A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy (Communications in Computational Physics, 2018)
Numerical schemes for systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We considera system modeling the steady transport of photons and electrons in the field of radiotherapy. Naive discretizations of such a system are commonly constrained by avery restrictive CFL condition. This issue is circumvented by constructing an implicitscheme based on a relaxation approach.We use an entropy-based moment model, namely the $M_1$ model. Such a systemof equations possesses the non-linear flux terms of a hyperbolic system but no timederivative. The flux terms are well-defined only under a condition on the unknowns,called realizability, which corresponds to the positivity of an underlying kinetic distribution function.The present numerical approach is applicable to non-linear systems which possess no hyperbolic operator, and it preserves the realizability property. However, thediscrete equations are non-linear, and we propose a numerical method to solve suchnon-linear systems.Our approach is tested on academic and practical cases in 1D, 2D, and 3D and it isshown to require significantly less computational power than reference methods.