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

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  15 January 2016

Teddy Pichard ,
Denise Aregba-Driollet ,
Stéphane Brull ,
Bruno Dubroca  and
Martin Frank
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
*
*Corresponding author. Email addresses:pichard@celia.u-bordeaux1.fr (T. Pichard), denise.aregba@math.u-bordeaux1.fr (D. Aregba-Driollet), stephane.brull@math.u-bordeaux1.fr (S. Brull), bruno.dubroca@math.u-bordeaux1.fr (B. Dubroca), frank@mathcces.rwth-aachen.de (M. Frank)
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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.

Keywords

RadiotherapyMoments modelsrelaxation modelsmethod of characteristics

MSC classification

Secondary: 82C40: Kinetic theory of gases 35L65: Conservation laws 65M25: Method of characteristics
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 168 - 191
Copyright
Copyright © Global-Science Press 2016 

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