Long time stability of a classical efficient scheme for two dimensional Navier–Stokes equations

# Long time stability of a classical efficient scheme for two dimensional Navier–Stokes equations

S. Gottlieb Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747 (sgottlieb@umassd.edu)    F. Tone Department of Mathematics and Statistics, University of West Florida, Pensacola, FL 32514 (ftone@uwf.edu)    C. Wang Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747 (cwang1@umassd.edu)    X. Wang Department of Mathematics, Florida State University, Tallahassee, FL 32306–4510 (wxm@math.fsu.edu)    D. Wirosoetisno Department of Mathematical Sciences, Durham University, Durham  DH1 3LE, United Kingdom (djoko.wirosoetisno@durham.ac.uk)
###### Abstract

We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier–Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time step is sufficiently small in the case with periodic boundary conditions. The long time stability in the and norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the NSE itself at vanishing time step. Both semi-discrete in time and fully discrete schemes with either Galerkin Fourier spectral or collocation Fourier spectral methods are considered.

Key words. 2d Navier–Stokes equations, semi-implicit schemes, global attractor, invariant measures, spectral and collocation

AMS subject classifications. 65M12, 65M70, 76D06, 37L40

## 1 Introduction

The celebrated Navier–Stokes system for homogeneous incompressible Newtonian fluids in the vorticity–streamfunction formulation in two dimensions takes the form

 ∂ω∂t+∇⊥ψ⋅∇ω−νΔω =f, \hb@xt@.01(1.1) −Δψ =ω,

where denotes the vorticity, is the streamfunction, and represents (given) external forcing. For simplicity we will assume periodic boundary condition, i.e., the domain is a two dimensional torus , and that all functions have mean zero over the torus.

It is well-known that two dimensional incompressible flows could be extremely complicated with possible chaos and turbulent behavior [13, 11, 29, 5, 27, 38]. Although some of the features of this turbulent or chaotic behavior may be deduced via analytic means, it is widely believed that numerical methods are indispensable for obtaining a better understanding of these complicated phenomena. For analytic forcing, it is known that the solution is analytic in space (in fact Gevrey class regular [12]), and hence Fourier spectral is the obvious choice for spatial discretization. As for time discretization, one of the popular schemes [2, 30] is the following semi-implicit algorithm, which treats the viscous term implicitly and the nonlinear advection term explicitly

 ωn+1−ωnΔt+∇⊥ψn⋅∇ωn−νΔωn+1=fn. \hb@xt@.01(1.2)

ere is the time step, and are the approximations of the vorticity at the discrete times , respectively. The convergence of this scheme on any fixed time interval is standard and well-known [16, 17, 18, 19, 34]. There are many off-the-shelf efficient solvers of (LABEL:scheme), since it essentially reduces to a Poisson solver at each time step.

It is also well-known that the NSE (LABEL:NSE) is long time enstrophy stable in the sense that the enstrophy is bounded uniformly in time, and it possesses a global attractor and invariant measures [5, 11, 38]. In fact, it is the long time dynamics characterized by the global attractor and invariant measure that are central to the understanding of turbulence. Therefore a natural question is if numerical schemes such as (LABEL:scheme) can capture the long time dynamics of the NSE (LABEL:NSE) in the sense of convergence of global attractors and invariant measures. To say the least, we would require that the scheme inherit the long time stability of the NSE.

There is a long list of works on time discretization of the NSE and related dissipative systems that preserve the dissipativity in various forms [31, 32, 9, 10, 21, 33, 22, 39, 40]. It has also been discovered recently that if the dissipativity of a dissipative system is preserved appropriately, then the numerical scheme would be able to capture the long time statistical property of the underlying dissipative system asymptotically, in the sense that the invariant measures of the scheme would converge to those of the continuous-in-time system [44]. The main purpose of this manuscript is to show that the classical scheme (LABEL:scheme) is long time stable in and , and that the global attractor as well as the invariant measures of the scheme, converge to those of the NSE at vanishing time step.

## 2 Long time behavior of the semi-discrete scheme

We first recall the well-known periodic Sobolev spaces on with average zero:

 ˙Hmper(Ω):={ϕ∈Hm(Ω)∣∣∣∫Ωϕ=0 and ϕ is 2π-periodic in each direction}. \hb@xt@.01(2.1)

is defined as the dual space of with the duality induced by the inner product. The adoption of is well-known [5, 37] since this space is invariant under the Navier–Stokes dynamics (LABEL:NSE), provided that the initial data and the forcing term belong to the same space.

### 2.1 Long time stability of the scheme

We first prove that the scheme (LABEL:scheme) is stable for all time.

###### Lemma 2.1

The scheme (LABEL:scheme) forms a dynamical system on .

Proof. It is easy to see that for , we have . Hence for all . Therefore, the classical scheme (LABEL:scheme), which can be viewed as a Poisson type problem , possesses a unique solution in (in fact in ) and the solution depends continuously on the data. Therefore it defines a (discrete) semi-group on .

Now we derive the long time stability of the scheme (LABEL:scheme) both in and in . Our proof relies on a Wente type estimate on the nonlinear term (see Appendix LABEL:s:wente), which may be of independent interest.

We first show that the scheme (LABEL:scheme) is uniformly bounded in , provided that the time step is sufficiently small. To this end, we take the scalar product of (LABEL:scheme) with and using the relation

 2(φ−ψ,φ)L2=∥φ∥22−∥ψ∥22+∥φ−ψ∥22, \hb@xt@.01(2.2)

where denotes the norm, we obtain

 ∥ωn+1∥22−∥ωn∥22+∥ωn+1−ωn∥22+2νΔt∥ωn+1∥2H1 +2Δtb(ψn,ωn,ωn+1) \hb@xt@.01(2.3) =2Δt(fn,ωn+1)L2

where

 b(ψ,ω,~ω):=(∇⊥ψ⋅∇ω,~ω)L2=−b(ψ,~ω,ω), \hb@xt@.01(2.4)

the last equality obtaining upon integration by parts. Using the Cauchy–Schwarz and the Poincaré inequalities, we majorize the right-hand side of (LABEL:1) by

 2Δt∥fn∥2∥ωn+1∥2≤2c0Δt∥fn∥2∥ωn+1∥H1≤νΔt∥ωn+1∥2H1+c20νΔt∥fn∥22. \hb@xt@.01(2.5)

Using the Wente type estimate (LABEL:q:wente2), we bound the nonlinear term as

 2Δtb(ψn,ωn,ωn+1) =2Δtb(ψn,ωn+1,ωn+1−ωn) \hb@xt@.01(2.6) ≤2CwΔt∥∇⊥ψn∥H1∥ωn+1∥H1∥ωn−ωn+1∥2 ≤12∥ωn+1−ωn∥22+2C2wΔt2∥∇⊥ψn∥2H1∥ωn+1∥2H1 ≤12∥ωn+1−ωn∥22+2C2wΔt2∥ωn∥22∥ωn+1∥2H1.

Relations (LABEL:1)–(LABEL:3) imply

 ∥ωn+1∥22−∥ωn∥22+12∥ωn+1−ωn∥22+(ν−2C2wΔt∥ωn∥22)Δt ∥ωn+1∥2H1 \hb@xt@.01(2.7) ≤c20νΔt∥fn∥22.

Here and in what follows, and denote generic constants whose value may not be the same each time they appear. Numbered constants, e.g., , have fixed values.

We are now able to prove the following:

###### Lemma 2.2

Let and let be the solution of the numerical scheme LABEL:scheme. Also, let and set . Then there exists such that if

 Δt≤ν4C2wM20, \hb@xt@.01(2.8)

then

 ∥ωn∥2≤M0,∀n≥0, \hb@xt@.01(2.9)
 ∥ωn∥22≤(1+ν2c20Δt)−n∥ω0∥22+2c40ν2∥f∥2∞⎡⎣1−(1+ν2c20Δt)−n⎤⎦,∀n≥0, \hb@xt@.01(2.10)

and

 ν2Δtm∑n=i∥ωn∥2H1≤∥ωi−1∥22+c20ν∥f∥2∞(m−i+1)Δt,∀i=1,⋯,m. \hb@xt@.01(2.11)

Proof. We will first prove (LABEL:q:bdv) by induction on . It is clear that (LABEL:q:bdv) holds for . Assuming that (LABEL:q:bdv) holds for , we then have that (LABEL:q:bdinh) holds for , where

 M20=M20(∥ω0∥2,ν,∥f∥∞)=∥ω0∥22+2c40ν2∥f∥2∞. \hb@xt@.01(2.12)

Then (LABEL:4) and (LABEL:5a) yield

 ∥ωn+1∥22−∥ωn∥22+12∥ωn+1−ωn∥22+ν2Δt∥ωn+1∥2H1≤c20νΔt∥fn∥22 \hb@xt@.01(2.13)

for all . Using again the Poincaré inequality, the above inequality implies

 ∥ωn+1∥22≤1α∥ωn∥22+c20ανΔt∥fn∥22, \hb@xt@.01(2.14)

where

 α=1+ν2c20Δt. \hb@xt@.01(2.15)

Using recursively (LABEL:8), we find

 ∥ωm+1∥22 ≤1αm+1∥ω0∥22+c20νΔtm+1∑i=11αi∥fm+1−i∥22 \hb@xt@.01(2.16) ≤(1+ν2c20Δt)−m−1∥ω0∥22+2c40ν2∥f∥2∞[1−(1+ν2c20Δt)−m−1],

and thus (LABEL:q:bdv) holds for . We therefore have that (LABEL:q:bdv) holds for and (LABEL:q:bdinh) follows right away.

Now adding inequalities (LABEL:7) with from to and dropping some positive terms, we find

 ν2Δtm∑n=i∥ωn+1∥2H1 ≤∥ωi∥22+c20νΔtm∑n=i∥fn∥22 \hb@xt@.01(2.17) ≤∥ωi∥22+c20ν∥f∥2∞(m−i+1)Δt,

which is exactly (LABEL:6). This completes the proof of Lemma LABEL:t:bdh.

###### Corollary 1

If

 0<Δt≤min{ν4C2wM20,2c20ν}=:k0, \hb@xt@.01(2.18)

then

 ∥ωn∥22≤2ρ20,∀nΔt≥T0(∥ω0∥2,∥f∥∞):=8c20νln(∥ω0∥2ρ0), \hb@xt@.01(2.19)

where .

Proof. From the bound (LABEL:q:bdv) on , we infer that

 ∥ωn∥22≤(1+ν2c20Δt)−n∥ω0∥22+ρ20,

and using assumption (LABEL:q:k0) on and the fact that if , we obtain

For , the last inequality implies the conclusion (LABEL:q:tabs) of the Corollary.

Now we show that the norm is also bounded uniformly in time under the same kind of constraint as for the estimate. To this end, we first prove that is bounded for , for some , and then, with the aid of a version of the discrete uniform Gronwall lemma, we show that is bounded for all .

More precisely, we have the following:

###### Lemma 2.3

Let and let be the solution of the numerical scheme (LABEL:scheme). Also, let , with as in Corollary LABEL:C1, and let be arbitrarily fixed. Then, for ,

 ∥ωn∥2H1≤4(2C2w/ν)M20(T0+r)(∥ω0∥2H1+1C2wM20∥f∥2∞) \hb@xt@.01(2.20)

where , with and that in Corollary LABEL:C1.

Proof. Taking the scalar product of (LABEL:scheme) with , we obtain

 ∥ωn+1∥2H1−∥ωn∥2H1 +∥ωn+1−ωn∥2H1+2νΔt∥Δωn+1∥22 \hb@xt@.01(2.21) −2Δtb(ψn,ωn,Δωn+1)=−2Δt(fn,Δωn+1)L2.

We bound the right-hand side of (LABEL:13) using the Cauchy–Schwarz inequality,

 −2Δt(fn,Δωn+1)L2≤2Δt∥fn∥2∥Δωn+1∥2≤ν2Δt∥Δωn+1∥22+2νΔt∥fn∥22. \hb@xt@.01(2.22)

Using the Wente type estimate (LABEL:q:wente2), we bound the nonlinear term as

 2Δtb(ψn,ωn, Δωn+1)=2Δtb(ψn,ωn−ωn+1,Δωn+1) to80.0pt+2Δtb(ψn,ωn+1,Δωn+1) ≤2CwΔt∥∇⊥ψn∥H1∥ωn+1−ωn∥H1∥Δωn+1∥2 +2CwΔt∥∇⊥ψn∥H1∥ωn+1∥H1∥Δωn+1∥2 ≤12∥ωn+1−ωn∥2H1+2C2wΔt2∥∇⊥ψn∥2H1∥Δωn+1∥22 +ν2Δt∥Δωn+1∥22+2C2wνΔt∥∇⊥ψn∥2H1∥ωn+1∥2H1 ≤12∥ωn+1−ωn∥2H1+2C2wΔt2∥ωn∥22∥Δωn+1∥22 +ν2Δt∥Δωn+1∥22+2C2wνΔt∥ωn∥22∥ωn+1∥2H1. \hb@xt@.01(2.23)

Relations (LABEL:13)–(LABEL:15) imply

 (1−2C2wν∥ωn∥22Δt)∥ωn+1∥2H1 −∥ωn∥2H1+12∥ωn+1−ωn∥2H1 \hb@xt@.01(2.24) +(ν−2C2wΔtM20)Δt∥Δωn+1∥22≤2νΔt∥fn∥22,

from which we find

 ∥ωn+1∥2H1≤1α∥ωn∥2H1+2ανΔt∥f∥2∞, \hb@xt@.01(2.25)

where

 α=1−2C2wνΔtM20>0. \hb@xt@.01(2.26)

Using recursively (LABEL:17), we find

 ∥ωn+1∥2H1 ≤ 1αn+1∥ω0∥2H1+2νΔt∥f∥2∞n+1∑i=11αi \hb@xt@.01(2.27) ≤ (1−2C2wνΔtM20)−1−n[∥ω0∥2H1+1C2wM20∥f∥2∞].

Since by hypothesis (LABEL:q:k0) and

 1−x≥4−x if x∈(0,1/2),

relation (LABEL:19) gives conclusion (LABEL:12) of Lemma LABEL:finite. Thus, the lemma is proved.

In order to obtain a uniform bound valid for , we need the following discrete uniform Gronwall lemma, which has been proved in [39] and we repeat here for convenience.

###### Lemma 2.4

We are given , positive integers , and positive sequences , , such that

 Δtηn+1<12,∀n≥n0, \hb@xt@.01(2.28) (1−Δtηn+1)ξn+1≤ξn+Δtζn+1,∀n≥n0. \hb@xt@.01(2.29)

Assume also that

 Δtn2+n1+1∑n=n2ηn≤a1, \hb@xt@.01(2.30) Δtn2+n1+1∑n=n2ζn≤a2, Δtn2+n1+1∑n=n2ξn≤a3,

for all . We then have,

 ξn+1≤(a3Δtn1+a2)e4a1,∀n>n0+n1. \hb@xt@.01(2.31)

Proof. Let and be such that . Using recursively (LABEL:gronseq2), we derive

 ξm1+n1+1 ≤m1+n1+1∏n=m211−Δtηnξm2−1+Δtm1+n1+1∑n=m2ζnm1+n1+1∏j=n11−Δtηj. \hb@xt@.01(2.32)

Using the fact that , and recalling assumptions , and the first and second conditions in , we obtain

 ξm1+n1+1≤(ξm2−1+a2)e4a1.

Multiplying this inequality by , summing from to and using the third assumption gives conclusion (LABEL:ugronest) of the lemma.

We are now able to derive a uniform bound for valid for sufficiently large . More precisely, we have the following:

###### Lemma 2.5

Let and let be the solution of the numerical scheme (LABEL:scheme). Also, let , with as in Corollary LABEL:C1. Then there exist constants and such that

 ∥ωn∥H1≤M1,∀n≥N. \hb@xt@.01(2.33)

Proof. Let be as in the hypothesis, be as in Corollary LABEL:C1, as in Lemma LABEL:finite and set . We will apply Lemma LABEL:dugronwall to (LABEL:16), with , , , , . For , we compute (taking into account that, by (LABEL:q:tabs), , for ):

 Δt n2+n1+1∑n=n2ηn=Δtn2+n1+1∑n=n22C2wν∥ωn−1∥22≤4C2wνρ20r:=a1, \hb@xt@.01(2.34) Δt n2+n1+1∑n=n2ζn=Δtn2+n1+1∑n=n22ν∥f∥2∞≤2ν∥f∥2∞r:=a2, \hb@xt@.01(2.35) Δt n2+n1+1∑n=n2ξn=Δtn2+n1+1∑n=n2∥ωn∥2H1(by (???)) \hb@xt@.01(2.38) ≤2ν(∥ωn2−1∥22+c20ν∥f∥2∞(n1+2)Δt)(by (???)) ≤2ν[2ρ20+c20ν∥f∥2∞r]=:a3.

By (LABEL:ugronest), we obtain

 ∥ωn∥2H1 ≤[4ν(2ρ20r+1νλ1∥f∥2∞)+2ν∥f∥2∞r]exp(16C2wνρ20r) \hb@xt@.01(2.39) =:M21(ν,∥f∥∞),∀n≥N0+Nr. \hb@xt@.01(2.40)

Taking , we obtain conclusion (LABEL:20) of Lemma LABEL:5.

We can summarize the above results in the following:

###### Theorem 2.6

The classical scheme (LABEL:scheme) defines a discrete dynamical system on that is long time stable in both and norms. More precisely, for any , there exist constants , , and such that

 ∥ωn∥2 ≤ M0,∀n≥0,∀k∈(0,k0), \hb@xt@.01(2.41) ∥ωn∥H1 ≤ M1,∀n≥N,∀k∈(0,k0). \hb@xt@.01(2.42)

### 2.2 Convergence of long time statistics

Here we show that, with time-independent forcing, the long time statistical properties as well as the global attractors of the scheme (LABEL:scheme) converge to that of the Navier–Stokes sytem (LABEL:NSE) at vanishing time step size. This is a straightforward application of the abstract convergence result (Prop. LABEL:abs_conv_stat) in Appendix LABEL:s:conv, which itself is a slight modification of the results presented in [44].

###### Theorem 2.7

Let . The global attractor and the long time statistical properties of the classical scheme (LABEL:scheme) converge to that of the Navier–Stokes system (LABEL:NSE) at vanishing time step.

Proof. We use the abstract convergence result Prop. LABEL:abs_conv_stat, taking , i.e. a ball in centered at the origin with radius . (The size of the ball needs to be adjusted depending on the absorbing property of the scheme.)

The uniform continuity (H5) of the Navier–Stokes system (LABEL:NSE) is a classical result [5, 37]. The uniform dissipativity (H3) of the scheme (LABEL:scheme) for small enough time step with the choice of the phase space follows from Theorem LABEL:stability. The uniform convergence on finite time interval (H4) is proved in Lemma LABEL:t:error below.

###### Lemma 2.8

Let be the solution of the continuous system (LABEL:NSE) with and that of (LABEL:scheme) with . Assume that is sufficiently smooth so that

 MV:=supω∈A(∥∂ttω∥2H−1+∥ω∥2L2∥∂tω∥2L2)<∞, \hb@xt@.01(2.43)

and that Theorem LABEL:stability holds. Then for one has

 ∥ωn−ω(nΔt)∥22≤ΔtC(M0,MV;ν) \hb@xt@.01(2.44)

for all .

Proof. We follow the approach in [28, §17] and take . For notational convenience, we write and . Using the identity

 ∫(n+1)ΔtnΔt(t−nΔt)∂ttω(t)dt=Δt∂tω∣∣(n+1)Δt−ωn+1+ωn, \hb@xt@.01(2.45)

we have

 ωn+1−ωnΔt+∇⊥ψn⋅∇ωn−νΔωn+1=f+Rn+1. \hb@xt@.01(2.46)

Here and the local truncation error is

 −Rn+1:=∇⊥δψn+1⋅∇ωn−∇⊥ψn+1⋅∇δωn+1+1Δt∫(n+1)ΔtnΔt(t−nΔt)∂ttω(t)dt \hb@xt@.01(2.47)

with

 δωn+1:=ωn+1−ωn=∫(n+1)ΔtnΔt∂tω(t)dtand−Δδψn+1:=δωn+1. \hb@xt@.01(2.48)

We now consider the error , which satisfies

 en+1−enΔt−νΔen+1 =∇⊥ψn⋅∇ωn−∇⊥ψn⋅∇ωn+Rn+1 \hb@xt@.01(2.49) =−∇⊥ψn⋅∇en−∇⊥ϕn⋅∇ωn+Rn+1

with and . Multiplying by , we find

 ∥en+1∥22 −∥en∥22+∥en+1−en∥22+2νΔt∥en+1∥2H1 \hb@xt@.01(2.50) +2Δtb(ψn,en+1,en+1−en)+2Δtb(ϕn,ωn,en+1) =2Δt(Rn+1,en+1).

Bounding the nonlinear terms as

 2Δt(∇⊥ψn⋅∇en+1,en+1−en) ≤∥en+1−en∥22+Δt2∥∇⊥ψn⋅∇en+1∥22 \hb@xt@.01(2.51) ≤∥en+1−en∥22+C2wΔt2∥ωn∥22∥∇en+1∥22

where (LABEL:q:wente2) has been used for the second inequality, and

 2Δt(∇⊥ϕn⋅∇ωn,en+1) ≤2Δt∥∇⊥ϕn⋅∇en+1∥2∥ωn∥2 \hb@xt@.01(2.52) ≤2CwΔt∥en∥2∥en+1∥H1∥ωn∥2 ≤νΔt∥en+1∥2H1+C2w