Unconditional stability and error analysis of an Euler IMEX-SAV scheme for the micropolar Navier-Stokes equations

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Unconditional stability and error analysis of an Euler IMEX-SAV scheme

for the micropolar Navier-Stokes equations

Xiaodi Zhanga,b, Xiaonian Longc,∗

aHenan Academy of Big Data, Zhengzhou University, Zhengzhou 450052, China.

bSchool of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China.

cCollege of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450047, China.

Abstract

In this paper, we consider numerical approximations for solving the micropolar Navier-Stokes (MNS) equa-

tions, that couples the Navier-Stokes equations and the angular momentum equation together. By combin-

ing the scalar auxiliary variable (SAV) approach for the convective terms and some subtle implicit-explicit

(IMEX) treatments for the coupling terms, we propose a decoupled, linear and unconditionally energy stable

scheme for this system. We further derive rigorous error estimates for the velocity, pressure and angular

velocity in two dimensions without any condition on the time step. Numerical examples are presented to

verify the theoretical ﬁndings and show the performances of the scheme.

Keywords: micropolar Navier-Stokes equations; implicit-explicit schemes; energy stability; error estimates,

scalar auxiliary variable

1. Introduction

Let Ω be a convex polygonal/polyhedral with boundary Γ :=∂Ω in Rd, d = 2,3. In this paper, we

consider numerical approximation of the following MNS equations:

ut+u· ∇u−(ν+νr)∆u+∇p−2νr∇ × w=0in Ω ×J, (1a)

∇ · u= 0 in Ω ×J, (1b)

wt+u· ∇w−(ca+cd) ∆w−(c0+cd−ca)∇∇ · w+ 4νrw−2νr∇ × u=0in Ω ×J, (1c)

with boundary and initial conditions

u=0,w=0on Γ ×J,

u(x,0) = u0(x),w(x,0) = w0(x) in Ω,

where T > 0 is the ﬁnal time, J= (0, T ], (u, p, w) represent the the linear velocity, pressure and angular

velocity. All the material constants , ν, νr, ca, cdand c0are the kinematic viscosity which are assumed to

be constant, positive and satisfy c0+cd−ca>0. Moreover, νis the usual Newtonian viscosity, and νris

the microrotation viscosity. In order to simplify notation, we will set

ν0=ν+νr, c1=ca+cd, c2=c0+cd−ca.

Furthermore, there is a slight diﬀerence in two and three dimensions. Namely, if d= 2, we assume that the

velocity component in the z-direction is zero and the angular velocity is parallel to the z-axis [1]. That is,

u= (u1, u2,0) ,w= (0,0, w).

∗Corresponding author

Email addresses: zhangxiaodi@lsec.cc.ac.cn (Xiaodi Zhang), longxiaonian@lsec.cc.ac.cn (Xiaonian Long)

Preprint submitted to Elsevier October 7, 2022

arXiv:2210.02670v1 [math.NA] 6 Oct 2022

The MNS equations were ﬁrst introduced by Eringen [2] to describe the evolution of an incompressible

ﬂuid whose material particles possess both translational and rotational motions. The novelties of this system

are to reﬂect the eﬀects of microstructure on the ﬂuid via a microscopic dissipative evolution equations for the

angular momentum. Thus, this model is often used to describe the motion of blood, certain lubricants, liquid

crystals, ferroﬂuids, and some polymeric ﬂuids [1,3,4]. Given the signiﬁcant role it played in the microﬂuids,

numerical solving of the MNS system has drawn a considerable amount of attention. A penalty projection

method is proposed and optimal error estimates are proved in [5]. In [6], Nochetto et al. proposed and

analyzed ﬁrst-order and second-order semi-implicit fully-discrete schemes. These schemes decouple linear

velocity computation from angular velocity computation, while being energy-stable. Later, Salgado further

adopted the fractional time stepping technique to decouple the computation of pressure and velocity and

proved the rigorous error estimates in [7]. In these works, the nonlinear terms are treated either implicitly

or semi-implicitly so that one needs to solve a nonlinear system or a linear system with variable coeﬃcients

at each time step. It is desirable to treat the nonlinear term explicitly while maintaining energy stability.

With such treatment, the schemes only require the solution of linear system with constant coeﬃcients upon

discretization, which are very eﬃcient.

In recent years, SAV based schemes have attracted much attention due to their eﬃciency, ﬂexibility

and accuracy. The main idea is to introduce auxiliary variables to preserve the property of energy decay.

Several classes of energy stable numerical schemes have been developed for many dissipative systems, like

gradient ﬂows [8,9,10,11], NS equations [12,13,14], magnetohydrodynamic equations [15,16,17] and

Cahn-Hilliard-Navier-Stokes equations [18,19,20]. In particular, Shen et al. [21] proposed a new class of

eﬃcient IMEX BDFk(1 ≤k≤5) schemes combined with a SAV approach for general dissipative systems.

The distinct advantages are that their higher-order versions are also unconditionally energy stable and only

require solving one decoupled linear system with constant coeﬃcients at each time step.

For the MNS equations considered in this article, the energy structure is an inequality rather than an

equality like many dissipative systems. This fact makes the energy-equality based approaches [8,21] fail.

Thus, it is not trivial to construct eﬃcient SAV schemes for such systems. The aim of this work is to extend

the approach proposed in [13] to the MNS equations. Our main contributions are three-folds:

1. We propose a decoupled, linear and ﬁrst-order scheme for the MNS equations by combining the SAV

approach for the convective terms and some subtle IMEX treatments for the coupling terms. The

scheme only requires solving a sequence of diﬀerential equations with constant coeﬃcients at each

time step so it is very eﬃcient and easy to implement.

2. We establish rigorous unconditional energy stability and error analysis for the proposed scheme in two

dimensions.

3. We provide some numerical experiments to conﬁrm the predictions of the theory and demonstrate the

eﬃciency of the scheme.

Compared to the Navier-Stokes equations, the error analysis for the MNS equations is much more involved

due to the coupling terms. It is remarked that the present idea can be applies to the Boussinesq equations

and ferrohydrodynamics equations.

The rest of this paper is organized as follows. In Section 2, we introduce some notations and present the

energy estimate fo the MNS equations. In Section 3, we propose the Euler IMEX-SAV scheme and prove

the unconditional stability. In Section 4, we carry out a rigorous error analysis for the proposed scheme in

two dimensions. In Section 5, we present some numerical experiments. In Section 6, we conclude with a few

remarks.

2. Preliminaries

We start by introducing some notations and spaces. As usual, the inner product and norm in L2(Ω) are

denoted by (·,·) and k·k, respectively. Let Wm,p(Ω) stand for the standard Sobolev spaces equipped with

the standard Sobolev norms k·km,p. For p= 2, we write Hm(Ω) for Wm,2(Ω) and its corresponding norm

is k·km. For a given Sobolev space X, we write Lq(0, T ;X) for the Bochner space. Throughout the paper,

we use Cto denote generic positive constants independent of the discretization parameters, which may take

diﬀerent values at diﬀerent places.

For convenience, we introduce some notations for function spaces

X:=H1

0(Ω),V:={v∈X:∇ · v= 0}.

The following equation for the curl operator will be repeatedly used in our analysis

(∇ × w,u)=(w,∇ × u),∀u,w∈X.

Moreover, we recall that the following orthogonal decomposition of X,

k∇uk2=k∇ × uk2+k∇ · uk2,∀u∈X,

which implies

k∇ × uk ≤ k∇uk,k∇ · uk ≤ k∇uk,∀u∈X.(2)

To deal with the convection terms in (1a) and (1c), we deﬁne the following trilinear form,

b(u,v,w)=(u· ∇v,w).

It is easy to see that the trilinear form b(·,·,·) is a skew-symmetric with respect to its last two arguments,

b(u,v,w) = −b(u,w,v),∀u∈V,v,w∈X,(3)

and

b(u,v,v)=0,∀u∈V,v∈X.(4)

To end this section, we give the basic formal energy estimates for the model (1). By taking the L2-inner

product of (1a) with u, using the integration by parts and (1b), we get

dt kuk2+ν0k∇uk2+ (u· ∇u,u)=2νr(∇ × u,w).

Taking the L2-inner product of (1c) with w, and using the integration by parts, we have



dt kwk2+c1k∇wk2+(u· ∇w,w) + c2k∇ · wk2+ 4νrkwk2= 2νr(∇ × u,w).

Adding both ensuing equations and using (4), we obtain

dt 1

2kuk2+

2kwk2+ν0k∇uk2+c1k∇wk2+c2k∇ · wk2+ 4νrkwk2= 4νr(∇ × u,w).(5)

Invoking with the Cauchy-Schwarz inequality, Young inequality and (2), the right hand side of (5) can be

estimated as

4νr(∇ × u,w)≤4νrk∇ × uk kwk ≤ νrk∇ × uk2+ 4νrkwk2≤νrk∇uk2+ 4νrkwk2.(6)

Inserting (6) into (5), we have

dt 1

2kuk2+

2kwk2+νk∇uk2+c1k∇wk2+c2k∇ · wk2≤0.

Note that the energy dissipation law for the MNS equations is an inequality rather than an equality, which

is diﬀerent from the one for many other systems. The main reason is that the coupling terms can not be

canceled automatically in the process of deriving the energy estimates. We further ﬁnd that the nonlinear

terms do not contribute to the energy due to the skew-symmetric property in the above proof to obtain the

law of energy dissipation. The unique "zero-energy-contribution"property will be used to design eﬃcient

numerical schemes.

3. Numerical scheme

In this section, we propose a Euler IMEX scheme based on the SAV approach for the MNS equations

and show that it is unconditionally energy stable.

Inspired by the recent works [13], we introduce a scalar auxiliary variable

q(t):= exp(−t

T).(7)

Noticing that q(t)/exp −t

T= 1, we reformulate (1a) and (1c) into the equivalent forms as follows,

ut+q

exp(−t

T)u· ∇u−ν0∆u+∇p−2νr∇ × w=0,(8)

and

wt+q

exp(−t

T)u· ∇w−c1∆w−c2∇∇ · w+ 4νrw−2νr∇ × u=0.(9)

Diﬀerentiating (7) and using (4), we have

dt =−1

Tq+1

exp −t

T((u· ∇u,u) + (u· ∇w,w)) .(10)

The last term in this equation is added to balance the nonlinear terms in (8) and (9) in the discretized case.

Combining (8)-(10), we recast the original MNS equations as:

ut+q

exp(−t

T)u· ∇u−ν0∆u+∇p−2νr∇ × w=0,(11a)

∇ · u= 0,(11b)

wt+q

exp(−t

T)u· ∇w−c1∆w−c2∇∇ · w+ 4νrw−2νr∇ × u=0,(11c)

dt +1

Tq−1

exp −t

T((u· ∇u,u) + (u· ∇w,w)) = 0.(11d)

It is clear that provided with q(0) = 1, the exact solution of (11d) is given by (7). Therefore, the above

system is equivalent to the original system. Note that the SAV q(t) is related to the nonlinear part of the

free energy in the original SAV approach. However, the SAV q(t) in this paper is purely artiﬁcial, which

will allow us to construct unconditional energy stable schemes with fully explicit treatment of the nonlinear

terms.

Theorem 3.1. The expanded system (11)admits the following energy estimate,

dt 1

2kuk2+

2kwk2+1

2|q|2+νk∇uk2+c1k∇wk2+c2k∇ · wk2+1

T|q|2≤0.(12)

Proof. Taking the L2-inner product of (11a) with u, using the integration by parts and (11b), we get

dt kuk2+ν0k∇uk2+q

exp −t

T(u· ∇u,u)=2νr(∇ × u,w).(13)

Taking the L2-inner product of (11c) with w, and using the integration by parts, we have



dt kwk2+c1k∇wk2+q

exp(−t

T)(u· ∇w,w) + c2k∇ · wk2+ 4νrkwk2= 2νr(∇ × u,w).(14)

Multiplying (11d) with q, we obtain

dt |q|2+1

T|q|2−q

exp −t

T((u· ∇u,u) + (u· ∇w,w)) = 0.(15)

By combining (13)-(15), we derive

dt 1

2kuk2+

2kwk2+1

2|q|2+ν0k∇uk2+c1k∇wk2+c2k∇ · wk2+ 4νrkwk2+1

T|q|2= 4νr(∇ × u,w).

(16)

We ﬁnish the proof by using the estimate (6).

Remark 3.1.In this paper, the scalar auxiliary variable is only a time-dependent function q(t) = exp(−t/T )

not a energy-related function. With this treatment, the algebraic equation for the scalar auxiliary variable

is linear and unisolvent. Moreover, the ordinary diﬀerential equation for q(t) is linear and dissipative, which

makes our error estimates easier. In fact, the scalar auxiliary variable of this type admits a general form,

q(t) = Cq,0exp(−Cq,1t/T ) with Cq,06= 0 and Cq,1≥0. We refer to [22] for more details about this extension.

3.1. The SAV scheme

Let {tn=nτ :n= 0,1,· · · , N}, τ =T/N, be an equidistant partition of the time interval [0, T ].We

denote (·)nas the variable (·) at time step n. For any function v, deﬁne

δtvn+1 =vn+1 −vn

τ.

Combining the backward Euler method and some delicate implicit/explicit treatments for coupling terms,

we propose a ﬁrst-order SAV scheme for solving the system (11) as follows. Given the initial conditions u0,

w0and q0, compute un+1, pn+1,wn+1, qn+1,n= 0,1,· · · , N −1 by

δtun+1 +qn+1

exp(−tn+1

T)un· ∇un−ν0∆un+1 +∇pn+1 = 2νr∇ × wn,(17a)

divun+1 = 0,(17b)

δtwn+1 +qn+1

exp(−tn+1

T)un· ∇wn−c1∆wn+1 −c2∇∇ · wn+1 + 4νrwn+1 = 2νr∇ × un+1 (17c)

δtqn+1 +1

Tqn+1 −1

exp −tn+1

Tun· ∇un,un+1+un· ∇wn,wn+1= 0.(17d)

Before giving further stability estimates, we ﬁrst elaborate on how to implement the proposed scheme

eﬃciently. Since the auxiliary variable q(t) is a scalar number rather than a ﬁeld function, we can solve the

nonlocally coupled scheme in a decoupled fashion. Denote

Sn+1 :=qn+1 exp tn+1

T.(18)

We rewrite the ﬁrst three equations in (17) into

δtun+1 −ν0∆un+1 +∇pn+1 = 2νr∇ × wn−Sn+1un· ∇un,

divun+1 = 0,

δtwn+1 −c1∆wn+1 −c2∇∇ · wn+1 + 4νrwn+1 −2νr∇ × un+1 =−Sn+1(un· ∇)wn.

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UnconditionalstabilityanderroranalysisofanEulerIMEX-SAVschemeforthemicropolarNavier-StokesequationsXiaodiZhanga,b,XiaonianLongc,aHenanAcademyofBigData,ZhengzhouUniversity,Zhengzhou450052,China.bSchoolofMathematicsandStatistics,ZhengzhouUniversity,Zhengzhou450001,China.cCollegeofMathematicsandInform...

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Unconditional stability and error analysis of an Euler IMEX-SAV scheme for the micropolar Navier-Stokes equations

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