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 semi-Lagrangian 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 $L^1$-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.

在本文中，我们提出了一类新的动力学方程组的保守半拉格朗日方案。它们基于[1]中介绍的保守重建技术。该方法在空间和时间上都是高阶精度的。由于半拉格朗日性质，时间步长不受CFL类型条件的限制。将应用程序介绍给Vlasov-Poisson系统和稀有气体动力学的BGK模型。在第一种情况下，采用运算符拆分以获得时间上的高阶精度，并且使用保留函数最大值和最小值的保守重构。特别是对于最初的积极解决方案，这可以确保准确保留${\mathrm{大号}}^{\mathrm{1个}}$-规范。通过将保守重建与碰撞项的保守处理耦合，可以构造BGK模型的保守方案。时间上的高阶是通过Runge-Kutta或BDF对方程沿特征的时间离散化获得的。由于L稳定性和精确守恒性，BGK模型的结果方案是渐进式守恒底层的流体动力学极限。一维和二维空间中的几个测试案例证实了该方法的准确性和鲁棒性，以及该方案应用于BGK模型时的AP属性。