A spectral element Crank–Nicolson model to the 2D unsteady conduction–convection problems about vorticity and stream functions

In this study, a time semi-discretized Crank–Nicolson (CN) scheme of the two-dimensional (2D) unsteady conduction–convection problems for vorticity and stream functions is first built together with showing the existence and stability along with error estimates to the semi-discretized CN solutions. Afterwards, a fully discretized spectral element CN (SECN) model of the 2D unsteady conduction–convection problems as regards the vorticity and stream functions is set up together with showing the proof of the existence and stability along with error estimates of the SECN solution. Lastly, a set of numerical experiments are offered for checking the correctness of the theoretical conclusions.

where ∂ z = ∂/∂z (z = t, x, y), (u, v) T stands for the velocity vector of flow, p stands for the pressure, Q stands for the temperature or heat energy, T stands for the total time, μ = √ Pr/Re, Re stands for the Reynolds, Pr stands for Prandtl's number, γ 0 = 1/ √ RePr, (ϕ u (x, y, t), ϕ v (x, y, t)) T and Q 0 (t, x, y) stand, respectively, for the known boundary values to the flow velocity and temperature, and (u 0 (x, y), v 0 (x, y)) T , and Q 0 (x, y) stand, respectively, for the known initial values to the flow velocity and the temperature.
For convenience of the theoretic argumentation, we will presume that Q 0 (t, x, y) = ϕ u (t, x, y) = ϕ v (t, x, y) = 0 in the following.
The 2D unsteady conduction-convection problems possess very momentous physical background and can be applied for simulating the real-world natural phenomena (see [1][2][3][4]). But, due to the nonlinearity for Problem 1, most of all when the computational region for Problem 1 is of an irregular geometrical shape, one cannot usually find any genuine solution so one has to find numerical ones.
It is universally acknowledged that the spectral and finite element (FE) together with finite difference (FD) along with finite volume element (FVE) methods are four welcome numerical means (see [5][6][7][8][9][10]). Nevertheless, the spectral method possesses the highest precision among four numerical ones because the unknowns to the spectral method are approximated with the smooth functions, including trigonometric functions or the Chebyshev, Jacobi, and Legendre polynomials, but the unknowns to the FE and FVE methods are usually approximated by the classic polynomials, while the derivatives to the FD method are approached with difference quotients. Specially, the spectral element (SE) method possesses a similar principle to the FVE and FE ones so as to be adapt to the calculated regions of the non-regular shapes. Hence, it is more popular than the FE and FVE FD methods and has proverbially been applied for solving the various PDEs such as the hyperbolic and parabolic along with hydromechanics equations (see [11][12][13][14][15]).
Though the reduced-order extrapolating (SECN) method of the 2D unsteady conduction-convection problems to the vorticity and stream functions has been developed in [16], the SECN method has not been minutely developed. Specially, there have been no theoretic proofs as regards the existence along with stability as well as error estimates to the SECN solutions. Therefore, in Sect. 2, we firstly intend to set up a semi-discretized CN scheme as a function of time with second-order temporal precision to the 2D unsteady conduction-convection problems to the vorticity and stream functions, as well as a proof of the error estimates to the semi-discretized CN solutions. Afterwards, in Sect. 3, we intend to build the fully discretized SECN model of the 2D unsteady conduction-convection problems to the vorticity and stream functions, as well as prove the existence along with stability together with error estimates to the SECN solutions. In the end, in Sects. 4 and 5, we intend to pose a set of numeric experiments to verify the validity to the obtained theoretic consequences and give the primary conclusions and discussion, respectively.
What is noteworthy is that the SECN model of the 2D unsteady conduction-convection problems to the vorticity and stream functions is not only spilt into three sets of relatively linearly independent equations, but also that it possesses the second-order precision as a function of time. Specially, it is able to avoid the restriction for Babuska-Brezzi's stability conditions to spectral subspaces so as to be able to easily seek the SECN solutions, which is different from the previous other SE methods as stated above. As a consequence, the SECN model is fully distinguished from the spectral ones (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21]) and is a development or a supplement to the previous ones.

The generalized solution and semi-discrete solution as a function of time
Thanks to the connectivity and boundedness of Ω and ∂ x u + ∂ y v = 0, there is only a stream function θ fulfilling u = ∂ y θ and v = -∂ x θ . In additional, there is a vorticity function meeting = ∂v/∂x -∂u/∂y =θ .
Thereupon, Problem 1 may be turned into the next systems of equations: The Sobolev spaces along with norms adopted in the following are normative (see [1,22]). Set V = H 1 0 (Ω). Using the Green formula, we may gain the next weak format.
Let M > 0 stand for an integer, let t = TM -1 stand for the temporal step, let n (x, y), θ n (x, y), and Q n (x, y) stand, respectively, for the approximations of (t, x, y), θ (x, y, t), and Q(t, x, y) at t n = n t, as well as letφ = (ϕ n + ϕ n-1 )/2. If ∂ t and ∂ t Q are, respectively, approximated with ( nn-1 )/ t and (Q n -Q n-1 )/ t, then the semi-discretized CN scheme as a function of time with the second-order temporal accuracy is built in the following.
Problem 3 has the next consequence.
Proof (1) The existence along with uniqueness to solution of Problem 3 Firstly, it is easily seen that the bilinear functional a(θ , ϕ) to the left hand side in (19) is bounded and coercive in V × V for given n-1 ∈ V (1 ≤ n ≤ M + 1). Thus, from Lax-Milgram's theorem (see [1]) we conclude that Eq. (19) possesses only a series of solutions is the bilinear functional andF(ϕ) is the linear functional. By (9) and the Hölder inequality we havê . Therefore, the bilinear functionalÂ(·, ·) is bounded and coercive on V × V for obtained θ n-1 ∈ V . Moreover, by (9) and the Hölder inequality we havê Therefore, the linear functionF(ϕ) is bounded in V for the known θ n-1 and Q n-1 . Consequently, by the Lax-Milgram theorem (see [1]) we may assert that Eq. (21) possesses only a series of solutions {Q n } M n=1 ⊂ V for the known θ n-1 and Q n-1 .
Remark 1 The inequalities (22) and (23) to Theorem 2 signify that the sequence to solutions for Problem 3 is stable and convergent, respectively.

The SECN method for 2D unsteady conduction-convection problems
Let N stand for the quasi-uniform quadrangle partition forΩ. A spectral element space is defined as where N stands for the number of quadrangles and P 1 (K j ) is defined by the following: stands for an invertible mapping from the reference quadrangleK = [-1, 1] × [-1, 1] to K j ∈ N , and (x ij , y ij ) and (ξ i , η i ) are, respectively, the vertices of K j andK .
Let R N : Note that when N is the quasi-uniform quadrangle partition to Ω, the number of nodes equals the number of quadrangles (see [1]). Hence, R N shows the next consequence (see [7]).

Theorem 3 ∀ϕ ∈ H q (Ω) (m ≥ 2) meets
where C > 0 stands for a generic constant as well as N also stands for the number of nodes in N .
With the spectral element space, the SECN model is built in the following.
where 0 N = R N 0 and Q 0 N = R N Q 0 .
Problem 4 possesses the next result as regards existence, convergence, and stability.

Numerical examples
Here, we provide a set of experiments to check the correctness of the theoretical consequences.
Let the computational region Ω be a channel with a total length of 20 and a width of 6 that has two identical rectangular protrusions with a length of 4 and a width of 2 at the top and at the bottom (see Fig. 1). When the quadrilateral elements in N are the squares about edge length x = y = 0.01, N = 3 × 136 × 10 4 . In addition to the outflow velocity u(t, x, y) = u(20 -1/M, y, t) (20 -1/M ≤ x ≤ 20, 2 ≤ y ≤ 8, 0 ≤ t ≤ T) on the right boundary as well as the inflow velocity (u, v) = (0.1(y -2)(8y) sin 2πt, 0) (x = 0, 2 ≤ y ≤ 8) on the left boundary, the other boundary and initial values are chosen as 0. The temporal step t = 0.0001. In the case, the theoretic errors reach O (10 -8 ). Using the SECN model (Problem 4), we seek the SECN solutions at t = 4 and 8, painted in Figs. 2 to 5, respectively. The numerical test results are very ideal.
When 0 ≤ t ≤ 8, the errors of velocity and energy solutions are approximately estimated by u n-1 Nu n N 0 + v n-1 Nv n N 0 and Q n-1 N -Q n N 0 (1 ≤ n ≤ 80,000), painted in Figs. 6 and 7, respectively, which also accord with the theoretic consequences since two types of errors do not exceed O (10 -8 ). This signifies that the SECN method is reliable and valid for settling the 2D unsteady conduction-convection problems to the vorticity and stream functions.

Conclusions and discussions
Hereto, we have built the time semi-discretized CN and fully discretized SECN models of the 2D unsteady conduction-convection problems to vorticity and stream functions and analyzed the existence, convergence, and stability to the time semi-discretized CN along with SECN solutions, respectively. We have also posed a set of numeric experiments to verify the reliability and validity to the SECN method and to verify that the numeric consequences accord with the theoretic ones.
Though we here only dealt with the 2D unsteady conduction-convection problems to the vorticity and stream functions, the SECN method may be popularized to the threedimensional unsteady conduction-convection problems or more complicated flow dynamics problems, even to be used for the more complicated actual engineering computations. Thereupon, the SECN method shows an extensive prospect as regards applications.