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

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.