A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes

•A new numerical scheme solving the transport equation is giving for 2D geometries.•This scheme is based on the PN method and the discontinuous finite elements.•The method deal with unstructured, curved and non-conformal meshes.•The method is not limited neither in PN order nor in degrees of polynomials.•The coefficients matrices are evaluated exactly for arbitrarily-shaped regions. This document presents a new numerical scheme dealing with the Boltzmann transport equation. This scheme is based on the expansion of the angular flux in a truncated spherical harmonics function and the discontinuous finite element method for the spatial variable. The advantage of this scheme lies in the fact that we can deal with unstructured, non-conformal and curved meshes. Indeed, it is possible to deal with distorted regions whose boundary is constituted by edges that can be either line segments or circular arcs or circles. In this document, we detail the derivation of the method for 2D geometries. However, the generalization to 2D extruded geometries is trivial.