An asymptotic preserving scheme for the \(M_1\) model on polygonal and conical meshes (Q6536769)

This paper addresses the design and analysis of a two-dimensional numerical scheme for the \(M_1\) model, which approximates the first moments of the radiative transfer equation using an entropic closure on polygonal and conical meshes. The primary scientific problem addressed is the development of a numerical scheme that is asymptotic preserving (AP) and consistent with the diffusion limit as the cross-section goes to infinity, ensuring the model's applicability in highly scattering media over long timescales.\N\NThe methods employed to tackle this problem involve constructing the numerical scheme on polygonal meshes and then adapting it to conical meshes. The scheme's construction leverages the analogy between the \(M_1\) model and the Euler gas dynamics system. Key components of the methodology include the introduction of conical meshes described by rational quadratic Bezier curves, the reformulation of the \(M_1\) model as a gas dynamics system, and the implementation of a second-order reconstruction procedure to ensure accuracy. Additionally, the paper employs a partially implicit time discretisation to improve the scheme's stability and demonstrates the asymptotic preserving property of the scheme by proving its consistency with the diffusion equation in the limit.\N\NThe main findings of the manuscript include the successful extension of the asymptotic preserving scheme to conical meshes, which offers improved geometrical accuracy over standard polygonal meshes. Numerical tests demonstrate that the proposed scheme is consistent with the diffusion limit and outperforms the nodal scheme, particularly in avoiding the cross stencil phenomenon observed in nodal schemes. The authors also establish a new Courant-Friedrichs-Lewy (CFL) condition that ensures the positivity of the energy and the limitation of the flux, maintaining the scheme's stability without becoming overly restrictive in the diffusion regime.\N\NThis research is significant for the field of numerical analysis and computational mathematics, particularly in the context of radiative transfer equations. The development of an asymptotic preserving scheme that is applicable to both polygonal and conical meshes enhances the accuracy and robustness of simulations in radiative transfer problems. By addressing the limitations of previous schemes and providing a rigorous mathematical framework for the new scheme, this work contributes to advancing the numerical methods available for solving complex radiative transfer models in highly scattering media.