An asymptotic preserving scheme for the M1 model on polygonal and conical meshes

• Published in: Calcolo Vol. 61; no. 2
• Main Authors: Blanc, Xavier, Hoch, Philippe, Lasuen, Clément
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Asymptotic properties; Gas dynamics; Mathematical models; Mathematics; Mathematics and Statistics; Numerical Analysis; Polygons; Radiative transfer; Theory of Computation; 35L50; 35L60; 65L04; 65M08; Asymptotic preserving; Finite volume; Conical mesh; 65M12; 35L65; Diffusion limit; 76M12
• ISSN: 0008-0624; 1126-5434
• DOI: 10.1007/s10092-024-00574-4

This work focuses on the design of a 2
D
numerical scheme for the
M
1
model on polygonal and conical meshes. This model is nonlinear and approximates the firsts moments of the radiative transfer equation using an entropic closure. Besides, this model admits a diffusion limit as the cross section goes to infinity. It is important for the numerical scheme to be consistent with this limit, that is to say, it should be
asymptotic preserving
or
AP
. Such a scheme already exists on polygonal meshes and the present work consists in adapting it to conical meshes. After introducing conical meshes, we explain the construction of the scheme. It is based on an analogy between the
M
1
model and the Euler gas dynamics system. We also present a second order reconstruction procedure and we apply it on both polygonal and conical meshes. In the last section, some numerical test cases are given so as to compare the nodal and conical schemes. The limit scheme is studied and we observe numerically that it is consistent with the diffusion equation.