUMATH 2013, Lecture Notes in Computational Science and Engineering 103, DOI 10.1007/978-3-319-10705-9__61

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or .2r S u  p/ D 0

on @˝  Œ0; T ;

(3)

where  is the unit, outward pointing normal of @˝. In addition, the initial velocity u0 W ˝ ! Rd is prescribed, i.e. u.0; :/ WD u0 . At this point, we note that the extension to the Navier-Stokes system is treated similarly with the additional, but well known, techniques used to cope with the additional nonlinearity. Most projection methods are based on the original ideas of Chorin [1] and Temam [9], see also Goda [2]. We refer to [4] for an overview of projection methods. In this work, we obtain two different results regarding the so-called incremental pressure correction schemes studied for instance in [3, 5–7]: • The scheme proposed in [3] when the system is subject to open boundary conditions, see (3), is suboptimal with respect to the time discretization parameter. We propose and study a new scheme able to recover the optimal convergence rate, see Fig. 1. • We analyze a new scheme allowing for variable time stepping. It turns out that the straightforward generalization of constant time stepping to variable time stepping is unstable, see Fig. 2. To the best of our knowledge, projection schemes with variable time stepping have not been studied in the literature. Notice however, that no additional difficulty arises from having variable time stepping in the nonincremental scheme setting. Given a positive integer N , let 0 D t 0 < t 1 < t 2 <    < t N D T be a subdivision of the time interval Œ0; T  and set ıt n WD t n  t n1 . The norm in L2 .˝/ 1=2  is denoted by k:k0 and we equip H 1 .˝/ with the norm k:k1 WD k:k20 C kr:k20 .

Fig. 1 Decay of different error norms versus ıt for the original and modified standard pressure correction projection method. Suboptimal order of convergence O.ıt 1=2 / is observed for the standard method while the optimal order of convergence O.ıt / is recovered using the proposed scheme

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Fig. 2 (Left) Evolution of ku.t n ; :/kL2 .˝/ when using the standard scheme with ıt 1 D 0:025 and ıt n given by (16). (Right) Decay of the velocity and pressure errors versus ıt and with the time steps ıt n given by (16) when using the proposed scheme. The optimal order of convergence O.ıt / is observed

In addition, given a sequence of function 'ıt WD f' n gN nD0 , we define the following discrete (in time) norms: k'ıt kl 2 .E/ WD

N X

!1=2 ıt

n

k' n k2E

;

k'ıt kl 1 .E/ WD max .k' n kE /:

nD0

0nN

(4)

for E WD L2 .˝/ or H 1 .˝/.

2 Optimal Incremental Projection Scheme for Open Boundary Problem We consider the system (1) supplemented with the force condition at the boundary (3) and focus on the case of uniform (constant) time steps, i.e. ıt WD NT D ıt n , n D 0;    ; N . The case of variable time steps is discussed in Sect. 3. The approximations of u.t n ; :/ and p.t n ; :/, n D 0; : : : ; N , are denoted un and p n respectively. For clarity, we also denote by  n the pressure increment approximation, i.e. p n D p n1 C  n :

(5)

Together with the initial condition on the velocity u0 D u0 , the algorithm requires initial pressure p.0/ and we set p 1 WD p 0 WD p.0/, and so  0 WD 0. We seek recursively the velocity unC1 and the pressure p nC1 in three steps. First, given un ,  n and p n , the velocity approximation at t nC1 is given by 

  nC1  un u unC1  un  2div.r S unC1 / C r.p n C  n /  ˛rdiv D f .t nC1 ; :/; ıt ıt (6)

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in ˝, where ˛  1 is a stabilization parameter. As we shall see, the consistent “grad-div” term is instrumental to ensure the stability of the scheme by providing a control on k nC1   n kH 1 .˝/ , i.e. the second increment of the pressure; see (13). Equation (6) is supplemented by the boundary condition    nC1  un u D0 2r S unC1  .p n C  n / C ˛div ıt

on @˝:

(7)

The second step consist in seeking the new pressure increment approximation  nC1 as the solution to  ıt nC1 C ıt nC1 D div.unC1 /

in ˝

(8)

together with the boundary condition @ nC1  D0 @

on @˝:

(9)

Finally, the new pressure approximation is then given by (5). The novelty of this projection scheme is to impose a Neuman boundary condition on the pressure increment (and therefore on the pressure). Its aim is to reduce the boundary layer on the pressure and improve the convergence rate. Compare with [3] where a Dirichlet condition p nC1 D p n is proposed on the pressure. This is at the expense of adding (i) an harmless zero order term ıt nC1 in (8) to be able to recover the full l 2 .H 1 .˝// norm for the pressure and (ii) the more serious “grad-div” stabilization term in (6), which complicates the linear algebra. Notice that the boundary condition (9) proposed here corresponds to the standard boundary condition when the velocity is imposed at the boundary; refer to [3]. We now briefly discuss the stability and error estimates for the scheme (6)–(9). Theorem 1 (Velocity Stability) Set f  0 and assume ˛  1, then there holds kuıt k2l1 .L2 .˝// C 4kr S uıt k2l2 .L2 .˝// C ˛kdiv.uıt /k2l1 .L2 .˝// C .ıt /2 kpıt k2l1 .H 1 .˝//  ku0 k20 C ˛kdiv.u0 /k20 C .ıt /2 kp0 k21

provided u0 2 L2 .˝/d , div.u0 / 2 L2 .˝/ and p0 2 H 1 .˝/. Proof Multiplying (6) by 2ıtunC1 and integrating over ˝ one gets after integrating by parts and using the boundary condition (7)    kunC1 k20 C kunC1  un k20  kun k20 C 4ıtkr S unC1 k20   C ˛ kdiv.unC1 /k20 C kdiv.unC1  un /k20  kdiv.un /k20 Z  2ıt .p n C  n /div.unC1 /d x D 0: ˝

(10)

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The last term in the left hand side of the above relation is estimated upon multiplying (8) by 2ıt.p n C  n /, integrating over ˝ and using the boundary condition (9) Z Z n n nC1 2 2ıt .p C  /div.u /d x D 2.ıt/ r nC1  r.p n C  n /d x ˝

˝

Z

C 2.ıt/2

 nC1 .p n C  n /d x: ˝

In view of (5), we write p n C  n D  n   nC1 C p nC1 and realize that Z 2ıt ˝

.p n C  n /div.unC1 /d x D .ıt/2 k n k21  .ıt/2 k nC1   n k21 C .ıt/

2

kp nC1 k21

 .ıt/

2

(11)

kp n k21 :

It remains to derive a bound for k nC1   n k1 . Multiplying by  nC1   n the difference of two successive relations (8) and integrating over ˝ yield Z ıtk

nC1



 n k21

D

div.unC1  un /. nC1   n /d x;

(12)

˝

after an integration by parts and taking advantage of the boundary condition (9). Hence, we deduce that ıtk nC1   n k1  kdiv.unC1  un /k0 :

(13)

Gathering the estimate (13), (11) and (10), we obtain    kunC1 k20 C kunC1  un k20  kun k20 C 4ıtkr S unC1 k20   C ˛ kdiv.unC1 /k20  kdiv.un /k20 C .˛  1/kdiv.unC1  un /k20   C .ıt/2 kp nC1 k21  kp n k21 C k n k21  0: The desired bound follows after summing for n D 0 to N  1.

t u

We emphasize that the above proof is closely related to the case where Dirichlet boundary conditions are imposed on the velocity; refer for instance to [4, 8]. The difference resides on the fact that (13) can be circumvented using an integration by parts in (12). Hence following the techniques developed for the Dirichlet case

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together with the argumentation leading to (13) yields the optimal convergence rates N X

max ku.t ; :/  u kL2 .˝/ C n

n

nD1;:::;N

!1=2 ıtkr .u.t ; :/  u S

n

n

/k2L2 .˝/

nD1

C ˛ max kdiv.u.t ; :/  u /kL2 .˝/ C n

n

nD1;::;N

N X

!1=2 ıtkp.t /  n

p n k2L2 .˝/

 C ıt;

nD1

with a constant C independent of N and provided the exact velocity u and pressure p satisfy the appropriate regularity conditions. To illustrate the optimality of the proposed algorithm, we consider the exact solution   sin.t C x/ sin.t C y/ ; p.t; x; y/ D sin.t C x  y/ u.t; x; y/ WD cos.t C x/ cos.t C y/ defined ˝ WD .0; 1/2 . The behavior of the errors in velocity and pressure approximations versus the time step ıt used are depicted in Fig. 1. Suboptimal order of convergence O.ıt 1=2 / is observed for the standard method while the optimal order of convergence O.ıt/ is recovered using the proposed scheme. The space discretization is chosen fine enough not to interfere with the time discretization error.

3 Variable Time Stepping We now consider variable time steps ıt n satisfying ıt n  ıt;

1  n  N;

for a positive constant ıt independent of n. The incremental projection scheme with variable time stepping reads as follow. Given un ,  n and p n , the velocity approximation at t nC1 is defined by the relation 

unC1  un ıt

nC1

 2div.r S unC1 / C r.p n C

.ıt/2 n

ıt ıt

nC1

 n / D f .t nC1 ; :/:

in ˝ (14)

For simplicity, we consider the boundary condition u D 0 on @˝ but the techniques presented in Sect. 2 for the open boundary condition case apply in this context as

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well. The pressure increment  nC1 solves 

.ıt /2  nC1 D div.unC1 / ıt nC1

in ˝

and

@ nC1  D0 @

on @˝: (15)

Finally, the pressure is updated according to relation (5). The standard pressure correction schemes are derived from the original velocity prediction – projection scheme, see for instance [4]. When the same time step value is used for the velocity prediction and correction, the factors multiplying the 2

n

2

increment  n in (14) and (15) becomes ıtıtnC1 and ıt nC1 instead of ıt nıtıt nC1 and ıtıtnC1 as in the proposed scheme (14)–(15). This alternative is referred as the standard scheme but we emphasize that there is no reason for the projection step to use the velocity prediction time step as projection parameter. In fact, this choice turns out to be numerically unstable as illustrated now. We consider the same setting as in Sect. 2 but with variable time steps given by  ıt n D ıt 1 

1 102

when n is odd; when n is even;

(16)

for different values of ıt 1 . In this case, we set ıt WD ıt 1 . Figure 2 (left) illustrates the unstable behavior of kun kL2 .˝/ for n D 0; : : : ; N when using the standard scheme with ıt 1 D 0:025. However, the l 2 .H 1 .˝// and l 1 .L2 .˝// errors on the velocity decay like ıt when the proposed scheme (14)–(15) is used, see Fig. 2 (right). We now briefly discuss the stability and error estimates for the scheme (14)–(15). Theorem 2 (Velocity Stability) Set f  0, and assume ıt n  ıt, n D 1; ::; N , then there holds kuıt k2l1 .L2 .˝// C 4kr S uıt k2l2 .L2 .˝// C

1 .ıt/2 kpıt k2l1 .H 1 .˝// 

 ku0 k20 C .ıt/2 kp0 k21 provided u0 2 L2 .˝/d and p0 2 H 1 .˝/. Proof Multiplying (14) by 2ıt nC1 unC1 and integrating over ˝ one gets after integrating by parts and using the boundary condition u D 0,    kunC1 k20 C kunC1  un k20  kun k20 C 4ıt nC1 kr S unC1 k20 ! Z .ıt/2 n nC1 n ıt 2 p C  div.unC1 /d x D 0: ıt n ˝

(17)

The pressure increment relation (15) is invoked to derive a bound for the last term in the left hand side of the above relation. More precisely, multiplying (15) by

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/ n 2.ıt nC1 p n C .ıt ıt n  / , integrating over ˝ and using the boundary condition (9) we realize that

Z

 2

.ıt/2 n  /div.unC1 /d x ıt n ˝ Z Z 2 nC1 n r  rp d x C 2 r D 2.ıt/ .ıt nC1 p n C

˝

! .ıt/2 nC1 r  ıt nC1

˝

! .ıt/2 n  d x: ıt n

Relation (5) allows us to rewrite the right hand side of the above expression as   .ıt/2 krp nC1 k20  krp n k20  kr nC1 k20

ˇˇ ˇˇ .ıt/4 .ıt/4 ˇˇ nC1 2 n 2 kr k C kr k  C ˇˇr 0 0 ˇˇ .ıt nC1 /2 .ıt n /2

.ıt/2 nC1 .ıt/2 n    ıt nC1 ıt n

!ˇˇ2 ˇˇ ˇˇ ˇˇ : ˇˇ 0

Going back to (17), we get    kunC1 k20 C kunC1  un k20  kun k20 C 4ıt nC1 kr S unC1 k20

!   1 1 .ıt/2 2 nC1 2 n 2 2 C .ıt/ krp k0  krp k0 C .ıt/  1 kr nC1 k20   .ıt nC1 /2 ˇˇ !ˇˇ2 .ıt/2 nC1 .ıt/2 n ˇˇˇˇ 1 .ıt/4 1 ˇˇˇˇ n 2 C kr k0 D ˇˇr    ˇˇ : ˇˇ  .ıt n /2  ˇˇ ıt nC1 ıt n 0

The difference of two successive relations (15) together with the boundary condition un D unC1 D 0 on @˝ guarantee that ˇˇ ˇˇ ˇˇ ˇˇr ˇˇ

.ıt/2 nC1 .ıt/2 n    ıt nC1 ıt n

!ˇˇ ˇˇ ˇˇ ˇˇ  kunC1  un k0 : ˇˇ 0

Hence, using the assumption ıt nC1  ıt,     1  kunC1 k20  kun k20 C 4ıt nC1 kr S unC1 k20 C .ıt/2 krp nC1 k20  krp n k20  C

1 .ıt/4 kr n k20  0;  .ıt n /2

and the desired bound follows after summing for n D 0 to N  1.

t u

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Regarding the error decay we have that under the assumption ıt n  ıt, n D 1; : : : ; N , there exists a constant C independent of n and ıt such that max ku.t ; :/  u kL2 .˝/ C n

nD1;:::;N

n

N X

!1=2 ıt ku.t ; :/  n

n

un k2H 1 .˝/

 C ıt;

nD1

provided u and p are smooth enough and ıt is sufficiently small. The proof of the above claim is omitted but relies on the argumentations provided in the proof of Theorem 2. In addition, we emphasize that scheme (14)–(15) does not optimize the choice of ıt n in order to equi-distribute the time discretization errors and explain that the decay rate is dictated by ıt (and not ıt n , n D 1; : : : ; N ). Including such mechanism is out of the scope of this work. Moreover, the decay rate for the l 2 .L2 .˝// error on the pressure is still an open problem but the numerical results provided in Fig. 2 indicate an optimal rate. Acknowledgements Partially supported by NSF grant DMS-1254618 and award No. KUS-C1016-04 made by King Abdullah University of Science and Technology (KAUST).

References 1. A.J. Chorin, On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comput. 23, 341–353 (1969) 2. K. Goda, A multistep technique with implicit difference schemes for calculating two- or threedimensional cavity flows. J. Comput. Phys. 30, 76–95 (1979) 3. J.L. Guermond, P. Minev, J. Shen, Error analysis of pressure-correction schemes for the timedependent stokes equations with open boundary conditions. SIAM J. Numer. Anal. 43(1), 239– 258 (2005). (electronic) 4. J.L. Guermond, P. Minev, J. Shen, An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44–47), 6011–6045 (2006) 5. J.-L. Guermond, L. Quartapelle, On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80(2), 207–238 (1998) 6. J.L. Guermond, J. Shen, Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41(1), 112–134 (2003). (electronic) 7. J.L. Guermond, J. Shen, On the error estimates of rotational pressure-correction projection methods. Math. Comput. 73, 1719–1737 (2004) 8. J.-L. Guermond, A.J. Salgado, A splitting method for incompressible flows with variable density based on a pressure poisson equation. J. Comput. Phys. 228(8), 2834–2846 (2009) 9. R. Témam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II. Arch. Rational Mech. Anal. 33, 377–385 (1969)