Comparative Study of the Adomian Decomposition Method and Alternating Direction Implicit (ADI) for the Resolution of the Problems of Advection-Diffusion-Reaction

In this paper, we use the Adomian decomposition 
method (ADM), the finite differences method and the Alternating Direction 
Implicit method to estimate the advantages and the weakness of the above 
methods. For it, we make a numerical simulation of the different solutions 
constructed with these methods and compare the error investigated case.


Introduction
The advection-diffusion-reaction equation is a combination of the advection, ( ) , f x t the source function and 1, 2, 3, m =  . The equations of advection-diffusion-reaction are used to describe the problems of transport (the transport of pollutants, flows in the conduits, the modeling of the air pollution, etc.) [1] [2] [3].

The Adomian Decomposition Method
Suppose that we need to solve the following equation: Fu f = (2) in a real Hilbert space H, where is a linear or a nonlinear operator, f H ∈ and u is the unknown function. The principle of the ADM is based on the decomposition of the operator F in the following form [2] [4] [5] [6] F L R N = + + where L R + is a linear part, N nonlinear operator. We suppose that L is an invertible operator in the sense of Adomian with 1 L − as inverse.
Using that decomposition, Equation (2) is equivalent to where θ verifies 0 Lθ = . (4) is called the Adomian's fundamental equation or Adomian's canonical form. We look for the solution of (2) in the following series expansion form where λ is a parameter used by "convenience". Thus Equation (4) can be rewritten as follows: We suppose that the series In practice it is often difficult to calculate all the terms of an Adomian series solution, so we approach the series solution by the truncated series: Let's consider the following advection-diffusion: [7] [8] [9] ( ) ( ) ( ) and , e sin π 1 1 , e sin π πe e sin π 2 2 , , π π 5000 2 50 5000 2 50 1 10000π 484 2500π 121 n π 11 121 1 550π exp π 50 5000 2 cos π 2500π 121 The exact solution of the problem (9) by the Adomian decomposition method is: and we remark that ( ) 1, 0. u t =

Resolution by ADI Method
Grid of the field { }

Semi-Discretization in Relation to the Space
Let's consider the following equation: where f is a function that depends on x (26) and (27) give The discretisation of the Equation (24) is: that is equivalent to ( ) 2  2  2  2  2  4   1  0  12 12 Let's note 1 12 and we obtain ( ) that is equivalent to ( ) Finally the semi-discretisation of Equation (24) is: One obtains the following diagram of the finite differences In the matrix form (44) becomes that is equivalent to: and A. Bitsindou et al.
The numerical solution is represented for 1 . 50 h = In the following, we give the numerical simulation of the approximate solution, the exact solution and the error between these two solutions in three-dimensional space.
On Figure 1(1) and Figure 1(2) we have the respective curves of diffusion of the exact and of the solution approached.

Resolution by the Finite Difference Method
Discretisation of the space The discretised problem is: Journal of Applied Mathematics and Physics point.

Conclusion
In this paper, two examples have been investigated. In the first example, we got the exact solution, using the ADM and the comparison has been done with the numerical solution obtained by ADI method. We find that the solution by the ADI method approaches the exact solution quite well, and the error is consisted between 0 and 0.005. In the second example, using the ADM, we got the approached solution; we remark that, the error between the solution gotten by the ADM and the one gotten by the finite differences method is very minimal.