Characteristic Methods for Fokker-Planck and Fermi Pencil Beam Equations

Paper in proceeding, 1999

We design an efficient and accurate numerical method for the pencil beam equations based on the principle of solving i) An {\sl exact transport} problem on each collision free spatial segment: Let $x$ be the beam penetration direction, $\{x_{n}\}$ an increasing sequence of discrete points indicating collision sites and $\{\mathcal{V}_{n}\}$ be a corresponding sequence of piecewise polynomial spaces on meshes $\{\mathcal{T}_{n}\}$ on the transversal variable $x_{\perp}$. Then given the approximate solution $J^{h,n}\in\mathcal{V}_{n}$ at the collision site $x_{n}$ solve the pencil beam equation exactly on the collision free interval $(x_{n},x_{n+1})$ with the data $J^{h,n}$ to give the solution $J^{h,n+1}_{-}$ at the next collision site $x_{n+1}$, before the collision. ii) A {\sl projection}: Compute $J^{h,n+1}=\mathcal{P}_{n+1}J^{h,n+1}_{-}$, with $\mathcal{P}_{n+1}$ being a projection into $\{\mathcal{V}_{n+1}\}$.