Paper in proceedings, 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 $\{\mathcal{V}_{n+1}\}$.

Fokker-Planck equation

Finite element methods

Characteristic methods

Exact transport+Projection

Department of Mathematics

University of Gothenburg

2854-2849 (ISSN)

Vol. 2 205-212Computational Mathematics

285428499-2