Abstract
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.
1. Introduction
In this paper, we consider the following time-dependent natural convection problem in whose coupled equations governing viscous incompressible flow and heat transfer for the incompressible fluid are Boussinesq system: where is a bounded convex polygonal domain, is the fluid velocity, is the pressure, is the temperature, is the viscosity, is the Grashof number, , is the Prandtl number, is the vector of gravitational acceleration, is the final time, and and are forcing functions.
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 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.
Under the conditions of (A1)–(A6) which are presented in Section 2, the numerical solution of the decoupled method at the time level satisfies the following error estimates for all time step : where is a generic constant depending on the data independent of and it may stand for different values at different places:
Under the conditions of (A1)–(A6), the numerical solution of the coupled method satisfies the following error estimates for all :
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.
2. Preliminaries
In this section, we will construct the variable formulation for problem (1) and develop some necessary assumptions which will be frequently used in this paper. To fix the idea, we set In this paper, we adopt and to denote the inner product and norm in or . The spaces and are equipped with the usual scalar product and norm . We define the continuous bilinear forms , , and , respectively, by for all , , and .
Next, we introduce the closed subset of given by and denote the to be the closed subset of (see [11, 12]):
We denote by the unbounded linear operator on or given by or and assume that the domain of is given by (see [13, 14]) For instance, (9) holds if is of class or if is a convex plane polygonal domain.
Moreover, the trilinear terms for all and can be defined as follows:
With above notations, for a given with and with , the variational formulation of problem (1) reads as follows: for all , find a pair with such that
We make the regularity assumptions on the following problems [15–17].
(A1) Assume that is smooth such that there exists a unique solution of the following Stokes problem: for any prescribed . Furthermore, the solution satisfies Form (13), it follows that and where is the dual space of and () denotes a positive constant depending on .
(A2) Assume that is smooth such that there exists a unique solution of the following elliptic problem: for any prescribed . Furthermore, the solution satisfies Form (16), it follows that and where is the dual space of .
Assume that , , and , . Problem (12) has at least one solution satisfying and . Uniqueness and regularity of the solution can also be proved by strengthening the assumptions on the data. In particular, we assume that , and satisfy the following
(A3) , , .
(A4) , .
(A5) , .
(A6) , .
(A7) , .
Note that all such assumptions are feasible. For example, (A3) and (A4) can be proved with assumptions , , , and , . When is of class of 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 . 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.
We recall the following discrete Gronwall lemma, which can be found in [11, 18].
Lemma 1. Let and , , , , for integers , be nonnegative numbers such that Then,
Following the proofs provided in [1, 12, 14, 21], we can obtain that problem (12) possesses a unique solution which satisfies the following regularity results under some nonlocal compatibility assumptions at .
Theorem 2. Let , , , , and , . Then for all and the solution of problem (12) satisfies
We introduce the following inequalities:
We end this section by recalling some properties of the trilinear forms and , which can be found in [1, 11, 13, 14, 22].
Lemma 3. The trilinear forms and satisfy the following.(1)In view of , one has where (2)Under the condition of , (3)The following estimates about trilinear terms and hold:
3. The Coupled and Decoupled Algorithms for Time-Dependent Natural Convection Problem
In this section, let be the time step and ; 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.
3.1. 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 . The superscript denotes the time level . The system (27) is a nonlinear problem; the weak form of (27) can be formulated as for all
The existence and uniqueness of , , and 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).
3.2. Decoupled Algorithm for Time-Dependent Natural Convection Problem
The decoupled and linearized semidiscrete scheme for time-dependent natural convection problem (1) based on the backward Euler scheme can be presented as follows: Scheme (29) is a mixed problem that contains a generalized Stokes problem and an elliptic problem. Namely, problem (29) can be split into two subproblems:
The weak form of problem (29) consists of finding , , and such that for all , If satisfies the so-called inf-sup condition, we can obtain the existence and uniqueness of , , and by [13, 14]. From the expressions of (30) we know that the variables , , and can be solved separately with two small linear algebra systems (30). Furthermore, the coefficient matrices of subproblems (30) are symmetric. It is much easier to solve these two subproblems than the original one.
4. 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.
Lemma 4. Under assumptions (A3)–(A7), the following results of scheme (28) hold for all : where , .
Proof. Taking , , and in (28) we derive that By using the identities problem (34) can be transformed into For the right-hand side terms of (36) we have Combining above estimates with (36) and summing them for from 0 to yield We complete the proof by substituting (39) into (38) and using (22).
Lemma 5. Under assumptions (A3)–(A7), the following results of scheme (28) hold for all : where and .
Proof. In order to simplify the notations, we set , taking or . Choose and in (28) to obtain Using Lemma 3 and Cauchy inequality we arrive at Combining above inequalities with (41) one finds Summing (43) for from 0 to , using Lemmas 1 and 4, we have which together with Lemma 4 implies the desired results.
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).
Lemma 6. Under assumptions (A3)–(A7), the following results of problem (31) hold: for all , where and .
5. 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 by where
Firstly, we present the estimates of and which show that both and are order approximations to and in and in , respectively.
Lemma 7. Assume that assumptions (A3)–(A6) hold. Then for all one has
Proof. By subtracting (27) from (47) we have
Taking the inner product of (50) with , , and , we can transform problem (50) into
Now, the right-hand side terms of (51) can be treated as follows:
For the nonlinear term, thanks to Lemma 3, we have
From above inequalities we arrive at
Summing (55) from to one finds
Combining Lemma 1 and Theorem 2 with (22) yields the desired results.
Remark 8. In particular, Lemma 7 shows that the coupled finite element method provides uniformly stable velocity and temperature in and , respectively. Using the fact that , and , , we know that there exists a positive constant independent of the time step such that for all ,
Lemma 9. For all , under assumptions (A3)–(A6), one has
Proof. Taking the inner product of (50) with and , using the fact that and the self-adjointness of , we obtain Taking in (13) and in (16). For the terms and , we have For the right-hand side terms of (59), we can estimate them as follows: For the nonlinear term, using Lemma 3, one finds Combining above inequalities with (59) and adding them from to , one finds Thanks to Lemmas 1, 4 and Theorem 2, we derive that Substituting (65) into (64) and using Lemma 1, we complete the proof.
Lemma 10. Assume that assumptions (A3)–(A6) are valid. Then, for all , one has
Proof. Based on Lemma 9, we reestimate the trilinear terms and right-hand side terms of (51) as follows: Thanks to (67), taking into account (53) and (55) and summing up (51) from to , we have We obtain the desired results with application of Lemma 1 at above inequalities.
Lemma 11. Under assumptions (A3)–(A6), for all , one has
Proof. Taking the inner product of (50) with and , using the fact that , we obtain We treat the right-hand side terms of (70) as follows: By Lemma 3, for the trilinear terms, one finds Combining above inequalities with (70) and summing from 0 to we obtain Thanks to Lemmas 10 and 1 we complete the proof.
(b) Error estimates for pressure in scheme (28) are as follows.
Now, we give the estimates for which shows that is order 1 approximation to in both and norms. In order to achieve this aim, we firstly provide some estimates for and .
Lemma 12. Under assumptions (A3)–(A7), for all , one has
Proof. From problem (50) we obtain that for all and Choosing and in (75) we deduce Now, we estimate the right-hand side terms of (76)-(77) separately. For and , using the techniques that are adopted by He in [20], we have for all . We deduce from above equalities that In the same way, we obtain For the nonlinear terms, with the help of Lemma 3, one finds Combining above inequalities with (76)-(77) and summing from to , we arrive at 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 and 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 and 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 norm based on the results presented in Lemmas 10, 11, and 12.
Theorem 14. Under assumptions (A3)–(A7) for all one has
Proof. We rewrite the first equation of (50) as follows: Take the inner product of (85) with an arbitrary and use Poincare inequality to obtain For the nonlinear terms, use the results provided in Lemma 3 to arrive at Thanks to (32), we have With the results obtained in Lemmas 5, 10, 11, and 12, we complete the proof.
6. 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
Lemma 15. Under assumptions (A3)–(A6), for all , one has
Proof. By subtracting (29) from (47) we have Taking the inner product of (94) with , , and , problem (94) can be transformed into Now, we estimate the right-hand side terms of (95) as follows: For the nonlinear term, thanks to Lemma 3, we arrive at Combining above inequalities with (52) and (53) and summing from to one finds We complete the proof by using Lemma 1 and (22).
Lemma 16. For all , under assumptions (A3)–(A6), one has
Proof. Taking the inner product of (94) with and , using the fact that and the self-adjointness of , we get
For the right-hand side terms of (100), we have
For the other terms, we can treat them as we have done in Lemma 9. By combining above inequalities and adding (100) from to , one finds
Applying Lemmas 1, 15 and assumptions (A3)–(A7) we finish the proof.
Lemma 17. Under conditions (A3)–(A6), for all , one has
Proof. Taking the inner product of (94) with and , using the fact that , we obtain (95). Based on (67), and the estimates presented in Lemma 7. Taking these inequalities of Lemmas 7 and 9 into (95), summing (95) from to , we arrive at We obtain the desired results with application of Lemma 1 at above inequalities.
Lemma 18. Under assumptions (A3)–(A6), for all , one has
Proof. Taking the inner product of (94) with and , using the fact that , we get For the trilinear terms, using Lemma 3, we have Combining above inequalities with (106) and summing from 0 to we obtain Thanks to Lemmas 1, 6, and 17 we complete the proof.
Lemma 19. Under assumptions (A3)–(A7), for all , one has
Proof. From problem (94) we can obtain that for all and Choosing and in (110) one gets For the nonlinear terms, thanks to Lemma 3, we deduce that Combine above inequalities with (79)-(80) with (111), summing them from to , to arrive at Thanks to Lemma 1, we obtain the desired results.
Finally, we present the optimal error estimate for pressure in norm based on the results presented in Lemmas 18 and 19.
Theorem 20. For all , under assumptions (A3)–(A7), one has Furthermore, if and , then
Proof. We can rewrite the first equation of problem (94) as
Take the inner product of (116) with an arbitrary and use Poincare inequality to obtain
For the nonlinear terms, use Lemma 3 to arrive at
Thus, thanks to (32), (86)–(88) and above inequalities, one finds
Squaring (119) and summing it from to , with the results obtained in Lemmas 6, 18, and 19, we obtain the desired result.
Furthermore, under the assumptions of and , (119) can be transformed into
With the help of Lemmas 6, 18, and 19, we obtain the optimal error estimate for pressure in norm.
7. 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 (28) and (31). In all experiments, the time-dependent natural convection problem is defined on a convex domain . The mesh consists of triangular elements that are obtained by dividing into subsquares of equal size and then drawing the diagonal in each subsquare. The model parameters , , and are simply set to 1. We fix the mesh size and use the MINI element which satisfies the discrete inf-sup condition to approximate the velocity and pressure and the linear polynomial to approximate the temperature . The boundary and initial conditions and right-hand side functions and are selected such that the exact solutions are given by where the components of are denoted by for convenience.
Firstly, we compare the errors and CPU times for the coupled and decoupled numerical schemes with varying time step . 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 due to the approximation errors . For example, assuming thus we have Here, can take , , and can be 0 or 1. While 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 and varying time steps . 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 , which implies that the error estimates for the -norm and -norm of , , and in the decoupled algorithm (31) are optimal. Our numerical results confirm the established theoretical analysis very well.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the NSF of China (no. 11301157), the Doctor Fund of Henan Polytechnic University (B2012-098), the Natural Science Foundation of Education Department of Henan Province (no. 14A110008), and the Foundation of Distinguished Young’s Scientists of Henan Polytechnic University (J2015-05).