The Euler implicit/explicit scheme for the Boussinesq equations

R E S E A R C H

Open Access

The Euler implicit/explicit scheme for the

Boussinesq equations

Tong Zhang

_{, Jiaojiao Jin}

_{and Shunwei Xu}

*_{Correspondence:} [email protected] 1_{School of Mathematics and} Information Science, Henan Polytechnic University, Jiaozuo, 454003, P.R. China

2_{Departamento de Matemática,} Centro Politécnico, Universidade Federal do Paraná, Curitiba, 81531-990, Brazil

Abstract

In this article, we consider the stability and convergence of the first-order implicit/explicit scheme for the Boussinesq equations. The finite element spatial discretization is based on a MINI element for the velocity and pressure, which satisfies the discrete inf-sup condition, and a linear polynomial for the temperature. The temporal terms are treated by the Euler implicit/explicit scheme, which is implicit for the linear terms and explicit for the nonlinear terms. The advantage of using the implicit/explicit scheme is that a linear system with constant coefficient matrix is obtained, which can save a lot of computational cost. The main novelties of this work are the stability of numerical solutions under the conditionsk1

with two positive constantsk1,k2and the optimal error estimates of numerical solutions in different norms. Finally, some numerical results are provided to verify the performances of the Euler implicit/explicit scheme.

MSC: 65N15; 65N30; 76D07

Keywords: Boussinesq equations; Euler implicit/explicit scheme; stability; error estimates

1 Introduction

In this paper, we consider the following Boussinesq equations inRwith coupled equa-tions governing the viscous incompressible flow and heat transfer:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ut–νu+ (u· ∇)u+∇p= –jθ+f in×(,T],

divu=  in×(,T],

θt–λνθ+u· ∇θ=g in×(,T],

u= , θ=  on∂×(,T],

u(x, ) =u, θ(x, ) =θ, on× {},

(.)

whereis a bounded convex polygonal domain,u= (u,u)Tis the fluid velocity,pis the

pressure,θ is the temperature,ν>  is the viscosity,λ=Pr–_{,Pr is the Prandtl number,}

j= (, )T_{is the vector of gravitational acceleration,T>  is the final time, andf andgare} the forcing functions.

The Boussinesq equations (.) are an important dissipative nonlinear model in the at-mospheric dynamics (see []). This system not only contains the velocity and pressure but also includes the temperature filed, and therefore finding a numerical solution of problem

(.) becomes a difficult task. There are numerous works devoted to the development of ef-ficient schemes for this model, for example, the standard Galerkin finite element method (FEM) [], the projection-based stabilized mixed FEM [], the precondition techniques [, ], the two-level algorithms [–], and the references therein. In the literature men-tioned, the suitable stability condition is a key issue for these developed schemes. Gener-ally speaking, we can adopt the fully implicit, semi-implicit, and implicit/explicit schemes to treat the nonlinear equations. The fully implicit schemes are unconditionally stable. However, we have to solve a system of nonlinear equations at each time level. Explicit schemes are much easier in computation, but they suffer from a sever restriction of time step by the stability requirement. A popular approach is based on an implicit scheme for the linear terms and a semi-implicit scheme or an explicit scheme for the nonlinear terms. The semi-implicit scheme for the nonlinear terms results in a linear system with variable coefficient matrix of time, and an explicit treatment for the nonlinear term gives a constant matrix. Many researchers have studied the stability and convergence of these schemes for the Navier-Stokes equations [–]. The main results are summarized by He in his recent works [, ].

In this paper, we consider a first-order scheme for the Boussinesq equations. In view of the advantages of the explicit scheme for the nonlinear terms, we adopt the im-plicit/explicit scheme for the Boussinesq equations (.). Under the conditionskt≤

andkt≤ with two positive constantsk,k, we present some new stabilities and

estab-lish the corresponding convergence for velocity, pressure, and temperature by the Taylor expansion and other skills. This report can be considered as an extension of the existing results [, , , ] from the Navier-Stokes equations to the more complex Boussinesq equations. Our main results can be stated as follows:

u–un_h+θ–θ_hn

≤C

t+h, (.)

_∇

u–un_h+∇θ–θ_hn

+p–p

n

h≤C(t+h), (.)

whereC>  is a constant depending on the parametersf∞,b∞,u,θ,,ν, andλ, but

in-dependent ofhandt, wheref∞=sup_t≥{|f|+|ft|},ft=df_dt,g∞=sup_t≥{|g|+|gt|},gt=dg_dt. Here and thereafter,Cdenotes a general positive constant, which may take different values at different places. From (.)-(.) we can see that our results are optimal for both space lengthhand time stept.

The outline of this article is as follows. Some basic notation and results for problem (.) are recalled in Section . Section  is devoted to develop the Euler implicit/explicit scheme. Stabilities and optimal error estimates are established in Sections  and , respectively. Finally, a series of numerical results are provided to verify the efficiency and effectiveness of the Euler implicit/explicit scheme.

2 Preliminaries

In this section, we construct a variable formulation for problem (.) and recall some clas-sical results, which will be frequently used in this paper. To fix the idea, we set

X=H_(), W=H_(), Y=L(), Z=L(),

M=L_() = ϕ∈L();

ϕdx= 

Throughout this paper, we adopt (·,·) and · to denote the inner product and norm onL_{() orL}₍₎_{. The spacesH}

() andXare equipped with the usual scalar product

and norm∇u_= (∇u,∇u). Define the continuous bilinear formsa(·,·),d(·,·), anda(·,·) by

a(u,v) =ν(∇u,∇v), d(v,q) = (q,divv), a(θ,ψ) =λν(∇θ,∇ψ)

for allu,v∈X,q∈M, andθ,ψ∈W.

Next, we introduce the closed subsetVofXgiven by

V=v∈X,d(v,q) = ,∀q∈M={v∈X,∇ ·v=  in}

and denote byHthe closed subset ofY(see [, ]) given by

H={v∈Y,∇ ·v= ,v·n|∂= }.

We denote the Stokes operator byA=P, wherePis theL_{-orthogonal projection ofY}

ontoHor ofZontoW. Assume thatis such that the domain ofAis given by (see [, , , ])

D(A) =H()∩X or E(A) =H()∩W. (.)

For instance, (.) holds ifis of classC_{or ifis a convex plane polygonal domain.}

Moreover, we can define the trilinear forms for allu,v,w∈Xandθ,ψ∈Was follows:

b(u,v,w) =(u· ∇)v,w+ 

(divu)v,w= 

(u· ∇)v,w– 

(u· ∇)w,v,

b(u,θ,ψ) =(u· ∇)θ,ψ+ 

(divu)θ,ψ=  

(u· ∇)θ,ψ– 

(u· ∇)θ,ψ.

With these notations, for givenf ∈L∞(R+;Y) withu∈D(A)∩Vandg∈L∞(R+;Z) with

θ∈E(A), the variational formulation of (.) reads as follows: For all (v,q,ψ)∈X×M×

W, find (u,p,θ)∈X×M×Wwith

u∈L∞R+;X∩L(,T;V), ut∈L,T;V, θt∈L

,T;W,

p∈L(,T;M), θ∈L∞R+;W∩L(,T;Z), ∀T> ,

such that

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(ut,v) +a(u,v) –d(v,p) +b(u,u,v) = (f,v) – (jθ,v), d(u,q) = ,

(θt,ψ) +a(θ,ψ) +b(u,θ,ψ) = (g,ψ), u(x, ) =u, θ(x, ) =θ.

(.)

Assuming thatf ∈L_{(,T;X),g∈L}_{(,T;W), andu}

∈V,θ∈W, problem (.) has

L_{(,T;W). The uniqueness and regularity of the solution (u,p,θ) can also be proved by}

strengthening the assumptions on the data; see [] for details.

We recall the following discrete Gronwall lemma, which can be found in [, ].

Lemma . Let Cand ak,bk,ck,dkfor integer k≥be nonnegative numbers such that

an+t n

k=

bk≤t n–

k=

dkak+t n–

k=

ck+C ∀n≥.

Then,

an+t n

k=

bk≤

t n–

k=

ck+C

exp

t n–

k=

dk

∀n≥.

Following the proofs provided in [, , , ], we can obtain that problem (.) pos-sesses a unique solution (u,p,θ) with the following regularity properties.

Theorem . Let f ∈L∞(R+_;Y),ft∈L_{(,T;Y),g∈L}∞_(R+_;Z),gt∈L_{(,T;Z)and u} ∈

D(A)∩V,θ∈E(A).Then the solution(u,p,θ)of problem(.)satisfies

Au(t)_+∇ut(t)_+Aut(t)_+Aθ(t)_+∇θt(t)_+Aθt(t)_≤C.

Introduce the following Poincaré inequalities:

v≤C∇v ∀v∈X or W; ∇v≤CAv ∀v∈D(A)or H(). (.)

We end this section by recalling some properties of the trilinear formsb(·,·,·) andb(·,·,·), which can be found in [, –, , , , ].

Lemma . The trilinear forms b(·,·,·)and b(·,·,·)satisfy: ()Under the conditiondivu= ,we have that

b(u,v,v) =  ∀u,v∈X; b(u,θ,θ) =  ∀u∈X,θ∈W.

()We have the following estimates for trilinear terms b(·,·,·)and b(·,·,·):

b(u,v,w)≤Cu/_ A/u/_ A/v/_ Av/_ w ∀u∈V,v∈D(A),w∈X,

b(u,v,w)≤Cu/_ Av_/vw ∀u∈V,v∈D(A),w∈X,

b(u,θ,ψ)≤Cu/_ A/u/_ A/θ/

 Aθ /

 ψ ∀u∈V,θ∈E(A),ψ∈W, b(u,θ,ψ)≤Cu/_ Au/_ A/θ_ψ ∀u∈D(A),θ,ψ∈W.

3 The Euler implicit/explicit scheme for the Boussinesq equations

for allK∈Th, wherehK andρhdenote the diameter ofKand the diameter of the largest ball that can be inscribed intoK, respectively.

The finite element subspaces of interest in this paper are defined by the so-called MINI element with the continuous piecewise finite element subspace for the approximation of velocity and pressure and the linear polynomial for temperature, respectively:

Xh=vh∈Xh:vh|K∈P(K)⊕span{λλλ} ∀K∈Th

,

Mh=q∈M:q|K∈P(K)∀K∈Th

,

Wh=

ψ∈W:ψ|K∈P(K)∀K∈Th

.

Note thatλ,λ, andλare the barycentric coordinates of the reference element.

We define the subspaceVhofXhby

Vh=vh∈Xh:d(vh,qh) = ∀qh∈Mh.

LetPh:Y→VhorZ→Whdenote theL-orthogonal projection defined by

(Phω,χh) = (ω,χh) ∀ω∈YorZ,χh∈VhorWh.

We introduce a discrete analogueAh= –Phh of the Stokes operatorAthrough the condition (Ahφh,ψh) = (∇φh,∇ψh) for allφh,ψh∈XhorWh. The restriction ofAh to Vh is invertible with the inverseA–

h . SinceA–h is self-adjoint and positive definite, we may define the “discrete” Sobolev norms onVhof any orderr∈Rby setting

ωhr=Ahr/ωh_, ωh∈Vh.

These norms will be assumed to have various properties similar to their continuous coun-terparts, implicitly imposing conditions on the structure of the spacesXh,Mh, andWh. In particular,

ωh=∇ωh, ωh=Ahωh ∀ωh∈VhorWh.

The discrete Laplace operatorAhis first introduced in [] to analyze and obtain the op-timal estimates for the transient Navier-Stokes equations. Furthermore, from [, ] we know that there exists a positive constantβ>  independent ofhsuch that, for allvh∈Xh andqh∈Mh,

b(vh,qh)≥βvhqh. (.)

The finite element discretization applied to problem (.) leads to spatial discrete equa-tions as follows: Find (uh,ph,θh)∈Xh×Mh×Whsuch that, for all (vh,qh,ψh)∈Xh×Mh× Wh,

⎧ ⎪ ⎨ ⎪ ⎩

(uht,vh) +a(uh,vh) +b(uh,uh,vh) –d(vh,ph) = (f,vh) – (jθh,vh), b(uh,qh) = ,

(θht,ψ) +a(θh,ψh) +b(uh,θh,ψ) = (g,ψh).

For the stability and convergence of problem (.), we have the following results.

Theorem . (see []) Let f ∈L∞(R+_{;Y),ft ∈L}_{(,T;Y),g∈L}∞_(R+_;Z),gt∈L_(,T;Z),

u∈D(A)∩V,θ∈E(A)and assume that uh() =u,θh() =θ.By Theorem.problem

(.)admits a unique solution(uh,ph,θh)∈Xh×Mh×Whsatisfying

uh_+ν

t



∇uh_ds≤ u_+C

 

ν

t



f_ds+C

 

λν

θ+

C 

λν

t

 g_ds

,

∇uh_+ν

t



Ahuh_ds≤ ∇u_+ν

_C 

λ θ

 +

νC 

λ t



g_ds+ ν

t



f_ds,

θh_+λν

t

 ∇

θh_ds≤ θ_+C

 

λν

t



g_ds,

∇θh+λν t



Ahθhds≤ ∇θ+

 λν

t



g_ds,

uht+ν t



∇uhds+θht+λν t



∇θhds≤C,

ν Ahuh

 +

t



∇uht_ds+λν  Ahθh

 +

t



∇θhtds≤C,

uhtt_+ν

t



∇uht_ds+θhtt+λν t



∇θhtds≤C,

∇uht+

ν 

t



Ahuhtds+∇θht+

λν 

t



Ahθhtds≤C,

ν

Ahuht

 +

t



∇uhtt_ds+λν  Ahθht

 +

t



∇θhttds≤C,

_∇(u–uh)_+p–ph+∇(θ–θh)_≤Ch, u–uh+θ–θh≤Ch.

Lett>  be the time-step, and lettn=nt(≤n≤N= [T_t]),un_h,pn_h, andθ_hndenote the numerical solutions of uh,ph, andθh attn, respectively. We consider the Euler im-plicit/explicit scheme for the Boussinesq equations (.). As we have pointed out in Sec-tion , the advantage of adopting the Euler implicit/explicit scheme is that a linear system with constant coefficient matrix is obtained, and then a lot of computational cost can be saved.

The Euler implicit/explicit scheme for problem (.) reads as follows:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(u

n h–unh–

t ,v) +ν(∇unh,∇v) +b(unh–,uhn–,v) – (v,∇pnh) = (f(tn),v) – (jθhn,v), (∇ ·un

h,q) = , (θ

n h–θ

n–

h

t ,ψ) +λν(∇θ n

h,∇ψ) +b(unh–,θhn–,ψ) = (g(tn),ψ),

(.)

with ≤n≤N. From (.) we can see that the discrete system (.) is a linear system; for the existence and uniqueness ofun_h,pn_h, andθ_hn, we refer to [].

Forn= , equations (.) can be rewritten as

⎧ ⎪ ⎨ ⎪ ⎩

(u

h,v) +tν(∇uh,∇v) –t(v,∇ph) =t(f(t),v) –t(jθh,v), (∇ ·u_h,q) = ,

(θ_h,ψ) +tλν(∇θ_h,∇ψ) =t(g(t),ψ).

In order to simplify the expressions, we setdtωn_h=ω

n h–ωnh–

t , whereωisuorθ. Choosing ψ=θ_h,vh=u_h, andqh=p_hin (.), we have the following estimates:

u_h_≤tf+tg, θ_h_≤tg, (.)

_∇u

h≤

C

ν f+ C



λνg, ∇θ 

h≤

C

λνg. (.)

From (.)-(.) we can see that scheme (.) is stable.

4 Stability of the numerical solutions

In this section, we establish the stability of the numerical solutionsun

h,pnh, andθhnin the Euler implicit/explicit scheme (.) for the Boussinesq equations. The mathematical in-duction has been used to obtain the desired results; this technique has also been used to other problems, for example, for the Dirichlet problems with (p,q)-Laplacian [] and the mixed initial-boundary value problems [].

Theorem . Under the conditions of Theorem.and the stability conditions kt≤

and kt≤,the solutions un_h,pn_h,andθ_hnare bounded for any integer≤n≤[T_t]:

un_h_+νt n

i=

_∇ui_h_≤γ_, θ_hn

+λνt

n

i= _∇θ_hi

≤γ 

, (.)

_∇un_h_+νt n

i= Ahuih



≤k, ∇θ

n h

 +λνt

n

i= Ahθhi



≤k, (.)

dtun_h_+νt n

i=

_∇dtui_h_≤k, dtθhn

 +νt

n

i=

_∇dtθ_hi

≤k, (.)

Ahun_h_≤k= ν–

k+f∞

+γ+ ν–Ckγ, (.) _Ahθ_hn_≤k= (λν)–

k+b∞

+ (λν)–C_k_γ_, (.)

where

γ_= u_h+ ( + t)θ_h

+ T(T+t)g 

∞+ ( +t)f∞,

γ_= θ_h

+ T(T+t)g  ∞,

k=exp



νC  γ

∇u_h_+ν Au



h

 

+ ν–

T sup

≤t≤T

f(t)_+t n

i= θ_hi



,

k=exp



λ_νC  γγ

∇θ_h_+λν Aθ

n h



t+ T(λν) – _sup

≤t≤T

g(t)_

k=exp

expν–C_k

exp

C k

λν

 λ_νC

 CkT

·expν–C_k

×dtu_h+ ν–C_T sup

≤t≤T

ft_+exp

C_k

λν

×dtθ_h_+ C

 T

λν sup≤t≤T

g(t)_

,

k=exp

C k

λν

dtθh

 +

C T

λν ≤supt≤T

g(t)_

×

 +C

 Ck

λν exp

C k

λν

,

k= ν–Ck, k= (λν)–Ck.

Proof We prove this theorem by induction. From (.)-(.) we know that (.)-(.) hold forn= . Assume that (.)-(.) hold forn= , . . . ,Jwith ≤J<N= [T_t]. We need to prove (.)-(.) forn=J+ .

First, takingvh= tun_h,qh= tpn_h, andψh= tθhnin (.), we obtain

un_h–un_h–, un_h+ νt∇un_h_+ tbun_h–,un_h–,un_h

= tf(tn),un_h– tjθ_hn,un_h (.)

and

θ_hn–θ_hn–, θ_hn+ λνt∇θ_hn_+ tbun_h–,θ_hn–,θ_hn= tg(tn),θ_hn. (.)

By using of the identities

(a–b, a) =|a|–|b|+|a–b| and (a,b) =|a|+|b|–|a–b|, (.)

equations (.)-(.) can be transformed into

un_h_–un_h–_+un_h–un_h–_+ tν∇un_h_+ tbun_h–,un_h–,un_h

= tf(tn),un_h– tjθ_hn,un_h (.)

and

θ_hn_–θ_hn–_+θ_hn–θ_hn–_+ tλν∇θ_hn_+ tbun_h–,θ_hn–,θ_hn

= tg(tn),θ_hn. (.)

For the right-hand side terms of (.)-(.), we have

tf,un_h=tf,u_hn–+ tf,un_h–un_h–

≤tfun_h–_+ u

n h–unh–

 + t

tjθ_hn,u_hn=tjθ_hn,un_h–+ tjθ_hn,un_h–un_h–

≤tθ_hn_un_h–_+ u

n h–unh–

 + t

_θn h

 , tg,θ_hn=tg,θ_hn–+ tg,θ_hn–θ_hn–

≤tgθ_hn–_+ θ

n h –θhn–

 + t

_g .

For the trilinear terms, thanks to Lemma ., we deduce that

bun_h–,u_hn–,un_h= bun_h–,un_h,un_h–un_h–

≤CAhunh–∇u

n hu

n

h–unh– ≤ν∇un_h+ν–C_Ahu_hn–_u_hn+–un_h_,

bun_h–,θ_hn–,θ_hn= bun_h–,θ_hn,θ_hn–θ_hn–

≤CAhunh–∇θ

n hθ

n

h –θhn–

≤λν∇θ_hn_+ (λν)–C_Ahu_hn–_θ_hn–θ_hn–_.

Combining these estimates with (.)-(.), we arrive at

un_h_–un_h–_+νt∇un_h_

≤

ν–C_Ahun_h–_t– 

un_h–un_h–_

+ tun_hf+θ_hn_+ tf_+θ_hn_, (.)

θ_hn_–θ_hn–_+λνt∇θ_hn_

≤

(λν)–C_Ahunh–

 t–

 

θ_hn–θ_hn–_

+ tθ_hn_g+ tg_, (.)

for all ≤n≤N. Under the stability conditionskt≤,kt≤ and the induction

assumption onn= , , . . . ,J, we have

ν–C_Ahu_hn–_t–  ≤ν

–_C kt–

  ≤

 tk–



≤, (.)

(λν)–C_Ahunh–

 t–

 ≤(λν)

–_C kt–

 ≤

 tk–



≤. (.)

Summing (.)-(.) fornfrom  toJ+  and using (.)-(.), we obtain

θ_hJ+_+tλν J+

n=

_∇θ_hn_≤θ_h_+ γTg∞+ Ttg∞≤γ

and

_uJ+

h

 +tν

J+

n= _∇un

h

 ≤u



h



+ γTf∞+ Ttf∞+ Tγγ+ Ttγ   ≤γ,

Next, takingvh= (_νdtun_h+Ahu_hn)t∈Vhandψh= (λν dtθ

n

h+Ahθhn)tin (.), we get

ν–tdtunh



+νtAhu

n h

 +∇u

n h

 –∇u

n–

h

 +∇

un_h–un_h–_

+b

un_h–,un_h–, νdtu

n h+Aunh

t

=f(tn),ν–dtun_h+Aun_ht–jθ_hn,ν–dtun_h+Aun_h (.)

and

(λν)–tdtθhn



+λνtAhθ

n h

 +∇θ

n h

 –∇θ

n–

h

 +∇

θ_hn–θ_hn–_

+bun_h–,θ_hn–, (λν)–dtθ_hn+Aθ_hnt

=g(tn), (λν)–dtθ_hn+Aθ_hnt. (.)

For the right-hand side terms and the trilinear terms, by using Lemma . we obtain

bun_h–,un_h–,ν–dtun_h+Ahun_h

≤CA/h unh–u

n–

h

/  Ahu

n–

h

/ 

ν–dtun_h+Ahun_h

≤ 

νdtu n h

 +

ν Ahu

n h  +  νC  A/h u

n–

h

 u

n–

h Ahu

n–

h 

≤ 

νdtu n h

 +

ν Ahu

n h

 +

ν Ahu

n– h  +    ν 

C_A/_h un_h–_un_h–_,

bun–

h ,θhn–, (λν)–dtθhn+Ahθhn

≤CA/h unh–

/  u

n–

h

/  A

/

h θhn–

/ 

×Ahθ_hn–/



(λν)–dtθ_hn

+Ahθ

n h

≤ 

λνdtθ n h

 +

λν  Ahθ

n h

 +

λν  Ahθ

n– h   +   λν 

C_A/_h un_h–_∇θ_hn–_un_h–_,

_f(tn),ν–dtun_h+Ahun_h≤  νdtu

n h

 +

ν Ahu

n h

 +

 ν f(tn)

 , jθ_hn,ν–dtunh+Ahunh≤

 νdtu

n h

 +

ν Ahu

n h

 +

 ν θ

n h

 , g(tn), (λν)–dtθhn+Ahθhn≤

 λνdtθ

n h

 +

λν  Ahθ

n h

 +

 λνg(tn)

 .

Combining these inequalities with (.)-(.), we find

(ν)–tdtun_h_+ ∇un_h_– ∇un_h–_+νt

 Ahu

n h

 –

 Ahu

n– h   ≤  ν 

C_∇un_h–_u_hn–_∇u_hn–_t+ ν f(tn)

 t+

 ν θ

n h



and

(λν)–tdtθ_hn_+ ∇θ_hn_– ∇θ_hn–_

+λνt

 Ahθ

n h

 –

 Ahθ

n–

h

 

≤

 λν



C_∇un_h–_∇θ_hn–

u

n–

h

 t+

 λνg(tn)



t. (.)

Summing (.)-(.) fornfrom  toJ+  and using Lemma ., we finish the proof of (.).

Moreover, for allv∈Vhandψ∈Whwith ≤n≤[T_t] – , we deduce from (.) that

dttun_h,v+adtun_h,v+bdtun_h–,u_hn–,v+bu_hn–,dtun_h–,v

=  t

tn

tn–

(ft,v)dt–jdtθ_hn,v, (.)

dttu_h,v+adtu_h,v= –jdtθ_h,v, (.)

dttθ_hn,ψ+adtθ_hn,ψ+bdtu_hn–,θ_hn–,ψ+bun_h–,dtθ_hn–,ψ

=  t

tn

tn–

(gt,ψ)dt, (.)

dttθ_h,ψ+adtθ_h,ψ= . (.)

Choosingv=dtu

htandψ=dtθhtin (.) and (.), respectively, we obtain

dtθh

 +dttθ



h

 t

_+λν∇d

tθh



t=dtθ 

h

 , dtu_h_+dttu_h_t+ν∇dtu_h_t≤dtu_h_+ C

λνdtθ



h

 .

Takingv= dtun_htin (.) andψ= dtθ_hntin (.) with ≤n≤[T_t], we get

dtunh

 –dtu

n–

h



+ νt∇dtu

n h

 + b

dtunh–,uhn–,dtunh

t

+ bun_h–,dtu_hn–,dtun_ht≤

tn

tn–

ft,dtun_hdt–jdtθ_hn,dtun_ht (.)

and

dtθ_hn_–dtθ_hn–_+ λνt∇dtθ_hn_

+ bdtun_h–,θ_hn–,dtθ_hnt+ bun_h–,dtθ_hn–,dtθ_hnt

≤

tn

tn–

gt,dtθhn

dt. (.)

By Lemma . and the Poincaré inequality we have

bdtun_h–,u_hn–,dtun_h+bun_h–,dtun_h–,dtun_h

≤CAhunh–+Ahu

n–

h ∇dtu

n hdtu

n–

≤ν

∇dtu n h

 + ν

–_C

Aunh–

 +Au

n–

h

 dtu

n–

h

 ,

bdtun_h–,θ_hn–,dtθ_hn+bun_h–,dtθ_hn–,dtθ_hn

≤CAhθhn–∇dtθ

n hdtu

n–

h + CAhu

n–

h ∇dtθ

n hdtθ

n–

h 

≤λν

∇dtθ n h

 +

C 

λν Ahθ n–

h

 dtu

n–

h

 +Ahu

n–

h

 dtθ

n– h   ,  tn

tn–

ft,dtun_hdt≤νt  ∇dtu

n h

 + ν

–_C 

tn

tn–

ft_dt,



tn

tn–

gt,dtθhn

dt≤λνt  ∇dtθ

n h

 + (λν)

–_C 

tn

tn–

gtdt,

jdtθhn,dtunh≤ ν ∇dtu

n h  +  νC  dtθhn

 .

It follows from these inequalities that (.) and (.) can be transformed into

dtunh

 –dtu

n–

h



+νt∇dtu

n h

 

≤ν–C_

tn

tn–

ft_dt+dtθ_hn



+ ν–C_Ahun_h–_+Ahun_h–_dtun_h–_ (.)

and

dtθ_hn_–dtθ_hn–_+λνt∇dtθ_hn_

≤(λν)–C_

tn

tn–

gtdt

+C

 

λν Ahθ n–

h

 dtu

n–

h

 +Ahu

n–

h

 dtθ

n– h   . (.)

Summing (.)-(.) fornfrom  toJ+  and using Lemma ., we finish the proof of (.).

Using again Lemma . and (.), we deduce

νAhunh≤dtu

n

h+f(tn)+ C∇u

n–

h u

n–

h

/  Ahu

n–

h

/ 

and

λνAhθ_hn_≤dtθ_hn_+g(tn)_+ Cunh–

/  ∇u

n–

h

/  ∇θ

n–

h

/  Ahθ

n–

h

/  .

IfAhun

h≤ Ahunh–andAhθhn≤ Ahθhn–, then by (.)-(.) we have that (.)-(.) hold. Otherwise, settingk∗=sup≤n≤J+Ahunhandk∗∗=sup≤n≤J+Ahθhnand using (.)-(.), the obtained inequalities give

AhuJ_h+_≤k_∗≤ ν

sup

≤n≤J+

dtun_h_+f_∞

+ ν–C_ sup

≤n≤J

_∇un_un_+γ, _Ahθ_hJ+_≤k_∗≤ 

(λν)

sup

≤n≤J+

_dtθ_hn_+g_∞

+ (λν)–C_ sup

≤n≤J+ _∇un

θ

n

Combining these estimates with (.)-(.), we finish the proof of (.)-(.) with n=

J+ .

5 Error estimates

This section is devoted to present the optimal error estimates of velocity, pressure, and temperature in the Euler implicit/explicit scheme (.). In order to simplify the descrip-tions, we denote

En_u=uh(tn) –unh, Epn=ph(tn) –pnh, Enθ=θh(tn) –θhn,

where (uh(tn),ph(tn),θh(tn)) and (unh,pnh,θhn) be the solutions of problems (.) and (.), respectively. Furthermore, we setE_u=Eθ= .

Let us define the truncation errorsRn

uandRnθby

⎧ ⎪ ⎨ ⎪ ⎩

uh(tn)–uh(tn–)

t –νuh(tn) + (uh(tn)· ∇)uh(tn) +∇ph(tn) =f(tn) –jθh(tn) +R n u,

∇ ·uh(tn) = ,

θh(tn)–θh(tn–)

t –λνθh(tn) + (uh(tn)· ∇)θh(tn) =g(tn) +R n

θ,

(.)

where

Rn u= –

 t

tn

tn–

(t–tn)uhtt(t)dt and Rnθ= –

 t

tn

tn–

(t–tn)θhtt(t)dt.

By subtracting (.) from (.) we obtain the following error equations:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

En u–Eun–

t –νEun+ (uh(tn)· ∇)uh(tn) – (uhn–· ∇)unh–+∇Enp=Rnu–jEθn,

∇ ·En u= , E_θn–E_θn–

t –λνE

n

θ+ (uh(tn)· ∇)θh(tn) – (uhn–· ∇)θhn–=Rnθ.

(.)

Now, we present the error estimates for En

u,Enp, andEθnin different norms. In order to

simplify the expressions, we denote

Zn

u=uh(tn) –uh(tn–) = tn

tn– uhtdt,

Znθ=θh(tn) –θh(tn–) = tn

tn– θhtdt.

Then

uh(tn) –un_h–=uh(tn) –uh(tn–) +uh(tn–) –unh–=Znu+Eun–,

θh(tn) –θhn–=θh(tn) –θh(tn–) +θh(tn–) –θhn–=Znθ+Enθ–.

As a consequence, we find

uh(tn)· ∇

uh(tn) –

un_h–· ∇un_h–

=En_u–· ∇un_h–+Zn_u· ∇uh(tn–) +

uh(tn–)· ∇

E_un–+uh(tn)· ∇

and

uh(tn)· ∇

θh(tn) –

un_h–· ∇θ_hn–

=En_u–· ∇θ_hn–+Z_un· ∇θh(tn–) +

uh(tn–)· ∇

Eθn–+

uh(tn)· ∇Zθn.

Lemma . Under the assumptions of Theorems.and.,we have

_EJ+

u

 +E

J+

θ

 +νt

J+

n= _∇En

u

 +λνt

J+

n= _∇En

θ



≤Ct

_.

Proof Taking the inner product of (.) with tEn

uand tEθnand using the fact that

∇ ·En

u= , we obtain

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ En

u–Enu–+ νt∇Enu+ tb(Eun–,unh–,Eun) + tb(Znu,uh(tn–),Eun) + tb(uh(tn–),Eun–,Eun) + tb(uh(tn),Zun,Enu)

≤t(Rn

u,Enu) – t(jEθn,Enu),

Enθ–Eθn–+ λνt∇Eθn+ tb(Eun–,θhn–,Enθ)

+ tb(Zn

u,θh(tn–),Enθ) + tb(uh(tn–),Eθn–,Enθ) + tb(uh(tn),Zθn,Enθ)

≤t(Rnθ,Eθn).

(.)

The right-hand side terms of (.) can be treated as follows:

tRn_u,En_u≤tC

 

ν

_ttn

n–

(t–tn–)uhttdt 



+νt  ∇E

n u

 

≤Ct

tn

tn–

untt_dt+νt  ∇E

n u

 ,

tRnθ,Eθn≤

tC_ λν

_ttn

n–

(t–tn–)θhttdt





+λνt  ∇E

n

θ

 

≤Ct

tn

tn–

θhttdt+

λνt  ∇E

n

θ

 ,

–tjEnθ,Enu≤tEθn_Eun≤

t ν E

n

θ

 +

νt  E

n u

 .

For the nonlinear terms, by Lemma . we have

tbEn_u–,un_h–,E_un≤CtEnu–Ahu

n–

h ∇E

n u

≤C

ν tE n–

u

 Ahu

n–

h

 +

νt  ∇E

n u

 , tbZn_u,uh(tn–),Eun≤CtZunAhuh(tn–)∇E

n u

≤C

ν tZ n u



Ahuh(tn–)  +

νt  ∇E

n u

 

≤Ct

tn

tn–

uht_dt+νt  ∇E

n u

tbuh(tn–),Eun–,Enu≤CAhuh(tn–)_Enu–∇E

n u

≤C

ν tE n–

u



Ahuh(tn–)  +

νt  ∇E

n u

 , tbuh(tn),Z_un,En_u≤CtZnuAhuh(tn)∇E

n u

≤C

ν tZ n u



Ahuh(tn)  +

νt  ∇E

n u

 

≤Ct

tn

tn–

uht_dt+νt  ∇E

n u

 , tbE_un–,θ_hn–,En_θ≤CtEnu–Ahθ

n–

h ∇E

n

θ

≤C

λν tE n–

u

 Ahθ

n–

h

 +

λνt  ∇E

n

θ

 , tbZ_un,θh(tn–),Enθ≤CtZunAhθh(tn–)∇E

n

θ

≤C

λν tZ n u



Ahθh(tn–)  +

λνt  ∇E

n

θ

 

≤Ct

tn

tn–

uhtdt+

λνt  ∇E

n

θ

 , tbuh(tn–),Eθn–,Eθn≤CAhuh(tn–)_Eθn–_∇Eθn_

≤C

λν tE n–

θ



Ahuh(tn–)  +

λνt  ∇E

n

θ

 , tbuh(tn),Zθn,Eθn≤CtZnθ_Ahuh(tn)_∇Eθn_

≤C

λν tZ n

θ



Ahuh(tn)  +

λνt  ∇E

n

θ

 

≤Ct

tn

tn–

θhtdt+

λνt  ∇E

n

θ

 .

From all these inequalities and Theorems . and . we obtain

E_un_–En_u–_+En_u–En_u–_+νt∇En_u_

≤CtEn_u–_+Ct

tn

tn–

uhtt_+uht_dt+ νtE

n

θ



 (.)

and

Eθn+

 –E

n

θ

 +E

n+

θ –Enθ



+λνt∇E

n+

θ

 

≤CtEn_u–_+Ct

tn

tn–

θhtt+θht+uht

dt. (.)

Summing (.) and (.) fromn=  toJ+  and using Lemma ., we finish the proof.

Lemma . Under the assumptions of Theorems.and.,we have

_∇EJ_u+_+∇EθJ+

 +νt

J+

n=

AhE_un_+λνt J+

n= AhEθn



≤Ct

Proof Taking the inner product of (.) with –tAhEn

u∈Vh and –tAhEθnand using

the fact that∇ ·En+

u = , we obtain

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∇En

u–∇Enu–+ νtAhEun

≤t(b(En_u–,u_nn–,AhE_un) +b(Zn_u,uh(tn–),AhEun) +b(uh(tn–),Enu–,AhEnu) +b(uh(tn),Zn

u,AhEun) + (jEnθ,AhEnu) – (Rnu,AhEnu)),

∇En

θ–∇Enθ–+ λνtAhEθn ≤t(b(En–

u ,θhn–,AhEnθ) +b(Zun,θh(tn–),AhEθn) +b(uh(tn–),Enθ–,AhEnθ)

+b(uh(tn),Znθ,AhEθn) – (Rnθ,AhEnθ)).

(.)

Now, we treat the linear terms in the right-hand side of (.) as follows:

tRn_u,AhEn_u≤tRn_uAhE_un_≤Ct

tn

tn–

uhtt_dt+νt  AE

n u

 ,

tRn_θ,AhEnθ≤tR

n

θAE

n+

θ ≤Ct

 tn

tn–

θhttdt+

λνt  AhE

n

θ

 ,

–tjEnθ,AhEun≤tEθn_AEnu≤ν

–_tEn

θ

 +

νt  AE

n u

 .

For the trilinear terms of (.), by applying Lemma . we have

tbEn–

u ,unh–,AhEun≤Ct∇Enu–Ahu

n–

h AhE

n u

≤C

ν t∇E n–

u

 Ahu

n–

h

 +

νt  AhE

n u

 , tbZn

u,uh(tn–),AhEun≤Ct∇ZnuAhuh(tn–)AhE

n u

≤C

ν t∇Z n u



Ahuh(tn–)  +

νt  AhE

n u

 

≤Ct

tn

tn–

∇uht_dt+νt  AhE

n u

 , tbuh(tn–),Eun–,AhEnu≤CAhuh(tn–)_∇Eun–AhE

n u

≤C

ν t∇E n–

u



Ahuh(tn–)  +

νt  AhE

n u

 , tbuh(tn),Z_un,AhEn_u≤Ct∇ZunAhuh(tn)AhE

n u

≤C

ν t∇Z n u



Ahuh(tn)  +

νt  AhE

n u

 

≤Ct

tn

tn–

∇uht_dt+νt  AhE

n u

 , tbE_un–,θ_hn–,AhEθn≤Ct∇Eun–Ahθ

n–

h AhE

n

θ

≤C

λν t∇E n–

u

 Ahθ

n–

h

 +

λνt  AhE

n

θ

 , tbZ_un,θh(tn–),AhEθn≤Ct∇ZnuAhθh(tn–)AhE

n

θ

≤C

λν t∇Z n u



Ahθh(tn–)  +

λνt  AhE

n

θ

≤Ct

tn

tn–

∇uht_dt+λνt  AhE

n

θ

 , tbuh(tn–),Eθn–,AhEnθ≤CAhuh(tn–)_∇Enθ–AhE

n

θ

≤C

λν t∇E n–

θ



Ahuh(tn–)  +

λνt  AhE

n

θ

 , tbuh(tn),Znθ,AhEθn≤Ct∇ZnθAhuh(tn)AhE

n

θ

≤C

λν t∇Z n

θ



Ahuh(tn)  +

λνt  AhE

n

θ

 

≤Ct

tn

tn–

∇θhtdt+

λνt  AhE

n

θ

 .

Combining these inequalities with (.) and summingnfrom  toJ+ , we obtain

_∇EJ+

θ

 +

λν  t

J+

n= _AhEn

θ



≤Ct



T



θhtt+∇θht+∇uht

dt

+Ct J+

n=

_∇En_u–_+Ct J+

n= _∇Enθ

 ,

_∇E_uJ+_+ν t

J+

n=

AEn_u+_≤Ct

T



uhtt+∇uht

dt

+Ct J+

n=

_∇En_u–_+ νt

J+

n=

_∇Eθn–

 .

By Lemma . we complete the proof.

Now, we present the error estimates forEn

p, which show thatpnh is first-order approxi-mations topin theL∞(L_{) norm. In order to achieve this aim, we provide some estimates}

fordtEun= Eun–Eun–

t anddtE n

θ=

En_θ–En_θ–

t .

Lemma . Under the assumptions of Theorems.and.,we have

dtEJu+

 +dtE

J+

θ

 +νt

J+

n=

_∇dtEun+

 +λνt

J+

n=

_∇dtEθn+



≤Ct

_.

Proof From problem (.) we obtain that, for allv∈Vandψ∈W,

dttE_un,v–νdtEn_u,v

=dtRn_u,v–jdtEθn,v

–bdtZ_un,uh(tn–),v

–bZ_hn–,dtuh(tn–),v

–bdtE_un–,u_hn–,v–bE_un–,dtun_h–,v–bdtuh(tn–),Enu–,v

–buh(tn–),dtEun–,v

–bdtu(tn),Zun,v

–bu(tn–),dtZun,v