Decoupled Scheme for Time-Dependent Natural Convection Problem II : Time Semidiscreteness

We study the numerical methods for time-dependent natural convection problem that models coupled fluid flow and temperature field. A coupled numerical scheme is analyzed for the considered problem based on the backward Euler scheme; stability and the corresponding optimal error estimates are presented. Furthermore, a decoupled numerical scheme is proposed by decoupling the nonlinear terms via temporal extrapolation; optimal error estimates are established. Finally, some numerical results are provided to verify the performances of the developed algorithms. Compared with the coupled numerical scheme, the decoupled algorithm not only keeps good accuracy but also saves a lot of computational cost. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the decoupled method for time-dependent natural convection problem.

The time-dependent natural convection problem ( 1) is an important system with dissipative nonlinear terms in atmospheric dynamics (see [1]).Since this system not only contains the velocity and pressure but also includes the temperature filed, finding the numerical solution of problem (1) becomes a difficult task.For the study of problem (1), many researchers have developed several kinds of efficient numerical schemes, for example, the standard Galerkin finite element method (FEM) [2], the projection-based stabilized MFEM [3,4], and the references therein.Here, we need to point out that all these numerical schemes for problem (1) are coupled.It means that we need to find the variables , , and  of (1) simultaneously; as a consequence, a large nonlinear algebra system is formed.In general, it is expensive to find the numerical solutions of the coupled nonlinear system directly in standard Galerkin FEM.
The decoupled algorithm is an efficient numerical scheme for the multivarious problems.There are many advantages for the decoupled method.For example, it allows us to search the algorithm components flexibly and conveniently in terms of physical, mathematical, and numerical properties for each variable.It is suitable for today's computing environment because it can efficiently and effectively exploit the existing computing resources, including both hardware and software.The decoupled method can be used in parallel in the conventional sense; other appealing reasons were discussed in [5].The decoupled algorithm has been successfully applied to the 2 Mathematical Problems in Engineering multidomain problem, for example, Mu and his coworkers [5,6] for the Stokes-Darcy problem, Layton and his coauthors [7,8] for the groundwater-surface water flows, and Zhang et al. [9,10] for coupling fluid flow with porous media flow.In view of the efficiency of the decoupled scheme, we try to extend it to solve the time-dependent natural convection problem (1).The decoupled time semidiscrete scheme is closely related to the usual temporal extrapolation method [8,11].Thanks to the decoupled scheme, we can decouple the complex and nonlinear problem into two small linear subproblems, and the coefficient matrix of each subproblem is symmetric; therefore, these subproblems can be solved easier than the origin problem.
In this paper we establish the optimal error estimates for velocity, pressure, and temperature for problem (1) in both coupled and decoupled numerical schemes.Firstly, problem (1) is discrete in standard Galerkin finite element formulation based on the backward Euler scheme; then a large and nonlinear algebraic system is formed.Secondly, in order to simplify the computation, we adopt the decoupled and linearized algorithm to solve problem (1).Namely, the temporal extrapolation technique is used to treat the nonlinear terms, and then problem (1) is split into two subproblems, each subproblem can be solved easier than the origin problem.Furthermore, compared with the coupled scheme, these two subproblems which were obtained by using the decoupled method can be solved in parallel.
From ( 2) and ( 4), we can see that the coupled and decoupled algorithms have the same order of approximation.While there are only two small linear subproblems that need to be solved in the decoupled algorithm, a lot of memory and computational work can be saved.
The outline of this paper is as follows.We recall some basic notations and results for problem (1) in Section 2. Section 3 is devoted to present the coupled and decoupled algorithms for problem (1).Stabilities of both the coupled and decoupled schemes are established in Section 4. Optimal error estimates of numerical solutions in both the coupled and decoupled numerical schemes are presented in Sections 5 and 6, respectively.Finally, we provide some numerical results to verify the efficiency and effectiveness of the decoupled algorithm for time-dependent natural convection problem.
Next, we introduce the closed subset  of  given by  = {V ∈  :  (V, ) = 0, ∀ ∈ } = {V ∈ , ∇ ⋅ V = 0 in Ω} (7) and denote the  to be the closed subset of  (see [11,12]): We denote by  the unbounded linear operator on  or  given by  = −Δ or  = −Δ and assume that the domain of  is given by (see [13,14]) For instance, (9) holds if Γ is of class  2 or if Ω is a convex plane polygonal domain.
(A4)   ∈  Note that all such assumptions are feasible.For example, (A3) and (A4) can be proved with assumptions or is a convex polygon, (A5) holds by [11,12].Furthermore, (A6) holds by Shen in [18,19] when he adds some nonlocal compatibility conditions at  = 0.A review of regularity results for Navier-Stokes equations and applications to error estimates for Euler-type scheme can be found in [20], where the proof of (A7) was given.

The Coupled and Decoupled Algorithms for Time-Dependent Natural Convection Problem
In this section, let Δ > 0 be the time step and   = Δ;   and   denote the numerical solutions of  and  at   , respectively.We consider the backward Euler time discretization schemes for problem (1).Our schemes consist of two kinds of numerical schemes.One is the coupled scheme; the other is the decoupled scheme; these numerical algorithms are formulated as follows.

Coupled Algorithm for Time-Dependent Natural Convection
Problem.The coupled time semidiscrete scheme for time-dependent natural convection problem (1) based on the backward Euler scheme can be written as with 0 ≤  <  = [/Δ].The superscript  denotes the time level   .The system (27) is a nonlinear problem; the weak form of (27) can be formulated as for all (V, , ) ∈ ×× The existence and uniqueness of  +1 ,  +1 , and  +1 have been established by Luo in [21].From the expression of (28), we can see that when we solve problem (28) numerically, a large nonlinear algebra system should be solved, and the coefficient matrix is asymmetric.In general, it is expensive to solve such a nonlinear and coupled system.In order to improve the computational efficiency, we develop a decoupled and linearized scheme for problem (1).

Stabilities of the Coupled and Decoupled Algorithms
In this section, we consider the stabilities of both the coupled and decoupled numerical schemes under some assumptions presented in Section 2.
Furthermore, following the proofs provided in [23,24], we obtain the following stability results for the numerical solutions  , and  , of the decoupled numerical scheme (31).

Error Estimates of the Coupled Numerical Scheme
This section is devoted to present the optimal error estimates of velocity, pressure, and temperature in the coupled numerical scheme (28) introduced in Section 3. In order to simplify the descriptions, we denote (a) Error estimates for velocity and temperature in scheme (28) are as follows.
Let us define the truncation errors    and where Firstly, we present the estimates of  +1  and  +1  which show that both  +1 and  +1 are order 1/2 approximations to  and  in  ∞ () and in  ∞ (), respectively.
(b) Error estimates for pressure in scheme (28) are as follows.
Now, we give the estimates for  +1 = ( +1 ) −  +1 which shows that  +1 is order 1 approximation to  in both  ∞ ( 2 ) and  2 ( 2 ) norms.In order to achieve this aim, we firstly provide some estimates for    Proof.From problem (50) we obtain that for all V ∈  and  ∈ Substituting ( 83) into (82) and using Lemma 1, we obtain the desired results.
Remark 13.In the estimates of trilinear terms, we used the bounds of ‖ +1  ‖ 0 and ‖ +1  ‖ 0 which can be proved by differentiating (12) with respect to time, using the backward Euler scheme to discrete the equations and following the proofs of Lemmas 4 and 5.In the same way, we can also obtain the bounds of ‖ ,+1  ‖ 0 and ‖ ,+1  ‖ 0 for the decoupled scheme.Here, we omit these proofs for simplification.Now, we are in the position of deriving the optimal error estimate for pressure in  ∞ ( 2 ) norm based on the results presented in Lemmas 10, 11, and 12. Proof.We rewrite the first equation of (50) as follows: Take the inner product of (85 } . ( With the results obtained in Lemmas 5, 10, 11, and 12, we complete the proof.

Error Estimates of the Decoupled Numerical Scheme
In this section, we try to establish the optimal error estimates for the decoupled algorithm (31).We just point out the differences between the coupled and decoupled numerical schemes in the following lemmas.In order to simplify the representation, we denote (94) Taking the inner product of (94) with 2Δ Proof.Taking the inner product of (94) with 2Δ −1  ,+1  and 2Δ −1  ,+1  , using the fact that ∇ ⋅  ,+1  = 0 and the self-adjointness of  −1 , we get For the right-hand side terms of (100), we have    (119) With the help of Lemmas 6, 18, and 19, we obtain the optimal error estimate for pressure in  ∞ ( 2 ) norm.

Numerical Experiments
In order to gain insights into the established convergence results in Sections 5 and 6, we present some numerical tests in this section.Our main interest is to verify and compare the performances of the coupled and decoupled algorithms where the components of  are denoted by ( 1 ,  2 ) for convenience.Firstly, we compare the errors and CPU times for the coupled and decoupled numerical schemes with varying time step Δ.From Tables 1 and 2, we can see that two kinds of numerical schemes almost get the same accuracy, but the decoupled scheme (31) spends much less CPU time than the coupled scheme (28).In other words, the decoupled scheme is comparable with the coupled scheme but cheaper and more efficient.
Secondly, we focus on examining the orders of convergence of the coupled and decoupled numerical schemes with respect to the time step.Following [6], we introduce a more accurate approach to examine the orders of convergence with respect to the time step Δ due to the approximation errors O(Δ  ).For example, assuming V Δ ≈ V (,   ) +  (,   ) Δ  , (122) Here, V can take , ,  and  can be 0 or 1.While  V,Δ, approach 4.0 or 2.0, the convergence order will be 2.0 or 1.0, respectively.In Tables 3 and 4, we present the convergence orders with the fixed spacing ℎ = 1/32 and varying time steps Δ = 0.1, 0.05, 0.025, 0.0125.From these results, we can see that the decoupled scheme almost gets the same accuracy with the coupled scheme.For the numerical solutions   and   of the coupled scheme (28), we can get the optimal orders of convergence; for the pressure   , the results are undesired.In contrast, the results in Table 4 strongly suggest that the orders of convergence in time are O(Δ), which implies that the error estimates for the  2 -norm and  1 -norm of  , ,  , , and  , in the decoupled algorithm (31) are optimal.Our numerical results confirm the established theoretical analysis very well.