论文标题
动力学方程的保守半拉格朗日方案第二部分：应用
Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications
论文作者
论文摘要
在本文中，我们为动力学方程式提供了一类新的保守半拉格朗日方案。它们基于[S. Y. Cho等人，动力学方程的保守半拉格朗日方案第一部分：重建，2020年]。这些方法在空间和时间上都是高阶精度。由于半拉格朗日的性质，时间步骤不受CFL型条件的限制。应用应用于刚体旋转，弗拉索夫泊松系统和稀有气体动力学的BGK模型。在前两种情况下，采用了操作员分裂，以及时获得高阶精度，并使用保守的重建，可保留函数的最大和最小值。特别是对于最初的阳性解决方案，这保证了L1-norm的确切保留。 BGK模型的保守计划是通过将保守的重建与对碰撞术语的保守处理结合来构建的。高阶是通过沿特征的方程式runge-kutta或BDF时间离散化获得的。由于L稳定性和确切的保守性，BGK模型的最终方案是渐近地保存基础流体动态极限。一个和两个空间维度中的几个测试用例证实了该方法的准确性和鲁棒性，以及应用于BGK模型时方案的AP特性。
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 [S. Y. Cho, et al., Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction, 2020]. 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 rigid body rotation, Vlasov Poisson system and the BGK model of rarefied gas dynamics. In the first two cases 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.
