Characteristic Methods for Fokker-Planck and Fermi Pencil Beam Equations

Paper i 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 $\{\mathcal{V}_{n+1}\}$.