Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications

In the first part of this paper [1], we introduce and analyse a new conservative reconstruction, which is now adopted to build conservative semi-Lagrangian (SL) schemes for various equations, including rigid rotation and kinetic equations. The basic idea of the semi-Lagrangian approach is to integrate the equations along the characteristics and update solutions on a fixed grid. This allows the use of larger time steps with respect to Eulerian methods, which suffer from CFL-type time step restrictions. For this reason, SL methods are suitable for the numerical treatment of kinetic equations, where the collision term may <comment>Superscript/Subscript Available</comment> ABSTRACT In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [1]. The methods are high order accurate both in space and time. Because of the semiLagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to the Vlasov-Poisson system and the BGK model of rarefied gas dynamics. In the first case, operator splitting is adopted to obtain high order accuracy in time, and a conservative reconstruction that preserves the maximum and minimum of the function is used. For initially positive solutions, in particular, this guarantees exact preservation of the L1-norm. Conservative schemes for the BGK model are constructed by coupling the conservative reconstruction with a conservative treatment of the collision term. High order in time is obtained by either Runge-Kutta or BDF time discretization of the equation along characteristics. Because of L-stability and exact conservation, the resulting scheme for the BGK model is asymptotic preserving for the underlying fluid dynamic limit. Several test cases in one and two space dimensions confirm the accuracy and robustness of the methods, and the AP property of the schemes when applied to the BGK model. (c) 2021 Elsevier Inc. All rights reserved.