Relaxation schemes for the M 1 model with space-dependent flux: application to radiotherapy dose calculation
Résumé
Because of stability constraints, most numerical schemes applied to hyper-bolic 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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