TY - JOUR T1 - A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy JO - Communications in Computational Physics VL - 4 SP - 1097 EP - 1126 PY - 2018 DA - 2018/12 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2017-0245 UR - https://global-sci.org/intro/article_detail/cicp/12892.html KW - Implicit scheme, relaxation scheme, $M_1$ model, radiotherapy dose computation. AB -

Numerical schemes for systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider a system modeling the steady transport of photons and electrons in the field of radiotherapy. Naive discretizations of such a system are commonly constrained by a very restrictive CFL condition. This issue is circumvented by constructing an implicit scheme based on a relaxation approach.
We use an entropy-based moment model, namely the $M_1$ model. Such a system of equations possesses the non-linear flux terms of a hyperbolic system but no time derivative. 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, the discrete equations are non-linear, and we propose a numerical method to solve such non-linear systems.
Our approach is tested on academic and practical cases in 1D, 2D, and 3D and it is shown to require significantly less computational power than reference methods.