Numerical simulation of the advection-diffusion-reaction equation using finite difference and operator splitting methods: Application on the 1D transport problem of contaminant in saturated porous media

. The combined advection-diffusion-reaction (ADR) equation, which describe the transport problem of a contaminant in porous medium, does not generally admit an analytical solution. In general, when solving the ADR equation, the numerical methods (such as finite differences, finite elements, splitting), for most practical problems, the ADR equation is too difficult to solve analytically. The finite difference method is the oldest and most commonly used method for the numerical solution of this kind of equation. Although newer techniques, such as those based on finite elements and splitting are appropriate for the solution of equilibrium-type problems, the finite difference remains the most appropriate for the solution of time-dependent phenomena. The transport of a contaminant can also be written by the ADR equation; hence, our objective is to choose the most efficient method to study the 1D transport problem of a contaminant and its evolution in a porous medium. In this work, we will simulate the ADR equation using two different methods: those of finite difference and splitting ones. The numerical result will be compared with the analytical solution in order to discuss the stability and the convergence of each of them using those two different methods. In the end, we will show that the splitting technical method is more efficient for solving this kind of problems in comparison with the finite difference method despite the fact that the latter is the most widely used by researchers. The validation of the efficiency of this method, implemented in this simulation, is tested on a 1D-transport problem of contaminant in a saturated porous medium.

The combination of the advection, diffusion and reaction equations describes physical phenomena, matter or other physical quantities that are transferred into a physical system because of these three processes: advection, diffusion and reaction.
This combination is named, advection-diffusion-reaction equation, and can be written in the following form [2]: In general, these equations do not admit any analytical solution except in very simplified cases. This is why a recourse to numerical solving methods is necessary.
The most commonly used methods for the numerical solution of this type of equation is the finite difference as mentioned above [6], [11], [9]- [8]. As there is also the splitting method, also called the operator splitting method, which is a powerful method for the numerical study of complex models [2], [9], [11]. The idea of operator splitting methods based on dividing a complex problem In this paper, we will introduce two different methods of dividing advection-diffusion-reaction equations. Those of the finite difference approach, which is a common technique for solving partial differential equations with an approximate solution. It consists in solving a system of relations (numerical scheme) linking the values of unknown functions at certain points sufficiently close to each other [3]. And the Splitting method, which is a method that replaces a single scheme to solve a complicated PDE (Partial differential equation) with a sequence of simpler schemes that solve the linked PDEs and solve together the original PDE (Partial differential equation) (up to a specified order of precision). Finally, we will compare between these two methods in terms of stability and convergence.
This document is organized as follows. In the next section, we will discuss the splitting, and finite difference methods used to solve our ADR equation. The third section will tackle the numerical result of our example.
We will afterwards compare between the exact and numerical solution.

METHODOLOGY
In this work, we will focus on two different methods of mathematical representation, that of splitting and that of the finite difference, discussing the stability of each method and spotting the most efficient of them in the following sections.

Mathematical description of the problem
As indicated in section 1, the problem to be solved is that of the numerical study of the advection-diffusionreaction problem, with two different techniques namely, finite difference and splitting methods. Which can be written in the following form, as mentioned above: Where C is the concentration of the mass transfer species, D is the diffusivity constant for mass; V is the component of the velocity, λ and is the decomposition rate, noting the temporal and spatial step sizes by ∆t and ∆x, respectively.

The finite difference method
We discretize the advection-diffusion-reaction equation using a first-order forward difference for the time derivative, and a symmetric (second-order) difference for the space derivative, and the equation can be written as: [1].
[8]The number of Current Cr for advection is calculated as V*∆t/∆x, and the stability for diffusion R is calculated as D*∆t/∆x2.

The splitting method
Using operator splitting method, our ADR equation will be divided into two sub-problems, which will be treated by the finite difference method. This mean that each part will be treated individually [14], [26]. The principle of this method is that, the first equation Using the previous finite difference schemes, the corresponding division scheme for solving the 1D advection-diffusion-reaction equation by the Splitting method is [4], [5], [6]: Noting that the equation is stable for Cr≤1 and R≤1/2 and As already mentioned, our sub-problem will be treated by the finite difference method, where our algorithm becomes as follows:

APPLICATION
In this part, a one-dimensional advection-diffusionreaction equation will be solved with the operator splitting method and the finite difference one. The result will be examined for different value of λ (the decomposition rate). Then it will be compared to the exact solution. In addition, the accuracy of the methods will be evaluated by calculating error standards. This kind of error is calculated as follows [4]: In our case, the data are as follows [4], [15]:  The calculations in Table 1 give the error value for the two different methods, with different values of λ:

EXAMPLE OF 1D CONTAMINANT TRANSPORT
In our work, we will deal with the case of the 1D transport of a contaminant within a soil-groundwater system, by changing the value of λ. To prove the results found before.
The following is the exact solution to this problem: [14] = �( 2 * ) 2 * ( + �√ 2 + 4 * * � * √4 * * + − � � 2 * � 2 * 2 * ( ( − �√ 2 + 4 * * � * √ * * With an initial, condition equal to: In addition boundary condition equal to : Hence, we will apply these two methods to our contaminant transport case, to confirm the efficiency of the Splitting method.    We demonstrated that the inaccuracy of the two methods is quite modest for very small values of λ during the simulation. It was also discovered that as the value of λ grows larger, the numerical scheme obtained using the finite difference method becomes unstable. The numerical scheme derived using the Splitting technique, on the other hand, remains stable, if not more so, for small values of λ, but converges to the precise solution for extremely small values of λ. As can be seen, the numerical and precise solutions for the Splitting approach are in agreement.
As a result, we can conclude that this method outperforms the finite difference method. For that reason, the splitting method is more useful for numerically simulating this type of transport as well as studying the movement of a contaminant in any medium.