Volume 25, Issue 4
A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy

Commun. Comput. Phys., 25 (2019), pp. 1097-1126.

Published online: 2018-12

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• Abstract

Numerical schemes for systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider a system modeling the steady transport of photons and electrons in the field of radiotherapy. Naive discretizations of such a system are commonly constrained by a very restrictive CFL condition. This issue is circumvented by constructing an implicit scheme based on a relaxation approach.
We use an entropy-based moment model, namely the $M_1$ model. Such a system of equations possesses the non-linear flux terms of a hyperbolic system but no time derivative. The flux terms are well-defined only under a condition on the unknowns, called realizability, which corresponds to the positivity of an underlying kinetic distribution function.
The present numerical approach is applicable to non-linear systems which possess no hyperbolic operator, and it preserves the realizability property. However, the discrete equations are non-linear, and we propose a numerical method to solve such non-linear systems.
Our approach is tested on academic and practical cases in 1D, 2D, and 3D and it is shown to require significantly less computational power than reference methods.

• Keywords

Implicit scheme, relaxation scheme, $M_1$ model, radiotherapy dose computation.

• AMS Subject Headings

35A35, 65M22, 35L65, 82C40

• Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-1097, author = {}, title = {A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {4}, pages = {1097--1126}, abstract = {

Numerical schemes for systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider a system modeling the steady transport of photons and electrons in the field of radiotherapy. Naive discretizations of such a system are commonly constrained by a very restrictive CFL condition. This issue is circumvented by constructing an implicit scheme based on a relaxation approach.
We use an entropy-based moment model, namely the $M_1$ model. Such a system of equations possesses the non-linear flux terms of a hyperbolic system but no time derivative. The flux terms are well-defined only under a condition on the unknowns, called realizability, which corresponds to the positivity of an underlying kinetic distribution function.
The present numerical approach is applicable to non-linear systems which possess no hyperbolic operator, and it preserves the realizability property. However, the discrete equations are non-linear, and we propose a numerical method to solve such non-linear systems.
Our approach is tested on academic and practical cases in 1D, 2D, and 3D and it is shown to require significantly less computational power than reference methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0245}, url = {http://global-sci.org/intro/article_detail/cicp/12892.html} }
TY - JOUR T1 - A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy JO - Communications in Computational Physics VL - 4 SP - 1097 EP - 1126 PY - 2018 DA - 2018/12 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2017-0245 UR - https://global-sci.org/intro/article_detail/cicp/12892.html KW - Implicit scheme, relaxation scheme, $M_1$ model, radiotherapy dose computation. AB -

Numerical schemes for systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider a system modeling the steady transport of photons and electrons in the field of radiotherapy. Naive discretizations of such a system are commonly constrained by a very restrictive CFL condition. This issue is circumvented by constructing an implicit scheme based on a relaxation approach.
We use an entropy-based moment model, namely the $M_1$ model. Such a system of equations possesses the non-linear flux terms of a hyperbolic system but no time derivative. The flux terms are well-defined only under a condition on the unknowns, called realizability, which corresponds to the positivity of an underlying kinetic distribution function.
The present numerical approach is applicable to non-linear systems which possess no hyperbolic operator, and it preserves the realizability property. However, the discrete equations are non-linear, and we propose a numerical method to solve such non-linear systems.
Our approach is tested on academic and practical cases in 1D, 2D, and 3D and it is shown to require significantly less computational power than reference methods.

Teddy Pichard, Stéphane Brull & Bruno Dubroca. (2020). A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy. Communications in Computational Physics. 25 (4). 1097-1126. doi:10.4208/cicp.OA-2017-0245
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