Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation
This work provides a provably convergent numerical scheme for a kinetic model used in rarefied gas dynamics, addressing stability and efficiency simultaneously.
The authors propose a new implicit semi-Lagrangian scheme for the ellipsoidal BGK model that achieves both stability and efficiency, and they derive an error estimate in a weighted L∞ norm that holds uniformly for all relaxation parameters, including the original BGK model.
The ellipsoidal BGK model is a generalized version of the original BGK model designed to reproduce the physical Prandtl number in the Navier-Stokes limit. In this paper, we propose a new implicit semi-Lagrangian scheme for the ellipsoidal BGK model, which, by exploiting special structures of the ellipsoidal Gaussian, can be transformed into a semi-explicit form, guaranteeing the stability of the implicit methods and the efficiency of the explicit methods at the same time. We then derive an error estimate of this scheme in a weighted $L^{\infty}$ norm. Our convergence estimate holds uniformly in the whole range of relaxation parameter $ν$ including $ν=0$, which corresponds to the original BGK model.