Relaxation schemes for the M 1 model with space-dependent flux: application to radiotherapy dose calculation

Teddy Pichard (1) , Denise Aregba-Driollet (2) , Stéphane Brull (3) , Bruno Dubroca (2) , Martin Frank

1 CELIA - Centre d'Etudes Lasers Intenses et Applications
2 IMB - Institut de Mathématiques de Bordeaux
3 IMT - Institut de Mathématiques de Toulouse UMR5219

Because of stability constraints, most numerical schemes applied to hyper-bolic 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.

Mots clés: Radiotherapy; Moments models; relaxation models; method of characteristics

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Teddy Pichard, Denise Aregba-Driollet, Stéphane Brull, Bruno Dubroca, Martin Frank. Relaxation schemes for the M 1 model with space-dependent flux: application to radiotherapy dose calculation. Communications in Computational Physics, 2016, 19 (1), pp.168-191. ⟨10.4208/cicp.121114.210415a⟩. ⟨hal-01934314⟩