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 conditions \(k_{1}\Delta t\leq1\) and \(k_{2}\Delta t\leq1\) with two positive constants \(k_{1},k_{2}\) and 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.
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1 Introduction
In this paper, we consider the following Boussinesq equations in \(\mathbb{R}^{2}\) with coupled equations governing the viscous incompressible flow and heat transfer:
where Ω is a bounded convex polygonal domain, \(u=(u_{1},u_{2})^{T}\) is the fluid velocity, p is the pressure, θ is the temperature, \(\nu>0\) is the viscosity, \(\lambda=Pr^{-1}, Pr\) is the Prandtl number, \(j=(0,1)^{T}\) is the vector of gravitational acceleration, \(T>0\) is the final time, and f and g are the forcing functions.
The Boussinesq equations (1.1) are an important dissipative nonlinear model in the atmospheric dynamics (see [1]). This system not only contains the velocity and pressure but also includes the temperature filed, and therefore finding a numerical solution of problem (1.1) becomes a difficult task. There are numerous works devoted to the development of efficient schemes for this model, for example, the standard Galerkin finite element method (FEM) [2], the projection-based stabilized mixed FEM [3], the precondition techniques [4, 5], the two-level algorithms [6–8], and the references therein. In the literature mentioned, the suitable stability condition is a key issue for these developed schemes. Generally 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 [9–15]. The main results are summarized by He in his recent works [16, 17].
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 implicit/explicit scheme for the Boussinesq equations (1.1). Under the conditions \(k_{1}\Delta t\leq1\) and \(k_{2}\Delta t\leq1\) with two positive constants \(k_{1},k_{2}\), we present some new stabilities and establish 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 [10, 16, 18, 19] from the Navier-Stokes equations to the more complex Boussinesq equations. Our main results can be stated as follows:
where \(C>0\) is a constant depending on the parameters \(f_{\infty },b_{\infty},u_{0},\theta_{0},\Omega,\nu\), and λ, but independent of h and Δt, where \(f_{\infty}=\sup_{t\geq0}\{|f|+|f_{t}|\}\), \(f_{t}=\frac{df}{dt}\), \(g_{\infty}=\sup_{t\geq0}\{|g|+|g_{t}|\}\), \(g_{t}=\frac{dg}{dt}\). Here and thereafter, C denotes a general positive constant, which may take different values at different places. From (1.2)-(1.3) we can see that our results are optimal for both space length h and time step Δt.
The outline of this article is as follows. Some basic notation and results for problem (1.1) are recalled in Section 2. Section 3 is devoted to develop the Euler implicit/explicit scheme. Stabilities and optimal error estimates are established in Sections 4 and 5, 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 (1.1) and recall some classical results, which will be frequently used in this paper. To fix the idea, we set
Throughout this paper, we adopt \((\cdot,\cdot)\) and \(\|\cdot\|_{0}\) to denote the inner product and norm on \(L^{2}(\Omega)\) or \(L^{2}(\Omega)^{2}\). The spaces \(H^{1}_{0}(\Omega)\) and X are equipped with the usual scalar product and norm \(\|\nabla u\|^{2}_{0}=(\nabla u,\nabla u)\). Define the continuous bilinear forms \(a(\cdot,\cdot),d(\cdot,\cdot)\), and \(\overline{a}(\cdot,\cdot)\) by
for all \(u,v\in X,q\in M\), and \(\theta,\psi\in W\).
Next, we introduce the closed subset V of X given by
and denote by H the closed subset of Y (see [17, 20]) given by
We denote the Stokes operator by \(A=P\Delta\), where P is the \(L^{2}\)-orthogonal projection of Y onto H or of Z onto W. Assume that Ω is such that the domain of A is given by (see [10, 17, 21, 22])
For instance, (2.1) holds if Γ is of class \(C^{2}\) or if Ω is a convex plane polygonal domain.
Moreover, we can define the trilinear forms for all \(u,v,w\in X\) and \(\theta,\psi\in W\) as follows:
With these notations, for given \(f\in L^{\infty}(R^{+};Y)\) with \(u_{0}\in D(A)\cap V\) and \(g\in L^{\infty}(R^{+};Z)\) with \(\theta_{0}\in E(A)\), the variational formulation of (1.1) reads as follows: For all \((v,q,\psi)\in X\times M\times W\), find \((u,p,\theta)\in X\times M\times W\) with
such that
Assuming that \(f\in L^{2}(0,T;X'),g\in L^{2}(0,T;W')\), and \(u_{0}\in V,\theta _{0}\in W\), problem (2.2) has at least one solution \((u,p,\theta)\) satisfying \(u\in L^{\infty}(0,T;\Omega)\cap L^{2}(0,T;V)\) and \(\theta\in L^{\infty}(0,T;\Omega)\cap L^{2}(0,T;W)\). The uniqueness and regularity of the solution \((u,p,\theta)\) can also be proved by strengthening the assumptions on the data; see [23] for details.
We recall the following discrete Gronwall lemma, which can be found in [17, 24].
Lemma 2.1
Let \(C_{0}\) and \(a_{k},b_{k},c_{k},d_{k}\) for integer \(k\geq0\) be nonnegative numbers such that
Then,
Following the proofs provided in [1, 20, 22, 23], we can obtain that problem (2.2) possesses a unique solution \((u,p,\theta)\) with the following regularity properties.
Theorem 2.2
Let \(f\in L^{\infty}(R^{+};Y),f_{t}\in L^{2}(0,T;Y),g\in L^{\infty}(R^{+};Z),g_{t}\in L^{2}(0,T;Z)\) and \(u_{0}\in D(A)\cap V,\theta_{0}\in E(A)\). Then the solution \((u,p,\theta)\) of problem (2.2) satisfies
Introduce the following Poincaré inequalities:
We end this section by recalling some properties of the trilinear forms \(b(\cdot,\cdot,\cdot)\) and \(\overline{b}(\cdot,\cdot,\cdot)\), which can be found in [1, 6–8, 10, 17, 21, 22].
Lemma 2.3
The trilinear forms \(b(\cdot,\cdot,\cdot)\) and \(\overline{b}(\cdot,\cdot,\cdot)\) satisfy:
(1) Under the condition \(\operatorname{div} u=0\), we have that
(2) We have the following estimates for trilinear terms \(b(\cdot,\cdot ,\cdot)\) and \(\overline{b}(\cdot,\cdot,\cdot)\):
3 The Euler implicit/explicit scheme for the Boussinesq equations
Let \(\mathcal{T}_{h}\) be a family of finite element partitions of Ω into triangles satisfying the usual compatibility conditions [21] with \(h=\max h_{K}\), where \(h_{K}\) is the diameter of an element \(K\in\mathcal{T}_{h}\). We assume that \(\mathcal{T}_{h}\) is shape regular, that is, there exists a constant \(\sigma> 0\) such that \(h_{K}<\sigma\rho _{K}\) for all \(K\in\mathcal{T}_{h}\), where \(h_{K}\) and \(\rho_{h}\) denote the diameter of K and the diameter of the largest ball that can be inscribed into K, 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:
Note that \(\lambda_{1},\lambda_{2}\), and \(\lambda_{3}\) are the barycentric coordinates of the reference element.
We define the subspace \(V_{h}\) of \(X_{h}\) by
Let \(P_{h}:Y\rightarrow V_{h}\) or \(Z\rightarrow W_{h}\) denote the \(L^{2}\)-orthogonal projection defined by
We introduce a discrete analogue \(A_{h}=-P_{h}\Delta_{h}\) of the Stokes operator A through the condition \((A_{h}\phi_{h},\psi_{h})=(\nabla\phi_{h},\nabla\psi_{h})\) for all \(\phi_{h},\psi_{h}\in X_{h} \mbox{or} W_{h}\). The restriction of \(A_{h}\) to \(V_{h}\) is invertible with the inverse \(A_{h}^{-1}\). Since \(A_{h}^{-1}\) is self-adjoint and positive definite, we may define the “discrete” Sobolev norms on \(V_{h}\) of any order \(r\in R\) by setting
These norms will be assumed to have various properties similar to their continuous counterparts, implicitly imposing conditions on the structure of the spaces \(X_{h},M_{h}\), and \(W_{h}\). In particular,
The discrete Laplace operator \(A_{h}\) is first introduced in [20] to analyze and obtain the optimal estimates for the transient Navier-Stokes equations. Furthermore, from [1, 22] we know that there exists a positive constant \(\beta>0\) independent of h such that, for all \(v_{h}\in X_{h}\) and \(q_{h}\in M_{h}\),
The finite element discretization applied to problem (2.2) leads to spatial discrete equations as follows: Find \((u_{h},p_{h},\theta _{h})\in X_{h}\times M_{h}\times W_{h}\) such that, for all \((v_{h},q_{h},\psi_{h})\in X_{h}\times M_{h}\times W_{h}\),
For the stability and convergence of problem (3.2), we have the following results.
Theorem 3.1
see [7]
Let \(f\in L^{\infty}(R^{+};Y),f_{t}\in L^{2}(0,T;Y),g\in L^{\infty}(R^{+};Z), g_{t}\in L^{2}(0,T;Z), u_{0}\in D(A)\cap V,\theta_{0}\in E(A)\) and assume that \(u_{h}(0)=u_{0},\theta_{h}(0)=\theta_{0}\). By Theorem 2.2 problem (3.2) admits a unique solution \((u_{h},p_{h},\theta_{h})\in X_{h}\times M_{h}\times W_{h}\) satisfying
Let \(\Delta t>0\) be the time-step, and let \(t_{n}=n\Delta t \) (\(0\leq n\leq N=[\frac{T}{\Delta t}]\)), \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\) denote the numerical solutions of \(u_{h},p_{h}\), and \(\theta_{h}\) at \(t_{n}\), respectively. We consider the Euler implicit/explicit scheme for the Boussinesq equations (1.1). As we have pointed out in Section 1, 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 (3.2) reads as follows:
with \(1\leq n\leq N\). From (3.3) we can see that the discrete system (3.3) is a linear system; for the existence and uniqueness of \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\), we refer to [25].
For \(n=0\), equations (3.3) can be rewritten as
In order to simplify the expressions, we set \(d_{t}\omega_{h}^{n}=\frac{\omega _{h}^{n}-\omega_{h}^{n-1}}{\Delta t}\), where ω is u or θ. Choosing \(\psi=\theta_{h}^{0},v_{h}=u_{h}^{0}\), and \(q_{h}=p_{h}^{0}\) in (3.4), we have the following estimates:
4 Stability of the numerical solutions
In this section, we establish the stability of the numerical solutions \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\) in the Euler implicit/explicit scheme (3.3) for the Boussinesq equations. The mathematical induction 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 [26] and the mixed initial-boundary value problems [27].
Theorem 4.1
Under the conditions of Theorem 3.1 and the stability conditions \(k_{1}\Delta t\leq1\) and \(k_{2}\Delta t\leq1\), the solutions \(u_{h}^{n},p_{h}^{n}\), and \(\theta_{h}^{n}\) are bounded for any integer \(0\leq n\leq[\frac{T}{\Delta t}]\):
where
Proof
We prove this theorem by induction. From (3.5)-(3.6) we know that (4.1)-(4.5) hold for \(n=0\). Assume that (4.1)-(4.5) hold for \(n=0,\ldots,J\) with \(0\leq J< N=[\frac{T}{\Delta t}]\). We need to prove (4.1)-(4.5) for \(n=J+1\).
First, taking \(v_{h}=2\Delta tu_{h}^{n},q_{h}=2\Delta t p_{h}^{n}\), and \(\psi _{h}=2\Delta t\theta_{h}^{n}\) in (3.3), we obtain
and
By using of the identities
equations (4.6)-(4.7) can be transformed into
and
For the right-hand side terms of (4.9)-(4.10), we have
For the trilinear terms, thanks to Lemma 2.3, we deduce that
Combining these estimates with (4.9)-(4.10), we arrive at
for all \(1\leq n\leq N\). Under the stability conditions \(k_{1}\Delta t\leq1,k_{2}\Delta t\leq1\) and the induction assumption on \(n=0,1,\ldots,J\), we have
Summing (4.11)-(4.12) for n from 1 to \(J+1\) and using (4.13)-(4.14), we obtain
and
which is (4.1) with \(n=J+1\).
Next, taking \(v_{h}=(\frac{1}{\nu}d_{t}u_{h}^{n}+A_{h}u_{h}^{n})\Delta t\in V_{h}\) and \(\psi_{h}=(\frac{1}{\lambda\nu}d_{t}\theta_{h}^{n}+A_{h}\theta_{h}^{n})\Delta t\) in (3.3), we get
and
For the right-hand side terms and the trilinear terms, by using Lemma 2.3 we obtain
Combining these inequalities with (4.15)-(4.16), we find
and
Summing (4.17)-(4.18) for n from 1 to \(J+1\) and using Lemma 2.3, we finish the proof of (4.2).
Moreover, for all \(v\in V_{h}\) and \(\psi\in W_{h}\) with \(2\leq n\leq[\frac {T}{\Delta t}]-1\), we deduce from (3.3) that
Choosing \(v=d_{t}u_{h}^{1}\Delta t\) and \(\psi=d_{t}\theta_{h}^{1}\Delta t\) in (4.20) and (4.22), respectively, we obtain
Taking \(v=2d_{t}u_{h}^{n}\Delta t\) in (4.19) and \(\psi=2d_{t}\theta _{h}^{n}\Delta t\) in (4.21) with \(2\leq n\leq[\frac{T}{\Delta t}]\), we get
and
By Lemma 2.3 and the Poincaré inequality we have
It follows from these inequalities that (4.23) and (4.24) can be transformed into
and
Summing (4.25)-(4.26) for n from 1 to \(J+1\) and using Lemma 2.3, we finish the proof of (4.3).
Using again Lemma 2.3 and (3.3), we deduce
and
If \(\|A_{h}u_{h}^{n}\|_{0}\leq\|A_{h}u_{h}^{n-1}\|_{0}\) and \(\|A_{h}\theta_{h}^{n}\|_{0}\leq\| A_{h}\theta_{h}^{n-1}\|_{0}\), then by (3.5)-(3.6) we have that (4.4)-(4.5) hold. Otherwise, setting \(k_{*}=\sup_{0\leq n\leq J+1}\|A_{h}u_{h}^{n}\|_{0}\) and \(k_{**}=\sup_{0\leq n\leq J+1}\|A_{h}\theta_{h}^{n}\|_{0}\) and using (4.1)-(4.3), the obtained inequalities give
Combining these estimates with (4.1)-(4.3), we finish the proof of (4.4)-(4.5) with \(n=J+1\). □
5 Error estimates
This section is devoted to present the optimal error estimates of velocity, pressure, and temperature in the Euler implicit/explicit scheme (3.3). In order to simplify the descriptions, we denote
where \((u_{h}(t_{n}),p_{h}(t_{n}),\theta_{h}(t_{n}))\) and \((u^{n}_{h},p^{n}_{h},\theta ^{n}_{h})\) be the solutions of problems (3.2) and (3.3), respectively. Furthermore, we set \(E_{u}^{0}=E_{\theta}^{0}=0\).
Let us define the truncation errors \(R_{u}^{n}\) and \(R_{\theta}^{n}\) by
where
By subtracting (3.3) from (5.1) we obtain the following error equations:
Now, we present the error estimates for \(E_{u}^{n},E_{p}^{n}\), and \(E_{\theta }^{n}\) in different norms. In order to simplify the expressions, we denote
Then
As a consequence, we find
and
Lemma 5.1
Under the assumptions of Theorems 3.1 and 4.1, we have
Proof
Taking the inner product of (5.2) with \(2\Delta tE_{u}^{n}\) and \(2\Delta tE_{\theta}^{n}\) and using the fact that \(\nabla\cdot E_{u}^{n}=0\), we obtain
The right-hand side terms of (5.3) can be treated as follows:
For the nonlinear terms, by Lemma 2.3 we have
From all these inequalities and Theorems 3.1 and 4.1 we obtain
and
Summing (5.4) and (5.5) from \(n=1\) to \(J+1\) and using Lemma 2.1, we finish the proof. □
Lemma 5.2
Under the assumptions of Theorems 3.1 and 4.1, we have
Proof
Taking the inner product of (5.2) with \(-2\Delta tA_{h}E_{u}^{n}\in V_{h}\) and \(-2\Delta tA_{h}E_{\theta}^{n}\) and using the fact that \(\nabla\cdot E_{u}^{n+1}=0\), we obtain
Now, we treat the linear terms in the right-hand side of (5.6) as follows:
For the trilinear terms of (5.6), by applying Lemma 2.3 we have
Combining these inequalities with (5.6) and summing n from 1 to \(J+1\), we obtain
By Lemma 2.1 we complete the proof. □
Now, we present the error estimates for \(E_{p}^{n}\), which show that \(p_{h}^{n}\) is first-order approximations to p in the \(L^{\infty}(L^{2})\) norm. In order to achieve this aim, we provide some estimates for \(d_{t}E_{u}^{n}=\frac {E_{u}^{n}-E_{u}^{n-1}}{\Delta t}\) and \(d_{t}E_{\theta}^{n}=\frac{E_{\theta}^{n}-E_{\theta}^{n-1}}{\Delta t}\).
Lemma 5.3
Under the assumptions of Theorems 3.1 and 4.1, we have
Proof
From problem (5.2) we obtain that, for all \(v\in V\) and \(\psi\in W\),
and
Choosing \(v=2\Delta td_{t}E_{u}^{n}\) and \(\psi=2\Delta td_{t}E_{\theta}^{n}\) in (5.7)-(5.8), respectively, we deduce that
and
Now, we estimate the right-hand side terms of (5.9)-(5.10) separately. For \((d_{t}R_{u}^{n},d_{t}E_{u}^{n})\) and \((d_{t}R_{\theta}^{n},d_{t}E_{\theta}^{n})\), using the techniques adopted by He [16], we arrive at
and
for all \(2\leq n\leq J\). We deduce from these equalities that
In the same way, we have
For the nonlinear terms, with the help of Lemma 2.3, we find that
In the same way, we have
Combining these inequalities with (5.9)-(5.10) and summing from \(n=1\) to \(J+1\), we obtain
and
Substituting (5.12) into (5.11) and using Lemma 2.1, we obtain the desired results. □
Remark 5.1
In the estimates of trilinear terms, the bounds of \(\int_{0}^{T}\|u_{httt}\|_{0}^{2}\,dt\) and \(\int_{0}^{T}\|\theta_{httt}\| _{0}^{2}\,dt\) are used. We can prove them by differentiating (3.2) twice with respect to time and following the proofs provided in [7]. As for the bounds of \(\|A_{h}u_{ht}^{n-2}\|_{0}\) and \(\|A_{h}\theta _{ht}^{n-2}\|_{0}\), we can obtain them as we have done in Section 4. Here, we omit these proofs for simplification.
Now, we are in the position of establishing the optimal error estimate for pressure in \(L^{\infty}(L^{2})\) norm based on the results presented in Theorems 3.1 and 4.1 and Lemmas 5.1-5.3.
Theorem 5.4
Under the assumptions of Theorems 3.1 and 4.1, we have
Proof
We rewrite the first equation of (5.2) as follows:
Taking the inner product of (5.13) with arbitrary \(v\in X\) and using the Poincaré inequality, we obtain
For the nonlinear terms, applying Lemma 2.3 and Theorems 3.1 and 4.1, we arrive at
By the discrete inf-sup condition (3.1) we obtain
With the results of Theorem 3.1 and Lemmas 5.1, 5.2, and 5.3, we complete the proof. □
6 Numerical experiments
In order to gain insights on the established convergence results in Section 5, in this section, we present some numerical tests. Our main interest is to verify and compare the performances of the Euler implicit/explicit scheme (3.3) for the Boussinesq equations. In all experiments, the Boussinesq equations are defined on the convex domain \(\Omega=[0,1]\times[0,1]\). The mesh consists of triangular elements that are obtained by dividing Ω into subsquares of equal size and then drawing the diagonal in each subsquare. The model parameters ν and λ are simply set to 1. We use the MINI element that satisfies the discrete inf-sup condition to approximate the velocity u and pressure p and the linear polynomial to approximate the temperature θ. The boundary and initial conditions and right-hand side functions f and g are selected such that the exact solutions are given by
where the components of u are denoted by \((u_{1},u_{2})\) for convenience.
First, we compare the errors and CPU time of the standard Galerkin finite element method using the backward Euler scheme and Newton iteration to treat the temporal term and nonlinear term and the Euler implicit/explicit scheme with varying time step Δt or mesh length h. From Tables 1-4 we can see that two kinds of numerical methods almost get the same accuracy, but the Euler implicit/explicit scheme takes less CPU time than the standard Galerkin FEM. In other words, the Euler implicit/explicit scheme is comparable with the standard Galerkin FEM but cheaper and more efficient.
Next, we focus on examining the orders of convergence for both standard Galerkin FEM and the Euler implicit/explicit scheme with respect to the time step Δt or the mesh size h. Following [28], we introduce the following way to examine the orders of convergence with respect to the time step Δt or the mesh size h due to the approximation errors \(\mathcal{O}(\Delta t^{\gamma})+\mathcal{O}(\Delta t^{\mu})\). For example, assuming that
we have
Here,v is \(u,p\), or θ, and j is 0 or 1. Since \(\rho _{v,h,j}\) and \(\rho_{v,\Delta t,j}\) approach 4.0 or 2.0, the convergent order will be 2.0 or 1.0, respectively.
In Tables 5-6, we present the convergent orders with fixed spacing \(h=\frac{1}{32}\) and varying time steps \(\Delta t=0.1,0.05,0.025,0.0125\). From these results we can see that the Euler implicit/explicit scheme almost gets the same accuracy with the standard Galerkin finite element method and shows optimal convergent orders on Δt. In Tables 7-8, we study the convergence orders with fixed time step \(\Delta t=0.01\) with varying spacing \(\frac {1}{h}=2,4,8,16\). Observe that \(\rho_{u,h,0}\) and \(\rho_{\theta,h,0}\) are close to 4.0 and \(\rho_{u,h,1}\), \(\rho_{\theta,h,1}\) approach 2.0, which suggests that the orders of convergence are \(\mathcal{O}(h^{2})\) for the \(L^{2}\)-norm of u and θ and \(\mathcal{O}(h)\) for the \(H^{1}\)-norm of u and θ in space. For the convergence order of pressure\(,\rho_{p,h,0}\) is close to 3.2, which shows the superconvergence. From these numerical results we can conclude that the Euler implicit/explicit scheme not only has a good accuracy, but also saves a lot of computational cost.
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Acknowledgements
The authors are grateful to the editor and two referees for a number of helpful suggestions, which have greatly improved our original manuscript. This research is supported by CAPES and CNPq of Brazil (No. 88881.068004/2014.01), the NSF of China (No. 11301157), and the Foundation of Distinguished Young Scientists of Henan Polytechnic University (J2015-05).
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TZ carried out the main theorem and wrote the paper, JJJ revised and checked the paper, and SWX checked the article, TZ, JJJ, and SWX read and approved the final version.
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Zhang, T., Jin, J. & Xu, S. The Euler implicit/explicit scheme for the Boussinesq equations. Bound Value Probl 2016, 181 (2016). https://doi.org/10.1186/s13661-016-0693-5
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DOI: https://doi.org/10.1186/s13661-016-0693-5