SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page B1057-B1080, January 2021.
It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximations to a range of kinetic equations. This is true especially if the solution is close to some asymptotic limit where it is known that the solution is low-rank. A particularly interesting case is the fluid dynamic limit that is commonly obtained in the limit of small Knudsen number. However, in this case the Maxwellian which describes the corresponding equilibrium distribution is not necessarily low-rank; because of this, the methods known in the literature are only applicable to the weakly compressible case. In this paper, we propose an efficient dynamical low-rank integrator that can capture the fluid limit---the Navier--Stokes equations---of the Boltzmann-Bhatnagar--Gross--Krook (Boltzmann-BGK) model even in the compressible regime. This is accomplished by writing the solution as $f=Mg$, where $M$ is the Maxwellian and the low-rank approximation is only applied to $g$. To efficiently implement this decomposition within a low-rank framework requires, in the isothermal case, that certain coefficients are evaluated using convolutions, for which fast algorithms are known. Using the proposed decomposition also has the advantage that the rank required to obtain accurate results is significantly reduced compared to the previous state of the art. We demonstrate this by performing a number of numerical experiments and also show that our method is able to capture sharp gradients/shock waves.

中文翻译：

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接近可压缩粘性流域的 Boltzmann-BGK 方程的一种高效动态低阶算法

SIAM 科学计算杂志，第 43 卷，第 5 期，第 B1057-B1080 页，2021 年 1 月。
最近已经证明，动态低秩算法可以提供对一系列动力学方程的稳健和有效的近似。这是真的，尤其是如果解接近某个渐近极限，其中已知解是低秩的。一个特别有趣的例子是流体动力学极限，它通常在小克努森数的极限中获得。然而，在这种情况下，描述相应均衡分布的麦克斯韦分布不一定是低秩的；因此，文献中已知的方法仅适用于弱可压缩的情况。在本文中，我们提出了一种有效的动态低秩积分器，它可以捕获流体极限---纳维-斯托克斯方程---玻尔兹曼-巴特纳加-格罗斯--克罗克（玻尔兹曼-BGK）模型，即使在可压缩状态。这是通过将解决方案写为 $f=Mg$ 来实现的，其中 $M$ 是麦克斯韦方程，低秩近似仅适用于 $g$。为了在低秩框架内有效地实现这种分解，在等温情况下，需要使用已知快速算法的卷积评估某些系数。使用所提出的分解还具有以下优点：与先前的技术状态相比，获得准确结果所需的等级显着降低。我们通过执行一些数值实验来证明这一点，并且还表明我们的方法能够捕获尖锐的梯度/冲击波。为了在低秩框架内有效地实现这种分解，在等温情况下，需要使用已知快速算法的卷积评估某些系数。使用所提出的分解还具有以下优点：与先前的技术状态相比，获得准确结果所需的等级显着降低。我们通过执行一些数值实验来证明这一点，并且还表明我们的方法能够捕获尖锐的梯度/冲击波。为了在低秩框架内有效地实现这种分解，在等温情况下，需要使用已知快速算法的卷积评估某些系数。使用所提出的分解还具有以下优点：与先前的技术状态相比，获得准确结果所需的等级显着降低。我们通过执行一些数值实验来证明这一点，并且还表明我们的方法能够捕获尖锐的梯度/冲击波。