A semi-discrete Boltzmann equation based on a finite volume scheme

In this work we consider the classical non-linear Boltzmann equation, where the unknown is the distribution function $f$, which depends on the time $t$, the vector $\mathbf{x}$ (the position of a molecule) and its velocity $\mathbf{\xi}$. From the Boltzmann equation we derive a semi-discrete model, which consists in a set of partial differential equations in time and space. The new unknowns are two moments of the distribution function $f$ and, according to a finite volume scheme, integrals of $f$ with respect to the velocity $\mathbf{\xi}$, over bounded and open sets. We do not introduce any approximation for the partial derivatives with respect to $\mathbf{x}$ and the time $t$. We also propose the use of truncated octahedra as bounded domains of integration. This reduces both the computing complexity and the number of the constant numerical coefficients arising from the collision operator. All the numerical parameters can be obtained by means of numerical quadrature.