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Volume 19, Issue 1
Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation

Teddy Pichard, Denise Aregba-Driollet, Stéphane Brull, Bruno Dubroca & Martin Frank

DOI: 10.4208/cicp.121114.210415a

Commun. Comput. Phys., 19 (2016), pp. 168-191.

Published online: 2018-04

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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 $M_1$ 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.

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@Article{CiCP-19-168, author = {Pichard , TeddyAregba-Driollet , DeniseBrull , StéphaneDubroca , Bruno and Frank , Martin}, title = {Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {19}, number = {1}, pages = {168--191}, 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 $M_1$ 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.121114.210415a}, url = {http://global-sci.org/intro/article_detail/cicp/11084.html} }
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 -

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.

Pichard , TeddyAregba-Driollet , DeniseBrull , StéphaneDubroca , Bruno and Frank , Martin. (2018). Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation. Communications in Computational Physics. 19 (1). 168-191. doi:10.4208/cicp.121114.210415a
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