On the stability of characteristic schemes for the Fermi equation

We study characteristic schemes for a model problem for the Fermi pencil beam equation. The objective is twofold: (i) To design efficient and accurate numerical schemes based on the principle of solving a particle transport problem, exactly, on each collision free spatial segment combined with a projection on each collision site, from a pre collision angle and energy coordinates (AE) to a post collision AE coordinates. (ii) To prove stability and derive a posteriori error estimates in $L_2$ and the maximum norms