Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation
Christophe Berthon 1, Martin Frank 2*, Celine Sarazin 1, Rodolphe Turpault 11 Universite de Nantes, Laboratoire de Mathematiques Jean Leray, UMR6629, 2 rue de la Houssiniere, BP 92208, 44322 Nantes Cedex 3, France.
2 RWTH Aachen University, Mathematics, Center for Computational Engineering Science, Schinkelstrasse 2, 52062 Aachen, Germany.
Received 2 August 2010; Accepted (in revised version) 17 December 2010
Available online 2 August 2011
The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method.AMS subject classifications: 35L65, 65N08, 92C99
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Key words: Radiotherapy, hyperbolic system of conservation laws, source term approximations, finite volume methods.
Email: firstname.lastname@example.org (C. Berthon), email@example.com (M. Frank), firstname.lastname@example.org (C. Sarazin), email@example.com (R. Turpault)