## 2 Materials and Methods
### Classical Model of the Equation  
Calling \(C(t,x,y)\) the concentration of the pollutant in the medium under study at the point \((x,y)\) in the plane and at the instant \(t\), for the diffusion-advection equation for \(C\), we obtain the following equation  
\[\frac{\partial C}{\partial t}+\{\text{div(flow)}\}+\{\text{decay}\}=\{\text{ source}\} \tag{1}\]  
More specifically, the pollutant flow (modeled in the equation by div(flow)) can be divided into two parts: diffusive flow (effective diffusion), which contains the effects of molecular and turbulent diffusion, and advective flow (transport), which is responsible for the macroscopic movements of the medium [7]:  
\[\overrightarrow{J}=\overrightarrow{J_{1}}+\overrightarrow{J_{2}}\]  
with \(\overrightarrow{J}\) representing the flow, \(\overrightarrow{J_{1}}\) the diffusive flow, and \(\overrightarrow{J_{2}}\) the advective flow. Making the separation between diffusive flow and advective flow, the Equation (1) can be rewritten as follows [7; 11; 12; 13; 14]:  
\[\frac{\partial C}{\partial t}=\{\text{diffusion}\}-\{\text{transport}\}-\{ \text{decay}\}+\{\text{source}\}. \tag{2}\]  
As the diffusive flow (\(\overrightarrow{J_{1}}\)) fulfills [PERSON]'s law [15], it is possible to state that matter tends to spread, moving from places of greater concentration to areas of lower concentration. It is proportional to the pollutant density gradient (\(\overrightarrow{J_{1}}=-\alpha\
abla C\)). The advective flow (\(\overrightarrow{J_{2}}\)) is the movement caused by external agents, such as the medium's velocity field. In this case, it is given by the product of velocity (\(\overrightarrow{v}\)) by concentration (\(\overrightarrow{J_{2}}=\overrightarrow{v}C\)) [7; 16]. The system's loss of mass, determined by the changes the substance undergoes over time due to the most diverse factors, determines the decay (or degradation). In many situations, we assume this loss is linearly proportional to the concentration. The means by which the substance is introduced (or removed) from the system are sources (or sinks) [7; 16]. It is important to note that the gradient, divergence, and Laplacian operators in the following descriptions only relate to the spatial variables, not the temporal variable. Thus, for the classical modeling of these phenomena in a space of arbitrary size, we obtain the following equations [15; 17; 18]:  
\[ \begin{array}{rcl}\{\text{diffusion}\}&=&\
abla\cdot(-\alpha\
abla C)\\ \{\text{transport}\}&=&\
abla\cdot\left(\overrightarrow{v}C\right)\\ \{\text{decay}\}&=&\sigma C\end{array} \]  
Thus, the equation that models the phenomenon of dispersion for a given substance in a two-dimensional domain is given by  
\[\frac{\partial C}{\partial t}=\
abla\cdot(\alpha\
abla C)-\
abla\cdot\left( \overrightarrow{v}C\right)-\sigma C+f,\;\;\;(x,y)\in\Omega\subset\mathbb{R}^{ 2},\;\;\;t\in(0,T]\subset\mathbb{R},\]  
being  
* \(\alpha=\alpha(t,x,y)\) the effective diffusion coefficient in the medium;
* \(\overrightarrow{v}=\overrightarrow{v}(t,x,y)=(v_{1}(t,x,y),v_{2}(t,x,y))\) the velocity field in the middle;
* \(\sigma=\sigma(t,x,y)\) the total decay coefficient in the medium;
* \(f=f(t,x,y)\) is the source term. The initial condition is given by \(C(0,x,y)=C_{0}(x,y),\ (x,y)\in\Omega\subset\mathbb{R}^{2}\). The following boundary conditions are considered, whose graphical representation are recorded in Figure 1:  
1. Non-homogeneous von Neumann-type for pollutant entry (in red) \(\alpha\left.\frac{\partial C}{\partial\eta}\right|_{\Gamma_{q}}=\theta(t,x,y)\), being \(\theta\) a given function and \(q=1\);
2. Homogeneous von Neumann-type for frontier without loss of pollutants (in blue) \(\alpha\left.\frac{\partial C}{\partial\eta}\right|_{\Gamma_{q}}=0\), being \(q=2\). Thus, the classical model of the diffusion-advection equation involves a diffusive term, an advective transport, a decay coefficient, and a source term. The diffusive term consists of the divergence of the product between the diffusion coefficient and the concentration gradient.