A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy

Abstract

Numerical schemes for the systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider here a system modeling the steady transport of photons and electrons in the field of radiotherapy, which leads to very stiff CFL conditions at the discrete level. We circumvent this issue by constructing an implicit scheme based on a relaxation approach. The physics is modeled by an entropy-based moment system, namely the M 1 model. This model is non-linear, possesses potentially no hyperbolic operator. It is furthermore only valid under a condition 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 potentially 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.

A moment closure based on a projection on the boundary of the realizability domain: 1D case

“…• This property is commonly used when constructing numerical schemes for moment equations in order to prove that such schemes preserve the realizability property from one step to another (see e.g. [3,52,53]).…”

Towards a Multigrid Method for the M1 Model for Radiative Transfer

“…In this work, we only consider the hyperbolic system, without source terms, and we present an implicit scheme that preserves the admissible states. [16,17] suggests solving the nonlinear system with a Jacobi method. Because this method is iterative, its convergence rate can be improved thanks to multigrid acceleration (e.g., [18,19]).…”