Convergence estimates of a semi-Lagrangian scheme for the ellipsoidal BGK model for polyatomic molecules

TL;DR

This paper introduces a semi-Lagrangian numerical scheme for the polyatomic ellipsoidal BGK model, combining explicit and implicit methods to ensure stability and efficiency across all Knudsen numbers, with proven convergence estimates.

Contribution

It presents a novel semi-Lagrangian scheme that explicitly solves the implicit relaxation step, improving stability and efficiency for the polyatomic ellipsoidal BGK model.

Findings

The scheme is stable for any Knudsen number.

Explicit error estimates are derived for convergence.

The method avoids time step restrictions from convection and relaxation.

Abstract

In this paper, we propose a new semi-Lagrangian scheme for the polyatomic ellipsoidal BGK model. In order to avoid time step restrictions coming from convection term and small Knudsen number, we combine a semi-Lagrangian approach for the convection term with an implicit treatment for the relaxation term. We show how to explicitly solve the implicit step, thus obtaining an efficient and stable scheme for any Knudsen number. We also derive an explicit error estimate on the convergence of the proposed scheme for every fixed value of the Knudsen number.