Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation (Communications in Computational Physics, 2018)
Because of stability constraints, most numerical schemes applied to hyperbolicsystems of equations turn out to be costly when the flux term is multiplied bysome very large scalar. This problem emerges with the $M_1$ system of equations inthe field of radiotherapy when considering heterogeneous media with very disparatedensities. Additionally, the flux term of the $M_1$ system is non-linear, and in order forthe model to be well-posed the numerical solution needs to fulfill conditions calledrealizability. In this paper, we propose a numerical method that overcomes the stabilityconstraint 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 differencequotient to obtain stability. The scheme is applied to a radiotherapy dose calculationexample.