In the KiT-RT, we employ the finite volume method (FVM) to model and compute the particle evolutions. It's a generic method for conservation laws. Consider the following PDE,
\frac{\partial \mathbf{u}}{\partial t}+\nabla \cdot \mathbf{f}(\mathbf{u})=\mathbf{0}
Here, \mathbf{u} represents any vector of states and \mathbf{f} represents the corresponding flux tensor. To solve the equation numerically, we can sub-divide the spatial domain into finite cells. For a particular cell i, we take the volume integral over the total volume of the cell, which gives,
\int_{v_{i}} \frac{\partial \mathbf{u}}{\partial t} d v+\int_{v_{i}} \nabla \cdot \mathbf{f}(\mathbf{u}) d v=\mathbf{0}.
On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields
v_{i} \frac{d \overline{\mathbf{u}}_{i}}{d t}+\oint_{S_{i}} \mathbf{f}(\mathbf{u}) \cdot \mathbf{n} d S=\mathbf{0},
where S_i represents the total surface area of the cell and \mathbf n is a unit vector normal to the surface and pointing outward. The equivalent formulation results
\frac{d \overline{\mathbf{u}}_{i}}{d t}+\frac{1}{v_{i}} \oint_{S_{i}} \mathbf{f}(\mathbf{u}) \cdot \mathbf{n} d S=\mathbf{0}.