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Relaxation Schemes for the M1 Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation

Published online by Cambridge University Press:  15 January 2016

Teddy Pichard
Affiliation:
Centre Lasers Intenses et Applications, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France Mathematics division, Center for Computational Engineering Science, Rheinisch-Westfälische Technische Hochschule, Schinkelstrasse 2, Aachen, 52062, Germany
Denise Aregba-Driollet
Affiliation:
Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France
Stéphane Brull
Affiliation:
Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France
Bruno Dubroca
Affiliation:
Centre Lasers Intenses et Applications, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France
Martin Frank
Affiliation:
Mathematics division, Center for Computational Engineering Science, Rheinisch-Westfälische Technische Hochschule, Schinkelstrasse 2, Aachen, 52062, Germany
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Abstract

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 M1 system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities. Additionally, the flux term of the M1 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 M1 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.

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Research Article
Copyright
Copyright © Global-Science Press 2016 

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Relaxation Schemes for the M1 Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation
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