High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation (Communications in Computational Physics, 2020)
In this paper, we present a conservative semi-Lagrangian finite-differencescheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large 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. Firstly, when using classical continuous Maxwellian, conservation 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 Maxwellian do not coincide with thediscrete moments. Secondly, the non-linear reconstruction used to prevent oscillationsdestroys the translation invariance which is at the basis of the conservation propertiesof the scheme. As a consequence, the schemes show a wrong shock speed in the limitof small Knudsen number. To treat the first problem and ensure machine precisionconservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellianintroduced in [22]. The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in [26]. In this way we can construct 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 numerical tests.