High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

In this paper, we present a conservative semi-Lagrangian finite-differencescheme for the BGK model. Classical semi-Lagrangian finite difference schemes, cou-pled with an L-stable treatment of the collision term, allowlarge time steps, for all therange of Knudsen number [17, 27, 30]. Unfortunately, however, such schemes are notconservative. Lack of conservation is analyzed in detail, and two main sources areidentified as its cause. First, when using classical continuous Maxwellian, conserva-tion error is negligible only if velocity space is resolved with sufficiently large numberof grid points. However, for a small number of grid points in velocity space such erroris not negligible, because the parameters of the Maxwelliando not coincide with thediscrete moments. Secondly, the non-linear reconstruction used to prevent oscillationsdestroys the translation invariance which is at the basis ofthe conservation propertiesof the scheme. As a consequence the schemes show a wrong shockspeed in the limitof small Knudsen number. To treat the first problem and ensuremachine precisionconservation of mass, momentum and energy with a relativelysmall number of veloc-ity grid points, we replace the continuous Maxwellian with the discrete Maxwellianintroduced in [22]. The second problem is treated by implementing a conservative cor-rection procedure based on the flux difference form as in [26]. In this way we can con-struct conservative semi-Lagrangian schemes which are Asymptotic Preserving (AP)for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness ofthe proposed scheme is demonstrated by extensive numericaltests.