An example of a linear non-stationary problem is the problem of a pollutant propagation simulation [1].
In this problem we calculate the scalar field of pollution concentration
\( {\cal R}^3 \ni (x,y,z)\times [0,T] \rightarrow c(x,y,z;t) \in {\cal R } \) such as
\( \frac{\partial c}{\partial t } + u \cdot \nabla c - \nabla \cdot (K \nabla c) = e \)
which, after rewriting, gives us
\( \frac{\partial c}{\partial t} + u_x\frac{\partial c}{\partial x}+u_y\frac{\partial c}{\partial y}+u_z\frac{\partial c}{\partial z} - \frac{\partial}{\partial x} \left( K_x \frac{\partial c}{\partial x}\right) -\frac{\partial}{\partial y} \left( K_y \frac{\partial c}{\partial y }\right) -\frac{\partial}{\partial z}\left( K_z \frac{\partial c}{\partial z } \right) \)
In the formula above, the first term \( \frac{\partial c}{\partial t } \) represents changes in the field of pollution concentration over time,
the second term
\( u_x\frac{\partial c}{\partial x}+u_y\frac{\partial c}{\partial y } +u_z\frac{\partial c}{\partial z } \) is a term of advection (also called convection) which means transport of pollutants by the wind, represented by a given vector field \( {\cal R}^3 \ni (x,y,z)\times [0,T] \rightarrow (u_x(x,y,z;t,,u_y(x,y,z;t);u_z(x,y,z;t))) \in {\cal R }^3 \), the third segment \( - \frac{\partial}{\partial x} \left( K_x \frac{\partial c}{\partial x} \right)-\frac{\partial}{\partial y} \left( K_y \frac{\partial c}{\partial y }\right) -\frac{\partial}{\partial z} \left( K_z \frac{\partial c}{\partial z } \right) \) means diffusion, i.e. spontaneous transport of pollutant particles on the principle of diffusion. All terms of the differential operator describing the contamination propagation process are linear, that is \( {\cal L}(c_1+c_2)={\cal L}(c_1)+{\cal L }(c_2) \). Thanks to this, we can build a system of linear equations and apply the solvers and algorithms described in the textbook.
The right-hand side indicates the emission sources, and it is positive where there is a pollution source (for example a chimney).
Diffusion coefficients
\( (K_x,K_y,K_z) \) are the diffusion velocity in the x, y and z directions.
In order to solve the advection-diffusion problem, we still need to determine what is happening on the boundary of the area. For example, we can assume the speed of the influx of pollutants at the edge of the area
\( \frac {\partial c} {\partial x}n_x+ \frac{\partial c}{\partial y }n_y+\frac{\partial c}{\partial z }n_z=g \) where \( (n_x,n_y,n_z) \) is the unit vector orthogonal to the edge. We also need to provide an initial condition (initial pollutant concentration)
\( c(x,0)=0 \).
Finally, we present two exemplary simulations, the first one showing the influx of pollutants from the west to the Krakowska Valley for the area obtained by generating a tetrahedral mesh based on topographic data from the NASA database, and the second one showing the dispersion of smoke from the chimney through the wind of variable direction.
They were counted with the codes described in the bibliographic articles
[2].