一种高阶中心化半拉格朗日方案用于中性气体流的多维动能模拟
A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows
AI总结：
本文提出一种高阶中心化半拉格朗日方案，通过改进的重映射方法减少数值扩散，保持守恒性和正性，适用于中性气体流的多维模拟。
摘要：
`Convected Scheme`（CS）指一类算法，通常用于求解玻尔兹曼方程，通过积分形式的特征方法将初始单元投影到最终单元组。因此，CS是一种`前向轨迹`半拉格朗日方案。对于中性气体流的多维模拟，中心化半拉格朗日（CCSL）方案因其实施简单、内存需求低和边界条件处理方便而具有优势。目前CCSL-CS的主要缺点是其在物理空间中的高数值扩散，由于每个时间步结束时的二次重映射。通过修改方程分析，证明可以提前获得高阶重映射误差估计，并在重映射后对单元的最终位置进行小修正，以完全补偿此误差。所得到的方案在空间上具有四阶精度，同时保留CS的有利特性：守恒性和正性保持，整体算法复杂度未显著增加。还提出了两种单调（即非振荡）的四阶CCSL-CS版本：一种使用公共通量限制器方法；另一种使用非多项式重建来评估密度函数的导数。该方法在简单的一维和二维示例中进行了说明，并展示了完全3D解的玻尔兹曼方程，描述气体通过圆柱管进入真空的膨胀过程。
英文摘要：
The term `Convected Scheme' (CS) refers to a family of algorithms, most usually applied to the solution of Boltzmann's equation, which uses a method of characteristics in an integral form to project an initial cell forward to a group of final cells. As such the CS is a `forward-trajectory' semi-Lagrangian scheme. For multi-dimensional simulations of neutral gas flows, the cell-centered version of this semi-Lagrangian (CCSL) scheme has advantages over other options due to its implementation simplicity, low memory requirements, and easier treatment of boundary conditions. The main drawback of the CCSL-CS to date has been its high numerical diffusion in physical space, because of the 2$^{\text{nd}}$ order remapping that takes place at the end of each time step. By means of a Modified Equation Analysis, it is shown that a high order estimate of the remapping error can be obtained a priori, and a small correction to the final position of the cells can be applied upon remapping, in order to achieve full compensation of this error. The resulting scheme is 4$^{\text{th}}$ order accurate in space while retaining the desirable properties of the CS: it is conservative and positivity-preserving, and the overall algorithm complexity is not appreciably increased. Two monotone (i.e. non-oscillating) versions of the fourth order CCSL-CS are also presented: one uses a common flux-limiter approach; the other uses a non-polynomial reconstruction to evaluate the derivatives of the density function. The method is illustrated in simple one- and two-dimensional examples, and a fully 3D solution of the Boltzmann equation describing expansion of a gas into vacuum through a cylindrical tube.
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