CENTRAL FINITE-DIFFERENCE OF NUMERICAL SOLUTION FOR THREE-DIMENSIONAL ATMOSPHERIC TRANSPORT EQUATION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The energy transport equation is fundamental for the meteorological analysis; in this work, we analyze this equation in three dimensions using the methods of central finite differences; the analysis of convergence, consistency, and stability of the scheme shows a strong dependence of space and temporal variables. In conclusion, with the central finite differences was possible to predict the three-dimensional dynamics of the temperature.


INTRODUCTION
The energy transport equation is fundamental in applications as meteorology, aerodynamics, oceanography, hydrology, and engineering [1].
Taking as the object of study the tree-dimensional transport equation for meteorology, we made the numerical interpretation of atmospheric dynamics. The energy transport equation has peculiar characteristics that are difficult to find the solution for different methods, its a consequence of advective term v ∂ φ ∂ x , because it explains the inertia of the model [7,10].
Therefore, we use the finite difference method by the simplicity in the numerical implementation for small and large scale in the domain. The advective term of the equation will be discretized using central finite differences scheme [4,5].
The model to describe the three-dimensional energy atmospheric transport is: where θ = θ (x, y, z,t) is the temperature variable , V = (v 1 , v 2 , v 3 ) is the velocity field and α i with i = 1, 3 represent the thermal diffusion coefficient, those in that work will be constants [4].
The boundary condition ∂ Ω are Dirichlet, in each face of the domain, the temperature G take values represented by f i with i = 1, 6.
The work is structured in different section as: section 2 we show the finite difference method for the model, section 3 we made the analysis of convergence, consistence and stability, section 4 we present the result and, some numerical experiment, section 5 and 6 we present the discussion and conclusion, and the references.

FINITE DIFFERENCE METHOD
The derivative of variables in the equation are replaced by finite differences for the dependent and independent variables, that process is called discretization and getting an algebraic system equation [8,6].
The discretization of the model, consists of the discretization of the domain, variables, and the equation. The domain in divided by number of finite number of rectangular sub-domains with dimension given by h x > 0, h y > 0, h z > 0, and h t > 0 for the time t > 0.
The sub-domains in some cases have different longitude, and the mesh is defined by: Given h x , h y , h z , h t positive numbers, a mesh is a set of points of the form (x i , y j , z k ,t n ) = (ih x , jh y , kh z , nh t ), called nodes, with i,j,k,n non negative integer numbers.
The solution is given at the nodes ξ i, j,k = (ih x , jh y , kh z ) ∈ R 3 of the discrete domain.

Discretization of variables.
The variables θ of the problem are discretized using the definition 2.1, for each node (ih x , jh y , kh z , nh t ) a value θ (ih x , jh y , kh z , nh t ) is designed and represented by (3) θ n i, j,k = θ (ih x , jh y , kh z , nh t ).
Definition 2.2. The discrete function φ is defined over a mesh and each point (x i , y j , z k ,t n ) have a real number φ n i, j,k .
The smooth function φ over Ω × R + is discretized on the mesh defined at 2.1. Taking φ n i, j,k := φ (x i , y j , z k ,t n ) in particular the solution θ of the problem (PVIC) given in (3) is discretized by θ n i, j,k = θ (x i , y j , z k ,t n ).
As the solution θ is unknown, so the discrete solution θ n i, j,k is approximated by a discrete function φ n i, j,k such that θ n i, j,k ≈ φ n i, j,k in each node of the mesh. The {φ n i, j,k } and {φ n+1 i, j,k } denote a discrete function at level n and n + 1 respectively, where : also, the discretized initial condition is and the discretized border conditions on the mesh i, j, k ∈ Z + are designed by 2.3. Discretization of the equation. The approximation of the terms advection and diffusion of the equation (1) using Taylor's series truncated after the first and second term, we get the finite differences for the first ans second derivative at point (i, j, k) ∈ Z + , of the form The discretized equation with central differences given in (6) for To simplified the expression, we defined the discrete operators S ± for = 1, 2, 3 as forward and backward displacement at = 1 for x, = 2 for y, and = 3 for z. For example, at x direction the forward operator is S 1+ and backward S 1− , those applied to discrete function φ t i, j,k , we have: S 1± φ n i, j,k := φ n i±1, j,k such that S 1± φ = {φ n i±1, j,k }. Given P differential and continuous operator and central finite differential scheme (7), we get the discrete operator as: is a local truncated error at the time n κ · We proof the consistency of the discretized equation (6), for this, we denote A, B and P differential and continuous operators as Strikwerda [11] , defined by: applying those operators to a smooth function θ (ξ ,t) for ξ ∈ R 3 , we have and the equation (6) is written as (10) P := A + B, and Pθ = Aθ + Bθ = 0.
Then, with the progressive Taylor's formula at the time, we have: and replacing the notation (3) in the Taylor's expansion (11), we have The Taylor's formula with spatial variable is: using the reduced notation θ n i±1, j,k = θ (x i ± h x , y j , z k ,t n ) in the Taylor's expansion (13), we have Using (12), (14) y (9) in the equation (6), the discrete operator (8) is of the equation (10) and (15), we have This form, the scheme (6) is consistent with (10), on the right hand side of (16), we have that the truncate local error goes to zero, when h x , h y , h k , h t → 0 goes to zero, that is : Therefore, we have contrast the definition 3.1.

Stability.
To proof the stability of the scheme (6), the Von Neumann criteria should be satisfied, for that, we have the follow definitions . if and only if the solution exist and is unique and it is dependent of initial conditions, that is, there are positives constants h x 0 , h y 0 , h z 0 , h t 0 and C > 0, t > 0, α > 0 with n ≥ 0, such that Definition 3.3. Given φ (ξ , n) a discrete function defined on Z, then the discrete Fourier transform φ (ξ , n) with n ∈ Z + , is denoted by φ (ξ , n) and defined by: We proceed to find the amplification factor that is necessary for the Von Neumann criteria, for that, we use the results of Strikwerda [11], and Rubio [5], there exist a biunivocal relation between discrete space 2 (Z) and the space this relation is called Parserval's relation [11], [12].
With the notation (23) and (24), the equation (22) is written as Finally, the equation (25) is expressed as Replacing (21) where the spectral radio ρ(β 1 , This equation is called amplification factor and shows the amplitude of the general solution for central finite differences scheme. Von Neumman criteria. Using Fourier analysis, we have the necessary and sufficient conditions for the stability of the finite differences scheme, this is called Von Neumann analysis. For this analysis we have the following theorems Remark. The central finite differences (6) satisfy the Von Neumann criteria as in Gary [13] , if Where κ > 0 is the step of the time and ρ(β 1 , β 2 , β 3 , h x , h y , h z , h t ) denote a spectral radio of amplification factor (29).
the stability condition (30) is replaced by a stability condition of the form In the next theorem, we use the Von Neumann criteria above to proof the stability of the central finite differences (6).
Theorem 3.1. The central finite differences (6) is stable with the norm 2 , if and only if, it satisfy the criteria (3.2) of Von Neumann.
The theorem (3.1) shows that only is necessary the amplification factor to determine the stability of central finite differences scheme (6).
Proof ⇐) If the Von Neumann criteria (3.2) is satisfied, the central finite differences scheme (6) is stable with the norm 2 .
Applying Fourier transform to the explicit scheme (6), we get simplifying this expression, we write as Using the Parseval's relation (17) of greater dimension, we have in effect φ n i, j,k replacing (33) in the Parselval's (35), we get Applying the criteria (3.2), such that |ρ(β 1 , whit the result 34, we have Given T > 0 sufficient greater such that nκ ≤ T , then n ≤ T κ , and 1 +Cκ ≤ e Cκ with C > 0 and κ ≤ e 2CT , and replacing in the inequality (37), we have Of the (37) and (38) the central finite differences (6)   Taking the initial condition φ 0 i and building a function such that Observe, that φ 0 = 1.

Convergence.
Definition 3.4. The central finite differences scheme of the equation (1) is convergent with some norm · if the partial differential solution θ (ξ ,t), and the solution of the finite differences scheme φ n i, j,k , such that φ 0 i, j,k converge to θ 0 (ξ ) when ih x , jh y , kh z , nh t converge to x, y, z,t respectively, then φ n i, j,k converge to θ (ξ ,t) when (ih x , jh y , kh z , nh t ) converge to (x, y, z,t) when h x , h y , h z , h t converge to 0; with ξ ∈ R 3 y t > 0.  (6) is convergent with the norm 2 , with the solution of partial differential equation (1), represented by θ (ξ ,t),

Proof
Given h x = h y = h z = h t = h the central finite differences (6) is written as Given θ (ξ ,t) a solution of equation (10), as the central differences is consistent with order of precision (1, 2), of the result (16), and considering h x = h y = h z = h t = h we have that that is and ϕ n i, j,k = θ n i, j,k − φ n i, j,k the error at n-th time step, The equation (50) and the equation (51) is expressed by using the equation (52), we have that and with the conclusion 3.2 of stability that inequality is written by Replacing the equation (51) in the inequality (53) have when T j ∼ t j = jh, we have that by the result of (55) the central finite differences is convergent with order (1,2).
We conclude that the criteria of consistence, convergence and stability are established.

Application 3.
The equation (7), for j = k = 0 and α l = v n = 0 with l = 2, 3 y n = 2, 3; is written as: For the case 1D of the equation (29) and the inequality (57), we have: , and taking (46), we have The transport equation (59) for the case 1D with h x = 0.2, h t = 0.1, allow us compare the central scheme getting for Fletcher [15] with boundary condition φ (−2,t) = 1 and φ (−2,t) = 0 for all t ∈ [0, 1] and initial condition under these conditions, an exact solution with methods of separation of variables, is where θ is the exact solution [15] of the equations (59) showed in the 5 and 6 and represent the change of temperature at 0s, 0.5s to 1s.

DISCUSSION
This research present the analysis of convergence, consistence and stability of central finite The equation (7) is stable if the constants of velocity are bounded by the equation of stability (46) in particular by (47).