TY - JOUR
T1 - Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation
AU - Pichard , Teddy
AU - Aregba-Driollet , Denise
AU - Brull , Stéphane
AU - Dubroca , Bruno
AU - Frank , Martin
JO - Communications in Computational Physics
VL - 1
SP - 168
EP - 191
PY - 2018
DA - 2018/04
SN - 19
DO - http://doi.org/10.4208/cicp.121114.210415a
UR - https://global-sci.org/intro/article_detail/cicp/11084.html
KW - 
AB - <p style="text-align: justify;">Because of stability constraints, most numerical schemes applied to hyperbolic
systems of equations turn out to be costly when the flux term is multiplied by
some very large scalar. This problem emerges with the $M_1$ system of equations in
the field of radiotherapy when considering heterogeneous media with very disparate
densities. Additionally, the flux term of the $M_1$ system is non-linear, and in order for
the model to be well-posed the numerical solution needs to fulfill conditions called
realizability. In this paper, we propose a numerical method that overcomes the stability
constraint and preserves the realizability property. For this purpose, we relax the
$M_1$ system to obtain a linear flux term. Then we extend the stencil of the difference
quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation
example.</p>