incremental unknowns for solving the incompressible navier–stokes equations

Mathematics and Computers in Simulation 52 (2000) 445–489

Incremental unknowns for solving the incompressibleNavier–Stokes equations

Salvador GarciaDepartment of Mathematics, University of California, Irvine, CA 92697-3875, USA

Received 1 March 1999; accepted 7 April 2000

Abstract

Incremental unknowns, earlier designed for the long-term integration of dissipative evolutionary equations, are in-troduced here for the incompressible Navier–Stokes equations in primitive variables when multilevel finite-differencediscretizations on a staggered grid are used for the spatial discretization; extending to this practical case, the notion ofsmall and large wavelengths that stems naturally from spectral methods when Fourier series expansions are consid-ered. Furthermore, for the temporal discretization, we use theθ -scheme, which allows to decouple the nonlinearityand the incompressibility in these equations; then we have to solve a generalized Stokes equation — we considerhere a leading preconditioned outer/inner iteration strategy — and a nonlinear elliptic equation — we linearize itsnonlinear terms to get an iterative process. Roughly, at each iterative stage several Poisson equations must be solved,incremental unknowns appear there as an efficient preconditioner. The incremental unknowns methodology appearswell suited to capture the turbulent behavior of the flow whose small eddies, even for moderate Reynolds numbers,never go to a steady state; instead they converge to a strange attractor, and keep always bringing imperceptiblekinetic energy to the flow. First, they wander around, then they converge to the strange attractor. © 2000 IMACS.Published by Elsevier Science B.V. All rights reserved.

Keywords:Finite differences; Staggered grids; Incremental unknowns; Hierarchical basis; Poisson equations; GeneralizedStokes equations; Nonlinear elliptic equations; Incompressible Navier–Stokes equations; Strange attractors

1. Introduction

Incremental unknowns appear [5,17] as the natural tool to study the long-term dynamic behaviorof nonlinear dissipative evolutionary equations, and to construct inertial manifolds and approximateinertial manifolds, when multilevel finite-difference discretizations of such equations are used for thespatial discretization; extending to this practical case the notion of small and large wavelengths thatstems naturally from spectral methods when Fourier series expansions are considered. The main purposeof this work is to introduce incremental unknowns to study the long-term dynamic behavior of the

E-mail address:[email protected] (S. Garcia)

0378-4754/00/$20.00 © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.PII: S0378-4754(00)00158-0

446 S. Garcia / Mathematics and Computers in Simulation 52 (2000) 445–489

two-dimensional (2D) time-dependent incompressible Navier–Stokes equations, with Dirichlet boundaryconditions for the velocity field, in primitive variables when multilevel finite-difference discretizationson a staggered grid are used for the spatial discretization; furthermore, for the temporal discretizationwe use theθ -scheme [11], which allows to decouple the nonlinearity and the incompressibility in theNavier–Stokes equations. This situation is considered here, since the extension of the overall methodologyto the three-dimensional case is then straightforward. Now, we have to solve a generalized Stokes equationand a nonlinear elliptic equation. For the numerical solution of the generalized Stokes equations, weconsider a leading preconditioned outer/inner iteration strategy, the preconditioned conjugate gradient(PCG) method is used to get the pressure field; at each iterative stage we have to solve two generalizedPoisson equations with Dirichlet boundary conditions for the velocity field — we use here the PCGmethod; that is, the preconditioned inner iteration strategy — and we have to solve a Poisson equationwith Neumann boundary conditions, as the preconditioner, for the pressure field — we use here again thePCG method; that is, the preconditioned outer iteration strategy. For the numerical solution of the nonlinearelliptic equation, we linearize its nonlinear terms to get an iterative process, at each iterative stage wehave to solve two generalized Poisson equations with Dirichlet boundary conditions for the velocity field— we use here the preconditioned Bi-CGSTAB method (PBi-CGSTAB). For the numerical solution ofthe linear symmetric elliptic equations before the use of the incremental unknowns appear throughout asan efficient preconditioner; the convergence behavior of the iterative methods is fast and smooth (nearlyoptimal). Altogether, only a few iterations of the leading PCG method are needed to obtain maximalaccuracy and the convergence behavior of the iterative process is acceptable. Now, there are two intrinsicdifficulties in the incremental unknowns methodology. The first one is to compute the products of thematricesSandST with a vector to put into action the iterative methods to solve the incremental unknownslinear systems, this is overcome by using the fast algorithms introduced in [7]; then only we need to knowthe explicit block-matrix structure of the transfer matrixS in the two-level case, which is an extremelysimple matrix, and such products are computed successively by considering two levels at each time andusing the adequate permutation matrices. The second one is to compute explicitly the block diagonal partof the incremental unknowns matrices to use block diagonal (scaling) preconditioning, this is overcomein the first part of this work by bringing forward the associated hierarchical basis; the computation of suchblock diagonal part is straightforward for the underlying incremental unknowns matrices associated tofinite-difference operators with constant coefficients. Otherwise, to do so, a matrix technique is set up. Forthe linear symmetric elliptic equations, the use of the incremental unknowns amounts to a change of basis— the use of hierarchical basis instead of nodal basis — in the finite-difference discretization spaceVh, which remains unchanged, and so by Céa’s Lemma ([4], p. 365) the accuracy of the discretizationremains unchanged; in the cases considered below, that is, second-order accuracy throughout. For thelong-term integration, we observe that the nodal basis functions are extremely localized step functionsand perceive oscillations happening only at their place; whereas, second-order (higher-order) hierarchicalbasis spread throughout the domain and may perceive oscillations happening far away from their place,giving in consequence more accurate and stable (able to traverse turbulent zones) numerical algorithms[8]. Thoroughly, this work presents some steps toward the long-term simulation of turbulent flows throughincremental unknowns. At last, we emphasize the fact that incremental unknowns are designed for thelong-term integration of dissipative evolutionary equations; whereas, the similar and usual hierarchicalbasis methods (finite element case) and multigrid methods have been developed mainly for the numericalsolution of stationary elliptic equations, even more the higher-order incremental unknowns do not fit intothe framework of these methods. The incremental unknowns methodology appears well suited to capture

S. Garcia / Mathematics and Computers in Simulation 52 (2000) 445–489 447

the turbulent behavior of the flow whose small eddies, even for moderate Reynolds numbers, never goto a steady state; instead they converge to a strange attractor, and keep always bringing imperceptiblykinetic energy to the flow. First, they wander around, then they converge to the strange attractor.

This article is organized as follows. In Section 2, we consider the numerical solution of the (gener-alized) Poisson equation with Dirichlet and Neumann boundary value conditions on several classicaland staggered spatial discretization grids through incremental unknowns; we use the finite-differencevariational framework to bring forward the associated hierarchical basis to compute the explicit blockdiagonal part of the finite-difference operators, which is needed to put into action block diagonal (scaling)preconditioning. In Section 3, we discuss the numerical solution of the 2D time-dependent incompressibleNavier–Stokes equations in primitive variables, with Dirichlet boundary conditions for the velocity field,when multilevel finite-difference spatial discretizations on a staggered grid are used, and with operatorsplitting methods (theθ -scheme) for the temporal discretization and incremental unknowns for the spa-tial discretization. In Section 4, we display computational experiments supporting the use of the overallincremental unknowns methodology. At last, in Section 5, we state further remarks.

2. The (generalized) Poisson equation

The numerical solution of the incompressible Navier–Stokes equations in primitive variables withDirichlet boundary value conditions on a staggered grid involves the numerical solution of the (gener-alized) Poisson equation with Dirichlet and Neumann boundary value conditions on a staggered grid.Hereafter, we introduce incremental unknowns on several classical and staggered discretization grids(with nested sequence of grids) to solve the underlying discrete equations; we use the finite-differencevariational framework to bring forward the associated hierarchical basis to compute the explicit blockdiagonal part of the finite-difference operators, which is needed to put into action block diagonal (scaling)preconditioning. First, we present the general setting, then we go through each specific case.

2.1. The continuous equations

Here, we consider in the domain�=]0, 1[×]0, 1[, the generalized Poisson equation with Dirichletboundary conditions

−ν 1u+ γ (xxx)u = f in �,

u = ϕ on 0 = ∂�,

whereν>0 is a given constant,γ (xxx)≥0,∀xxx∈� is a given function, and the Poisson equation with Neumannboundary conditions

−1u = f in �,

∂u

∂nnn= ϕ on 0 = ∂�,

where∂u/∂nnn is the normal derivative, and∫�f dxxx + ∫

0ϕ d0 = 0 (see [12], p. 35).


2.2. The incremental unknowns framework

The incremental unknowns were introduced by Temam [17] as the natural tool to study the long-termdynamic behavior of nonlinear dissipative evolutionary equations, and to construct inertial manifoldsand approximate inertial manifolds, when (spatial) multilevel finite-difference discretizations are used;extending to this practical case the notion of small and large wavelengths that stems naturally from spectralmethods when (spatial) Fourier series expansions are considered.

First of all, we summarize the main facts that characterize the incremental unknowns methodology andaccount for his efficiency at solving linear elliptic equations:

• There is a reordering of the equations and unknowns because a nested sequence of grids is considered.• There exist fast algorithms to compute the products of the matricesSandST with a vector, whereS

stands for the transfer matrix from the incremental unknowns to the nodal unknowns. There we needto know the explicit block-matrix structure of the transfer matrixS, in the two-level case, which is asimple matrix, then such products are computed successively by considering two levels at each timeand using the adequate permutation matrices.

• The incremental unknowns matricesSTAS, whereA is a nodal unknowns matrix, are never built upexplicitly because the iterative procedures requires only to compute the product of such matrices witha vector.

• The use of the incremental unknowns amounts to a change of basis — the use of the hierarchical basisinstead of the nodal basis — in the finite-difference discretization spaceVh, which remains unchanged,and so by Céa’s Lemma ([4], p. 365) the accuracy of the discretization remains unchanged; in the casesconsidered below, that is, second-order accuracy throughout.

• The associated hierarchical basis spread throughout the domain and those terms standing outside thedomain can be truncated.

• The block diagonal part of the incremental unknowns matricesSTAS, whereA is a nodal unknownsmatrix, can be computed explicitly by using the associated hierarchical basis.

• The incremental unknowns are a good preconditioner for the discrete equations.• The discrete equations are block-diagonal scaled.• Iterative solvers work efficiently since convergence is fast and smooth.

Now, we present in a systematic way the incremental unknowns methodology used in the spatialdiscretization. We consider the following finite-difference variational approach. Letn be a non-negativeinteger. In one space dimension, we consider the line segment�, which is either the line segment ]0, 1[ or]−1

2h, 1+12h[ or [0, 1] and set up the (staggered) uniform grid�h corresponding to the mesh sizeh=1/n.

Then, we introduce the finite-difference vector spaceWh that consists of restrictions to the line segment� of step functions that are constants on the line segments of lengthh linked to�h; in the vector spaceWh, we consider first, the nodal basisωk. In two space dimensions, we consider the domain� = �× �,where� and� are line segments as before, the finite-difference vector space isVh=Wh⊗ Wh, whereWh

andWh are finite-difference vector spaces as above; in the vector spaceVh, we consider first, the nodalbasisωk,l=ωk ⊗ ωl. We write a generic vectorvh∈Vh as

vh =∑k,l

vk,l ωk ⊗ ωl . (2.1)

Moreover, we introduce the finite-difference operators∇ ih, ∇ih


∇ihvh(x) = 1

h(vh(x + hei)− vh(x)), (2.2)

∇ihvh(x) = 1

h(vh(x)− vh(x − hei)), (2.3)

δihvh(x) = 1

2h(vh(x + hei)− vh(x − hei)), (2.4)

for i=1, 2, wheree1=(1, 0),e2=(0, 1) is the canonical basis ofR2.The discrete equation reads

−ν111huh + γhuh = fh,

in the former case, and

−111huh = fh,

in the latter case, where1h is the finite-difference Laplace operator,γ h is the finite-difference discretiza-tion of the given functionγ , uh is the finite-difference approximation of the unknown functionu, andfhis the finite-difference discretization of the right-hand sidef besides, near the boundary, terms comingfrom the finite-difference discretization of the Laplace operator and involving the boundary conditions(boundary and extrapolated values); the variational formulation of the discrete equation is

Find uh ∈ Vh such that((uh, vh))h = (fh, vh), for all vh ∈ Vh,where ((·,·))h is the associated finite-difference bilinear form and (·,·) is the scalar product of the HilbertspaceL2(�).

Then, the multigrid-like framework used to introduce incremental unknowns is as follows. Here, weconsidern = 2`N , where` = i − 1 andi andN are non-negative integers,i, N≥2, remaining fixed. Herethe parameteri defines the number of levels to be used and the parameterN establishes the size of thecoarsest grid to be used. Forj from ` down to 0, we introduce thejth-level (staggered) uniform grid�jcorresponding to the mesh sizehj = 2`−jh in both directions; therefore, we obtain the nested sequenceof grids

�` ⊃ �`−1 ⊃ · · · ⊃ �1 ⊃ �0. (2.5)

In Fig. 1, we present several discretization grids with nested sequence of grids, where the coarse gridis constituted by the squares, the intermediate grid is constituted by the squares and circles, and thefinest grid is constituted by the squares, circles, and crosses. These grids are components of the staggereddiscretization grid used to discretize the incompressible Navier–Stokes equations in primitive variables:a (classical grid)×(staggered grid) and a (staggered grid)×(classical grid) is used to discretize the firstand second component of the velocity field to solve then a generalized Poisson equation with Dirich-let boundary conditions, a (staggered grid)×(staggered grid) is used to discretize the pressure field tosolve then a Poisson equation with Neumann boundary conditions, as a preconditioner, and a (classicalgrid)×(classical grid) is used to discretize the stream function to solve then a generalized Poisson equationwith Dirichlet boundary conditions.


Fig. 1. Discretization grids with nested sequence of grids.

Next, we propose ahierarchical orderingof the nodal values (unknowns) ofuh at the nodes of thefinest grid�`.• Nodal values ofuh at the nodes of the fine grid�j that do not belong to the coarse grid�j−1 for j from` down to 1.

• Nodal values ofuh at the nodes of the coarsest grid�0.Then, we introduce theincremental unknownsrecursively from the finest level up to the coarsest level

(the coarse level being successively excluded). First, at the nodes of the fine grid�j that do not belongto the coarse grid�j−1 the jth-level incremental unknowns consist of the increment of the nodal valuesof uh to the average of the nodal values ofuh at the neighboring nodes in the coarse grid�j−1 (whichis considered the classical way) forj from ` down to 1. At last, at the nodes of the coarsest grid�0, theincremental unknowns consist of the nodal values ofuh.

An intrinsic (i.e. invariant under permutations) description of the transfer matrixS−1 from the nodalunknowns to the incremental unknowns is readily done with a picture of the associated directed graphof the transfer matrix from the (previous) nodal unknowns to thejth-level increment unknowns, such asillustrated on Fig. 2 (the left bottom corner of the grid�j ). Further, with the indication that the axialcoefficients are 1/2 and the oblique coefficients are 1/4 we have a complete definition of ann×n matrixDj , such that

S−1 = I −∑j=1

Dj. (2.6)


Fig. 2. Directed graph of the matrixDj .

The use of incremental unknowns provides an efficient way of solving the discrete equations that ingeneral we write

Ahuh = fh. (2.7)

This is the nodal linear system in lexicographical ordering. Next, we make the change of unknownsuh =Puh, whereP stands for the permutation matrix from hierarchical ordering to lexicographical ordering,and the tilde unknowns (·) stand for nodal unknowns in hierarchical ordering; we get the equivalent linearsystem

Ahuh = PTfh, (2.8)

whereAh = PTAhP. This is the nodal linear system in hierarchical ordering. At last, we make thechange of unknownsu = Su, whereSstands for the transfer matrix from the incremental unknowns tothe nodal unknowns, and hat unknowns (·) stand for incremental unknowns in hierarchical ordering; weget the equivalent linear system

Ahuh = STPTfh, (2.9)

whereAh = STAhS. This is the incremental unknowns linear system in hierarchical ordering. Now, weobserve that applying the conjugate gradient method to the linear system


is equivalent to applying the preconditioned conjugate gradient method to the linear system

Ahuh = PTfh, (2.11)

with preconditioning matrixK for Ah, whereK=(SST)−1; furthermore, we remark that applying theconjugate gradient method to the linear system


with preconditioning matrixK for Ah is equivalent to applying the preconditioned conjugate gradientmethod to the linear system

Ahuh = PTfh, (2.13)


with preconditioning matrixK for Ah, whereK = (SK−1ST)−1. Hereafter, we consider the latter pre-conditioner with block diagonal (scaling):K=blockdiagAh=blockdiagSTAhS. Now, we know [9] thatthe condition number of the incremental unknowns matrix

Ah = STAhS, (2.14)

is O(1/h20)O((logh)2), whereh0 is the mesh size of the coarsest grid andh is the mesh size of the finest

grid; furthermore, if block diagonal scaling is used, then the condition number of the preconditionedincremental unknowns matrix

K−1Ah = K−1STAhS, (2.15)

comes out to be O((logh)2). Instead of the condition number of the nodal unknowns matrix

Ah = PTAhP, (2.16)

is O(1/h2), which is much larger and becomes worst forh small.Now, the basisω(j)k,l of Vh that allows to recover the incremental unknowns introduced before is built

up from the one-dimensional hierarchical basis (finite-difference hat functions) by tensor products andtranslations [8]. This basis ofVh is the hierarchical basis. The coefficients of the incremental unknownsmatrix Ah = STAhS are built up from the coefficients((ω(r)k,l , ω

(s)p,q))h and(ω(r)k,l , ω

(s)p,q)h. Since the bi-

linear forms ((·,·))h and (·,·)h on Vh splits into the product of one-dimensional bilinear forms and scalarproducts over tensor-product functions, the one-dimensional computations and the setting above providein particular the coarse-level block diagonal part and the fine-level diagonal part of the finite-differenceoperators. Then, we need to compute in detail the matrices

[(∇hωck,∇h

ˆωcl )]k,l, [(ωck, ˆωcl )]k,l, (2.17)

where we designate byωck, ˆωcl the coarse-level hierarchical basis in one space dimension; besides tofine-level bilinear forms and scalar products, which are handled the same way. Then, we can put intoaction block diagonal (scaling) preconditioning to solve the incremental unknowns linear systems; thedetailed discussion of this issue is postponed up to Section 2.5.

2.3. The generalized Poisson equation with Dirichlet boundary conditions

For the discretization of the generalized Poisson equation with Dirichlet boundary conditions, weconsider first, a (classical grid)×(classical grid) and then a (classical grid)×(staggered grid) as statedbefore.

2.3.1. The classical gridHere,�h is the uniform grid corresponding to the mesh sizeh and the finite-difference vector space

Wh consists of restrictions to the line segment ]0, 1[ of step functions that are constants on the linesegments [kh, (k+1)h[ for k = 1, . . . , n − 1. The vector spaceWh is spanned by the nodal basisωk,k = 1, . . . , n − 1, which are equal to 1 on the line segment [kh, (k+1)h[ and vanish outside this linesegment. Now, we consider the domain�=]0, 1[×]0, 1[. The finite-difference vector space isVh=Wh ⊗Wh; it is spanned by the nodal basisωk,l=ωk ⊗ωl, k, l = 1, . . . , n− 1, which are equal to 1 on the planesegment [kh, (k+1)h[×[lh, (l+1)h[ and vanish outside this plane segment.


The finite-difference Laplace operator reads

111h =2∑i=1

∇ih ∇ih, (2.18)

its finite-difference matrix reads

111hvh = I ⊗1h +1h ⊗ I, (2.19)

where1h is the one-dimensional finite-difference Laplace operator whose finite-difference matrix is

1h = 1

h2

−2 11 −2 1

. . .

1 −2 11 −2

, (2.20)

and its associated finite-difference bilinear form reads

((uh, vh))h =2∑i=1

(∇ihuh,∇ihvh). (2.21)

Next, in Fig. 3, we display the hierarchical basis from the coarsest level down to the finest level whenthree levels are considered (classical grid).

Now, from the definition of the hierarchical basis [8] and from the formulae

l∑m=1

m = l(l + 1)

2,

l∑m=1

m2 = (2l + 1)

3

l(l + 1)

2, (2.22)

we obtain

(ωck, ωck) = h

(1 + (2` − 1)(2`+1 − 1)

32

), for k = 1, . . . , N − 1. (2.23)

Fig. 3. Hierarchical basis.


Indeed, we have that

(ωck, ωck) = h

1 + 2

2`−1∑m=1

(m2`

)2

, for k = 1, . . . , N − 1.

Now, since

2`−1∑m=1

(m2`

)2=(

1

2`

)2 2`−1∑m=1

m2 =(

1

2`

)2(2`+1 − 1)

3

(2` − 1)2`

2= (2` − 1)(2`+1 − 1)

32 +1,

we obtain

(ωck, ωck) = h

(1 + (2` − 1)(2`+1 − 1)

32

), for k = 1, . . . , N − 1.

Moreover, we get

(ωck, ωck) = h

((2` − 1)(2` + 1)

32 +1

), for k, l = 1, . . . , N − 1 with |k − l| = 1. (2.24)

Indeed, we have that

(ωck, ωc`) = h

2`−1∑m=1

m

2`2` −m

2`

, for k, l = 1, . . . , N − 1 with |k − l| = 1.

Now, since

2`−1∑m=1

m

2`2` −m

2`=(

1

2`

)2 2`−1∑m=1

m(2` −m) =(

1

2`

)22`

2`−1∑m=1

m−2`−1∑m=1

m2

=(

1

2`

)2{2`(2` − 1)2`

2− (2`+1 − 1)

3

(2` − 1)2`

2

}= (2` − 1)(2` + 1)

32 +1,

we obtain

(ωck, ωck) = h

((2` − 1)(2` + 1)

32 +1

), for k, l = 1, . . . , N − 1 with |k − l| = 1.

On the other hand, we get

(∇hωck,∇hω

ck) = 1

h

(2

2`

), for k = 1, . . . , N − 1, (2.25)

and

(∇hωck,∇hω

ck) = 1

h

(− 1

2`

), for k, l = 1, . . . , N − 1 with |k − l| = 1; (2.26)

otherwise, coarse-level scalar products are equal to 0.


2.3.2. The staggered gridHere,Wh is as before. Moreover,�h is the staggered uniform grid corresponding to the mesh sizeh

and the finite-difference vector spaceWh consists of restrictions to the line segment ]−12h, 1+1

2h[ of stepfunctions that are constants on the line segments [(l−1

2)h, (l+12)h[ for l = 1, . . . , n. The vector spaceWh

is spanned by the nodal basisωl, l = 1, . . . , n, which are equal to 1 on the line segment [(l−12)h, (l+1

2)h[ and vanish outside this line segment. Now, we consider the domain�=]0, 1[×]−1

2h, 1+12h[. The finite-

difference vector space isVh=Wh⊗Wh, it is spanned by the nodal basisωk,l=ωk⊗ ωl, k = 1, . . . , n−1,l = 1, . . . , n, which are equal to 1 on the plane segment [kh, (k+1)h[×[(l−1

2h, (l+12)h[ and vanish outside

this plane segment.The staggered finite-difference Laplace operator reads

1shvh =

2∑i=1

∇ih∇ihvh − 1

h2(vh(·, x1)+ vh(·, xn)), (2.27)

where we writexj=(j−12)h, j = 0, . . . , n+ 1 andx1, xn are the points near the boundary, because linear

extrapolation is used to get the exterior (fictitious) valuesvh(·, x0)) andvh(·, xn+1)), and its associatedstaggered finite-difference matrix reads

1shvh = I ⊗1h +1s

h ⊗ I, (2.28)

where1sh is the one-dimensional staggered finite-difference Laplace operator whose finite-difference

matrix is

1sh = 1

h2

−3 1

1 −2 1. . .

1 −2 1

1 −3

, (2.29)

and its associated finite-difference bilinear form reads

((uh, vh))sh =

2∑i=1

(∇ihuh,∇ihvh)− 1

h2

∫�h

uhvh1 ⊗ (ω1 + ωn)dx, (2.30)

where1 is the unity function.The terms of the hierarchical basis standing outside the domain aretruncated; in Fig. 3 (right bottom)

we present the truncated coarse-level hierarchical basis (staggered grid) when three levels are considered,truncation happens just then.

In addition, we need to compute here, in the vector spaceWh (staggered grid), the scalar product plus theevaluation near the boundary that comes from the definition of the associated staggered finite-differencebilinear form

(∇hˆωck,∇h

ˆωck)+ 1

h

(( ˆωck(x1))

2 + ( ˆωck(xn))2), (2.31)


for k,=1,N; then computations are the same as before except that

( ˆωcN, ˆωcN) = h

(1 + (2` − 1)(2`+1 − 1)

32 +1

), (2.32)

(∇hˆωcN,∇h

ˆωcN) = 1

h

(1 + 1

2`

). (2.33)

Moreover, we obtain

(∇hˆωc1,∇h

ˆωc1)+ 1

h

(( ˆωc1(x1))

2 + ( ˆωc1(xn))2)

= 1

h

(2

2`+(

1

2`

)2), (2.34)

(∇hˆωcN,∇h

ˆωcN)+ 1

h

(( ˆωcN(x1))

2 + ( ˆωcN(xn))2)

= 1

h

(2 + 1

2`

). (2.35)

2.4. The Poisson equation with Neumann boundary conditions

For the discretization of the Poisson equation with Neumann boundary conditions, we have to considera (staggered grid)×(staggered grid) as stated before. Here,�h is the staggered uniform grid correspondingto the mesh sizeh and the finite-difference vector spaceWh consists of restrictions to the line segment[0, 1] of step functions that are constants on the line segments [(k−1)h,kh[, for k = 1, . . . , n. The vectorspaceWh is spanned by the nodal basisωk, for k = 1, . . . , n, which are equal to 1 on the line segment[(k−1)h,kh[, for k = 1, . . . , n, and vanish outside this line segment. Now, we consider the domain�=[0, 1]×[0, 1]. The finite-difference vector space isVh=Wh ⊗ Wh, it is spanned by the nodal basisωk,l=ωk ⊗ ωl, k, l = 1, . . . , n. The terms of the nodal basis are now centered at the grid points of thestaggered grid.

The finite-difference Laplace operator reads

111nh =

2∑i=1

∇ih ∇ih, (2.36)

its staggered finite-difference matrix reads

111nh = I ⊗1n

h +1nh ⊗ I, (2.37)

where1nh is the one-dimensional staggered finite-difference Laplace operator (with Neumann boundary

conditions) whose finite-difference matrix is

1nh = 1

h2

−1 1

1 −2 1. . .

1 −2 1

1 −1

, (2.38)


its associated finite-difference bilinear form reads

((uh, vh))nh =

2∑i=1

(∇ihuh,∇ihvh). (2.39)

The terms of the hierarchical basis are also centered at the grid points of the staggered grid and thoseterms standing outside the domain are truncated.

The computations of the block diagonal part of finite-difference operators are the same as before exceptthat

(∇hωc1,∇hω

c1) = 1

h

(2

2`−(

1

2`

)2), (2.40)

(∇hωcN ,∇hω

cN) = 1

h

(1

2`

). (2.41)

To close, we emphasize the fact that the techniques above allow us to obtain explicitly the blockdiagonal part (coarse and fine levels) of finite-difference operators (with constant coefficients) and thento put into action block diagonal (scaling) preconditioning to solve the underlying incremental unknownslinear systems.

2.5. The block diagonal (scaling) preconditioning

Here, when the coarsest grid is not reduced to one point (nor is small enough) we will use left orien-tation block diagonal (scaling) preconditioning to solve the incremental unknowns linear systems. Thepreconditioning matrix for the incremental unknowns matrixSTAS, whereA is a nodal unknowns matrix,will be as follows:

(2.42)

whereL is the coarse-level block diagonal part, andLj , j = 1, . . . , `, is thejth-level (fine-level) diagonalpart, of the incremental unknowns matrixSTAS.

2.5.1. The generalized Poisson equation with Dirichlet boundary conditionsHerein, we consider on a (classical grid)×(staggered grid) the discrete equations

−ν111shuh + µuh + γhuh = fh. (2.43)

For the preconditioner, we neglect the variable functionγ h and we takeA=−ν111sh+µI and thenK in the

sense before, hereI is the identity matrix. The block diagonal part of the incremental unknowns matrixfor the staggered finite-difference Laplace operator is known from Section 2.3, and reads


L=

α β

β α β

β α β. . .

β α β

β αlast

N×N

⊗

γ δ

δ γ δ

δ γ δ. . .

δ γ δ

δ γ

(N−1)×(N−1)

+

γfirst δ

δ γ δ

δ γ δ. . .

δ γ δ

δ γlast

N×N

⊗

α β

β α β

β α β. . .

β α β

β α

(N−1)×(N−1)

,

where

α = 1 + (2` − 1)(2`+1 − 1)

32, αlast = 1 + (2` − 1)(2`+1 − 1)

32 +1, β = (2` − 1)(2` + 1)

32 +1,

γfirst = 2

2`+(

1

2`

)2

, γ = 2

2`, γlast = 2 + 1

2`, δ = − 1

2`.

The variable coefficients before can be considered into the preconditioning matrixK by observing thatSTDS ≈ DSTS for any diagonal matrixD. Now, we can use direct LU decomposition methods in thepreconditioner.

2.5.2. The Poisson equation with Neumann boundary conditionsHerein, we consider on a (staggered grid)×(staggered grid) the discrete equations

−111nhuh = fh. (2.44)

The matrix−111nh is symmetric positive semi-definite, we obtain that kernel(−111n

h) = R1and we concludethat the linear system before is solvable if and only if (fh, 1)=0, that is, if and only if

h2n∑

i,j=1

f

((i − 1

2

)h,

(j − 1

2

)h

)= 0.

This relation is the discrete analogous of the relation∫�f dxxx = 0, which is the condition for the general

existence of solutions of the continuous equations (see [12], p. 35). The block diagonal part of theincremental unknowns matrix for the staggered finite-difference Laplace operator is known from Section2.4, and reads


L=

α β

β α β

β α β. . .

β α β

β αlast

N×N

⊗

γfirst δ

δ γ δ

δ γ δ. . .

δ γ δ

δ γlast

N×N

+

γfirst δ

δ γ δ

δ γ δ. . .

δ γ δ

δ γlast

N×N

⊗

α β

β α β

β α β. . .

β α β

β αlast

N×N

,

where now

γfirst = 2

2`−(

1

2`

)2

, γlast = 1

2`.

The principal fact here is that the matrixL is a non-singular matrix. Now, we can use direct LUdecomposition methods in the preconditioner.

3. The incompressible Navier–Stokes equations

Now, we discuss the numerical solution of the 2D time-dependent incompressible Navier–Stokes equa-tions in primitive variables, with Dirichlet boundary conditions for the velocity field.

3.1. The continuous equation

Here, we consider in the domain�=]0, 1[×]0, 1[ the 2D time-dependent incompressible Navier–Stokesequations (non-dimensionalized equations) in primitive variables

∂uuu

∂t− ν 1uuu+ c(uuu,uuu)+ ∇p = fff in �, t > 0,

∇ · uuu = 0 in �,

uuu = ϕϕϕ on 0 = ∂�,

uuu(x,0) = uuu0(x) in �,

(3.1)

whereuuu is the velocity field,p the pressure field,ν>0 the kinematic viscosity, andfff is the external force,andc(uuu,vvv)=(uuu·∇)vvv represents the convection term and where we writec(uuu) instead ofc(uuu,uuu). Moreover,to make the formulation above well posed we assume{∇ · uuu0 = 0 in �

uuu0 = ϕϕϕ on 0 = ∂�(3.2)


and the compatibility condition∫0

ϕϕϕ · nnnd0 = 0, (3.3)

wherennn is the unit outward normal to0=∂�, which is the condition for the general existence of solutionsof the equation before (see [16], p. 31).

3.2. The temporal discretization

For the temporal discretization of the 2D time-dependent incompressible Navier–Stokes equations inprimitive variables, we consider the nonlinearθ -scheme introduced by Glowinski [11].

Algorithm. First, we setuuu0=uuu0. Then, for n≥0 and starting fromuuun , we solveuuun+θ − uuunθ 1t

− αν 1uuun+θ + ∇pn+θ = fff n+θ + βν 1uuun − c(uuun,uuun) in �,

∇ · uuun+θ = 0 in �,

uuun+θ = ϕϕϕn+θ on 0 = ∂�,uuun+1−θ − uuun+θ(1 − 2θ)1t

− βν 1uuun+1−θ + c(uuun+1−θ ) = fff n+1−θ + αν 1uuun+θ − ∇pn+θ in �,

uuun+1−θ = ϕϕϕn+1−θ on 0 = ∂�,uuun+1 − uuun+1−θ

θ 1t− αν 1uuun+1 + ∇pn+1 = fff n+1 + βν 1uuun+1−θ − c(uuun+1−θ ) in �,

∇ · uuun+1 = 0 in �,

uuun+1 = ϕϕϕn+1 on 0 = ∂�.

The definition of the parameters is given in Table 1, we refer to [15] for a theoretical discussion of thisissue.

Then, at each (temporal) iterative stage, we have to solve two generalized stationary Stokes equationsand one nonlinear elliptic equation.

3.3. The spatial discretization

Now, we discuss the numerical solution of the generalized stationary Stokes equations by using blockGaussian elimination and the numerical solution of the nonlinear elliptic equation by using uncenteredsecond-order finite-differences and incremental unknowns.

Table 1Definition of the parameters

Acronym Parameters

θ α β

θ 1−(1/√

2) (1−2θ )/(1−θ ) θ /(1−θ )


3.3.1. The generalized stationary Stokes equationNow, we describe the matrix framework associated to the (spatial) finite-difference discretization of the

generalized Stokes equation on a staggered grid [13,14]. Here, we consider in the domain�=]0, 1[×]0, 1[the scalar equations

−ν 1u+ γ u+ ∂p

∂x= f,

−ν 1u+ γ v + ∂p

∂y= g,

∂u

∂x+ ∂v

∂y= 0,

(3.4)

with Dirichlet boundary conditions:u|0 = ϕ, v|0 = ψ , where (u, v) is the velocity field,p is the pressurefield, (f,g) is the external force, andν>0,γ≥0 are given constants (Fig. 4).

Then, we set

Aγ = −ν(In ⊗1h +1sh ⊗ In−1)+ γ In ⊗ In−1, (3.5)

Aγ = −ν(In−1 ⊗1sh +1h ⊗ In)+ γ In−1 ⊗ In, (3.6)

B = In ⊗ δ(h/2), (3.7)

B = δ(h/2) ⊗ In, (3.8)

where

δ(h/2) = 1

h

−1 1−1 1

. . .

−1 1−1 1

. (3.9)

Fig. 4. The velocity/pressure staggered grid.


The finite-difference discretization of the generalized stationary Stokes equations, when lexicographicalordering of the unknowns is used, reads

Axxx = bbb, (3.10)

where

A =Aγ O B

O Aγ B

−BT −BTO

, xxx =

uvp

, bbb =

fgz

, (3.11)

andf andg stand for the finite-difference discretization of the RHS of the equations besides, near theboundary, terms coming from the finite-difference discretization of the Laplace operator and involv-ing the Dirichlet boundary conditions (boundary and extrapolated values), and wherez stands for thefinite-difference discretization of the zero function besides, near the boundary, terms coming from thefinite-difference discretization of the divergence operator and involving the Dirichlet boundary conditions(boundary values).

In particular, from a straightforward computation, we get

h2n∑

i,j=1

z

((i − 1

2

)h,

(j − 1

2

)h

)= h

n∑i=1

(ψ

((i − 1

2

)h,0

)− ψ

((i − 1

2

)h,1

))

+hn∑j=1

(ϕ

(0,

(j − 1

2

)h

)− ϕ

(1,

(j − 1

2

)h

)).

Here, the matrixA is indefinite. Since the bilinear forms associated to the linear operatorsAγ ,Aγ are

coercive and since kernel(B)∩kernel(B) = R1, we obtain that kernel(A) = R 0

01

.

Moreover, since range(A)=(kernel(A))⊥, we conclude that the linear system (3.10) is solvable if andonly if (z,1)=0, that is, if and only if

h

(n∑i=1

(−ψ

((i − 1

2

)h,0

)

+ψ((i − 1

2

)h,1

))+

n∑j=1

(−ϕ

(0,

(j − 1

2

)h

)+ ϕ

(1,

(j − 1

2

)h

)) = 0.

This relation is the discrete analogous of the relation∫0

ϕϕϕ · nnnd0 = 0,

whereϕϕϕ=(ϕ,ψ) andnnn is the unit outward normal to0=∂�, which is the condition for the generalexistence of solutions of the continuous equation (see [16], p. 31).


Fig. 5. Streamlines.

Now, we make the change of unknownsv = T v, whereT stands for the transpose permutation matrix;sinceT TAγT = Aγ , we can write the linear system before as

Aγ O B

O Aγ T TB

−BT −BTT O

u

v

p

=

f

T Tg

z

. (3.12)

To solve the linear system, we use the technique below.

3.3.1.1. Block Gaussian elimination.Here, we follow Atanga and Silvester [1] and we apply blockGaussian elimination to the linear system above; we get the linear system for the pressure field

Ap = bbb, (3.13)

where

A =[BTP B

TT P

] [Aγ OO Aγ

]−1 [PTBPTT TB

], (3.14)

bbb =[BTP B

TT P

] [Aγ OO Aγ

]−1 [PTf

PTT Tg

]+ z. (3.15)

The matrixA is symmetric positive semi-definite. Moreover, we use here the preconditioner introducedby Cahouet and Chabard [3], that is, instead of the linear system before we consider the equivalent linearsystem


Fig. 6. Pressure field.


Fig. 7. The generalized Stokes equation.

K−1Ap = K−1bbb, (3.16)

whereKp = γ ϕp + νp, with∫�p dx = 0, andϕp is the solution of the Poisson equation with Neumann

boundary conditions that follows:{−1ϕp = p in �,∂ϕp

∂nnn= 0 on 0 = ∂�,

∫�ϕp dx = 0.


Fig. 8. The nonlinear elliptic equation.

The linear system to compute the pressure field

K−1Ap = K−1bbb, (3.17)

is solved by the conjugate gradient method (see [10], p. 290). This is the leading preconditioned outer/inneriteration strategy. To put into action this iterative procedure requires the following:• To set up the right-hand side ofEq. (3.13). Here, it requires the numerical solution on a (classical

grid)×(staggered grid), of two generalized Poisson equations (the given functionγ is constant here)with Dirichlet boundary conditions.

• To initialize and to set up each iterative stage. Here, it requires one matrix-vector productAp and onematrix-vector productK−1p, in the former case, and one matrix-vector productK−1Ap, in the lattercase. In both cases, the following matrix-vector operations need to be realized. First, it requires onematrix-vector productAp, that is, it requires the numerical solution, on a (classical grid)×(staggeredgrid), of two generalized Poisson equations (the given functionγ is constant here) with Dirichlet bound-ary conditions. This is the preconditioned inner iteration strategy. Second, it requires one matrix-vectorproductK−1p, that is, it requires the numerical solution, on a (staggered grid)×(staggered grid), ofone Poisson equation with Neumann boundary conditions. This is the preconditioned outer iterationstrategy.The numerical solution, on a (classical grid)×(staggered grid), of the generalized Poisson equations

with Dirichlet boundary conditions is performed by the preconditioned conjugate gradient method [6],using incremental unknowns and the direct LU decomposition method in the preconditioner.

The numerical solution, on a (staggered grid)×(staggered grid), of the Poisson equation with Neumannboundary conditions is performed by the preconditioned conjugate gradient method (see [10], p. 290),using incremental unknowns and the direct LU decomposition method in the preconditioner.


Fig. 9. Streamlines.

The termination criteria are set as follows:1. Inner iterations:ε=h2.2. Outer iterations:ε=h.3. Leading iterations:ε=h.

Then, the velocity field is computed by

[u

v

]=[Aγ O

O Aγ

]−1[PT(f − Bp)

PTT T(g − Bp)

]. (3.18)

The conjugate gradient method (leading preconditioned outer/inner iteration strategy) reaches the accu-racyε=h in one iteration so that, in summary, to solve the generalized stationary Stokes equations, themethodology before requires the numerical solution of 10 Poisson equations: eight of these with Dirichletboundary conditions and two of these with Neumann boundary conditions.


Fig. 10. Pressure field.

3.3.2. The nonlinear elliptic equationNow, we describe the uncentered second-order finite-difference discretization and linearization of the

nonlinear elliptic equation. Two generalized Poisson equations (the given functionγ is variable here) withDirichlet boundary conditions on a (classical grid)×(staggered grid) need to be solved at each iterativestage.


Fig. 11. The generalized Stokes equation.

Here, we consider in the domain�=]0, 1[×]0, 1[, the equations

−ν 1u+ µu+ u∂u

∂x+ v

∂u

∂y= f,

−ν 1v + µv + u∂v

∂x+ v

∂v

∂y= g,

(3.19)


Fig. 12. The nonlinear elliptic equation.

with Dirichlet boundary conditions:u|0 = ϕ, v|0 = ψ , where (u, v) is the velocity field,p is the pressurefield, (f,g) is the external force, andν>0,µ≥0 are given constants.

For the discretization of the nonlinear terms of the incompressible Navier–Stokes equations, we useuncentered second-order finite-differences as follows [2]:

ρ(x)∂u

∂x(x) =

ρ(x)

3u(x)− 4u(x − h)+ u(x − 2h)

2h, if ρ(x) ≥ 0,

−ρ(x)3u(x)− 4u(x + h)+ u(x + 2h)

2h, if ρ(x) < 0.

(3.20)

Near the boundary, we use either centered second-order finite-differences or the formulae above as follows.For the classical grid (Dirichlet boundary conditions), we writexj=jh, j = −1, . . . , n+ 1, and we usenear the boundary the formulae

ρ(x1)∂u

∂x(x1) = ρ(x1)

3u(x1)︷︸︸︷−4u(x0)+ 3u(x−1)

2h,

whenρ(x1)≥0, and

ρ(xn−1)∂u

∂x(xn−1) = −ρ(xn−1)

3u(xn−1)︷︸︸︷−4u(xn)+ u(xn+1)

2h,

when ρ(xn−1)<0. Two ways to define the exterior (fictitious) valueu(x−1), u(xn+1) are consideredbelow.


Fig. 13. Kinetic energy.

3.3.2.1. Linear extrapolation. If linear extrapolation is used to get the exterior (fictitious) values, wehave

u(x−1) = 2u(x0)− u(x1), u(xn+1) = 2u(xn)− u(xn−1). (3.21)

Now, using these values and the formulae before we obtain

ρ(x1)∂u

∂x(x1) = ρ(x1)

u(x1)− u(x0)

h,

whenρ(x1)≥0, and

ρ(xn−1)∂u

∂x(xn−1) = −ρ(xn−1)

u(xn−1)− u(xn)

h,

whenρ(xn−1)<0. Here, the first-order derivatives are computed by first-order backward finite differencesin the former case and by first-order forward finite differences in the latter case. This choice introducesfirst-order accuracy.

3.3.2.2. Quadratic extrapolation.If quadratic extrapolation is used to get the exterior (fictitious) values,we have

u(x−1) = u(x2)− 3u(x1)+ 3u(x0), u(xn+1) = u(xn−2)− 3u(xn−1)+ 3u(xn). (3.22)


Fig. 14. Kinetic energy (detail).


ρ(x1)∂u

∂x(x1) = ρ(x1)

u(x2)− u(x0)

2h,

whenρ(x1)≥0, and

ρ(xn−1)∂u

∂x(xn−1) = ρ(xn−1)

u(xn)− u(xn−2)

2h,


Fig. 15. Phase diagram at the point( 18,

18).

whenρ(xn−1)<0. Here, the first-order derivatives are both computed by second-order centered finitedifferences. This choice maintains second-order accuracy and is used in the computations below.

For the staggered grid (Dirichlet boundary conditions), we writexj = (j − 1/2)h, j = 0, . . . , n+ 1,and we use near the boundary the formulae

ρ(x1)∂u

∂x(x1) = ρ(x1)

u(x2)− u(x0)

2h,

whenρ(x1)≥0, and

ρ(x2)∂u

∂x(x2) = ρ(x2)

3u(x2)︷︸︸︷−4u(x1)+ u(x0)

2h,

whenρ(x1)≥0, and

ρ(xn−1)∂u

∂x(xn−1) = −ρ(xn−1)

3u(xn−1)︷︸︸︷−4u(xn)+ u(xn+1)

2h,

whenρ(xn−1)<0, and

ρ(xn)∂u

∂x(xn) = ρ(xn)

u(xn+1)− u(xn−1)

2h,



18) (detail).

when ρ(xn)<0. Two ways to define the exterior (fictitious) valueu(x0), u(xn+1) are consideredbelow.

3.3.2.3. Linear extrapolation. If linear extrapolation is used to get the exterior (fictitious) values, wehave

u(x0) = 2u(0)− u(x1), u(xn+1) = 2u(1)− u(xn), (3.23)



18).


ρ(x1)∂u

∂x(x1) = ρ(x1)

3u(x1)︷︸︸︷+u(x2)− 2u(0)− 2u(x1)

2h,

whenρ(x1)≥0, and

ρ(x2)∂u

∂x(x2) = ρ(x2)

3u(x2)︷︸︸︷−5u(x1)+ 2u(0)

2h,

whenρ(x2)≥0, and

ρ(xn−1)∂u

∂x(xn−1) = −ρ(xn−1)

3u(xn−1)︷︸︸︷−5u(xn)+ 2u(1)

2h,


ρ(xn)∂u

∂x(xn) = ρ(xn)

3u(xn)︷︸︸︷+u(xn−1)− 2u(1)− 2u(xn)

2h,

whenρ(xn)<0. This choice reinforces the diagonal dominance of the underlying matrices and is used inthe computations below.



18) (detail).

3.3.2.4. Quadratic extrapolation.If quadratic extrapolation is used to get the exterior (fictitious) values,we have

u(x0) = u(x3)− 3u(x2)+ 3u(x1), u(xn+1) = u(xn−2)− 3u(xn−1)+ 3u(xn). (3.24)


ρ(x1)∂u

∂x(x1) = −ρ(x1)

3u(x1)− 4u(x2)+ u(x3)

2h,



78).

whenρ(x1)≥0, and

ρ(x2)∂u

∂x(x2) = ρ(x2)

u(x3)− u(x1)

2h,

whenρ(x2)≥0, and

ρ(xn−1)∂u

∂x(xn−1) = ρ(xn−1)

u(xn)− u(xn−2)

2h,


ρ(xn)∂u

∂x(xn) = ρ(xn)

3u(xn)− 4u(xn−1)+ u(xn−2)

2h,

whenρ(xn)<0. This choice reinforces the diagonal dominance of the underlying matrices.Now, to fall in with the formulae above we linearize the nonlinear terms of the Navier–Stokes equations;

furthermore, we keep into the LHS of the equation the term 3|ρ(x)|/2h (this reinforces the diagonaldominance of the underlying matrices) leading to two generalized Poisson equations (the given functionγ is variable here) with Dirichlet boundary conditions, other (overbrace) terms go into the RHS of theequation and so they are considered actual values (obtained in the previous iteration). Within the process



78) (detail).

beforev-velocity values are needed on theu-velocity discretization grid and vice versa; they are computedby second-order interpolation.

The numerical solution, on a (classical grid)×(staggered grid), of the generalized Poisson equations (thegiven functionγ is variable here) with Dirichlet boundary conditions is performed by the PBi-CGSTABmethod [18], using incremental unknowns and the direct LU decomposition method in the preconditioner.


Fig. 21. Kinetic energy.

The termination criteria are set as follows:1. PBi-CGSTAB iterations:ε=h2.2. Iterative process:ε=h2.

The number of iterations for the iterative process to reach the accuracyε=h2 is variable.

4. Computational experiments

First, we consider the 2D unregularized driven square cavity flow at Reynolds number Re=5000, that is,the 2D time-dependent incompressible Navier–Stokes equations,�=]0, 1[×]0, 1[, ν=1/Re, Re=5000,fff=0, with the boundary conditions

(u, v) ={(0,0), if x = 0,1 and y = 0,

(1,0), if y = 1 and 0≤ x ≤ 1.(4.1)

The finite-difference discretization before is set up with the parameters:N=32, i=3, so that we considera 127×128 staggered grid to discretize the velocity field and a 128×128 staggered grid to discretize thepressure field with three nested sequence of grids, the coarse grid associated with the velocity field beinga 31×32 staggered grid and the coarse grid associated with the pressure field being a 32×32 staggeredgrid. Here, the intermediate grid is used to do the pictures below.


Fig. 22. Kinetic energy (detail).

Then, we consider the 2D unregularized driven aspect ratio 2 cavity flow at Reynolds number Re=1000,that is, the 2D time-dependent incompressible Navier–Stokes equations,�=]0, 1[×]0, 2[, ν=1/Re,Re=1000,fff=0, with the boundary conditions

(u, v) ={(0,0), if x = 0,1 and y = 0,

(1,0), if y = 2 and 0≤ x ≤ 1.(4.2)



18).

The finite-difference discretization before is set up with the parameters:N=16 (coarsex-direction meshsize),M=2N (coarsey-direction mesh size), andi=4, so that we consider a 127×256 staggered grid todiscretize the velocity field and a 128×256 staggered grid to discretize the pressure field with four nestedsequence of grids, the coarse grid associated with the velocity field being a 15×32 staggered grid andthe coarse grid associated with the pressure field being a 16×32 staggered grid. The extension of theincremental unknowns framework to this case is straightforward. Here, the second intermediate grid isused to do the pictures below.

4.1. Long-term convergence behavior

Now, we consider the 2D unregularized driven square cavity flow at Reynolds number Re=5000.In Fig. 5, we display its streamlines and in Fig. 6 we display its pressure field. Next, we present theconvergence behavior of the iterative methods employed to solve the generalized Stokes equation and thenonlinear elliptic equation when starting from the velocity field and pressure field just shown.

As stated before, for the generalized Stokes equation, the PCG method is used to get the pressurefield, in Fig. 7 (top), we display its convergence behavior, that is, the leading iteration strategy. At eachiterative stage, we have to solve two generalized Poisson equations with Dirichlet boundary conditions(the given functionγ is constant here) for the velocity field: we use here the PCG method, in Fig. 7(left bottom), we display its convergence behavior, that is, the inner iteration strategy; and we have to



18) (detail).

solve a Poisson equation with Neumann boundary conditions, as a preconditioner, for the pressure field:we use here again the PCG method, in Fig. 7 (right bottom), we display its convergence behavior. Theconvergence behavior of the iterative methods before is fast and smooth (nearly optimal), only a fewiterations of the leading PCG method are needed to obtain maximal accuracy; for this to happen theappropriate termination criterion for the outer/inner iterations must be to reach maximal accuracy.

As stated before, to solve the nonlinear elliptic equation we linearize its nonlinear terms to get aniterative process; in Fig. 8 (left), we display the relative error (log 10) of two successive iterates. At each



18).

iterative stage, we have to solve two generalized Poisson equations with Dirichlet boundary conditions (thegiven functionγ is variable here) for the velocity field: we use the PBi-CGSTAB method, in Fig. 8 (right),we display its convergence behavior. The convergence behavior of the iterative process is acceptable andthe convergence behavior of the PBi-CGSTAB method is as happens before fast and smooth (nearlyoptimal). Here, we observe that the second component of the velocity is the hardest to compute.

Now, we consider the 2D unregularized driven aspect ratio 2 cavity flow at Reynolds number Re=1000.In Fig. 9, we display its streamlines and in Fig. 10, we display its pressure field.

Next, we present the convergence behavior of the iterative methods employed to solve the generalizedStokes equation and the nonlinear elliptic equation when starting from the velocity field and pressurefield just shown.

As stated before, for the generalized Stokes equation, the PCG method is used to get the pressurefield, in Fig. 11 (top), we display its convergence behavior, that is, the leading iteration strategy. At eachiterative stage, we have to solve two generalized Poisson equations with Dirichlet boundary conditions(the given functionγ is constant here) for the velocity field: we use here the PCG method, in Fig. 11(left bottom), we display its convergence behavior, that is, the inner iteration strategy; and we have tosolve a Poisson equation with Neumann boundary conditions, as a preconditioner, for the pressure field:we use here again the PCG method, in Fig. 11 (right bottom), we display its convergence behavior. Theconvergence behavior of the iterative methods before is fast and smooth (nearly optimal), only a fewiterations of the leading PCG method are needed to obtain maximal accuracy; for this to happen the



18) (detail).

appropriate termination criterion for the outer/inner iterations must be to reach maximal accuracy. Here,we observe that as the number of levels (nested sequence of grids) increases more iterations are neededto reach maximal accuracy.

As stated before, to solve the nonlinear elliptic equation we linearize its nonlinear terms to get aniterative process; in Fig. 12 (left), we display the relative error (log 10) of two successive iterates. At eachiterative stage, we have to solve two generalized Poisson equations with Dirichlet boundary conditions (the



78).

given function is variable here) for the velocity field: we use the PBi-CGSTAB method, in Fig. 12 (right),we display its convergence behavior. The convergence behavior of the iterative process is acceptableand the convergence behavior of the PBi-CGSTAB method is as happens before fast and smooth (nearlyoptimal). Here, we observe that the second component of the velocity is the hardest to compute.

In conclusion, the methodology proposed herein allows for an efficient (nearly optimal) solution al-gorithm of the generalized Stokes equation, and better solution algorithms should be used to solve thenonlinear elliptic equation; aside, the use of incremental unknowns appears throughout as an efficientpreconditioner.

4.2. Long-term dynamic behavior

Now, we report the long-term dynamic behavior of the flow whose kinetic energy is

K(uuu) = 12 ‖uuu‖`2

.

First, we consider the unregularized driven square cavity flow at Reynolds number Re=5000. In Fig. 13,we display the whole long-term dynamic behavior of the kinetic energy and in Fig. 14, we display thedetails. Here, we observe that for the long-term dynamic behavior the kinetic energy always keeps growingimperceptibly at the order of 10−3. Next, in Figs. 15–20 we display the phase diagram at the pointxxx,that is, in this point, we plot the values (u(xxx),v(xxx)), wherexxx=(1

8,18), (

78,

18), (

18,

78) successively; first,



78) (detail).

we display the whole long-term dynamic behavior of the flow and then the details following the detailsdisplay of the kinetic energy.

The small eddies of the flow never go to a steady state, instead they converge to a strange attractor,and always keep bringing imperceptibly kinetic energy to the flow; first, they wander around, then theyconverge to the strange attractor.


Next, we consider the unregularized driven aspect ratio 2 cavity flow at Reynolds number Re=1000. InFig. 21, we display the whole long-term dynamic behavior of the kinetic energy and in Fig. 22, we displaythe details. Here, we observe that for the long-term dynamic behavior, the kinetic energy always keepsgrowing imperceptibly at the order of 10−4. Next, in Figs. 23–28, we display the phase diagram at thepointxxx, that is, in this point, we plot the values (u(xxx),v(xxx)), wherexxx=(1

8,18), (

78,

18), (

18,

78) successively;

first, we display the whole long-term dynamic behavior of the flow and then the details following thedetails display of the kinetic energy.

The small eddies of the flow never go to a steady state, instead they converge to a strange attractor,and always keep bringing imperceptibly kinetic energy to the flow; first, they wander around, then theyconverge to the strange attractor.

A detailed comparison of quantities at various Reynolds numbers will be reported in a separate work.The calculations have been carried out in double precision arithmetic on the Silicon Graphics Octane.

blas andlapack from netlib have been used to do the numerical codes;TeXdraw, Maple V Release4, andgnuplot have been used to do the pictures.

5. Further remarks

First, to put into action the iterative methods to solve the incremental unknowns linear systems, we needin particular to compute the products of the matricesSandST with a vector; as stated before, this is oneintrinsic difficulty of the incremental unknowns strategy, which is overcome by using the fast algorithmsintroduced in [7]. Then, we only need to know the explicit block-matrix structure of the transfer matrixS in the two-level case; for instance, if a (staggered grid)×(staggered grid) is being used, this matrixreads

S =

IN ⊗ IN

Ds ⊗Q IN ⊗ I2N

IN ⊗Ds O IN ⊗ IN

, (5.1)

whereDs is the matrix of orderN×N

Ds =

1/21/2 1/2

· · ·1/21/2 1/2

,

andQ is a matrix of order 2N×N characterized by the fact that the matrix obtained fromQ by throwingaway its even rows is equal toDs , and the matrix obtained fromQ by throwing away its odd rows is equalto IN , hereIM is the identity matrix of orderM.

At last, it is possible to use throughout higher-order incremental unknowns [8] instead of second-orderincremental unknowns. Next, in Fig. 29, we display the hierarchical basis when two levels are considered(classical grid). From left to right and top to bottom, we present the coarse-level third-order hierarchi-cal basis, the coarse-level second-order hierarchical basis, the difference between them: coarse-levelthird-order hierarchical basis are small perturbation of coarse-level second-order hierarchical basis, and


Fig. 29. Higher-order hierarchical basis.

the truncated coarse-level third-order hierarchical basis. The finest-level hierarchical basis is a nodal basisfunction: the order of the hierarchical basis does not matter there. To truncate the hierarchical basis isallowed by the fact that the resulting functions are still linearly independent functions furnishing new(truncated) hierarchical basis functions. The nodal basis functions are extremely localized step functionsand perceive oscillations happening only at their place; whereas, second-order (higher-order) hierarchicalbasis spread throughout the domain and may perceive oscillations happening far away from their place,giving in consequence more accurate and stable (able to traverse turbulent zones) numerical algorithms [8].Furthermore, the incremental unknowns/operator splitting strategy presented herein can be used to solvethe three-dimensional time-dependent incompressible Navier–Stokes equations. Thoroughly, this workpresents some steps toward the long-term simulation of turbulent flows through incremental unknowns.

Acknowledgements

The author wishes to express his thanks to Prof. Roger Temam for inviting him to visit The Institutefor Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, USA, wherethe research reported herein grew up, and to Prof. Edriss S. Titi and the Department of Mathematics,University of California, Irvine, USA, for providing him with the supercomputing support that allowedthis work to be concluded. Moreover, this work was supported in part by the National Science FoundationGrant No. DMS-9706964.

Furthermore, this work was supported throughout by the Fondo Nacional de Desarrollo Cientıfico yTecnológico, Chile, through Proyecto Fondecyt 1940965 and Proyecto Fondecyt 1980656.

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incremental unknowns for solving the incompressible navier–stokes equations

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