An efficient dynamical low-rank algorithm for the Boltzmann-BGK equation close to the compressible viscous flow regime
An efficient dynamical low-rank algorithm for the Boltzmann-BGK equation close to the compressible viscous flow regime
复制标题
接近可压缩粘性流态的 Boltzmann-BGK 方程的高效动态低秩算法
DOI：
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发表时间：
2021
影响因子：
3.1
通讯作者：
Lexing Ying
中科院分区：
文献类型：
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作者：
L. Einkemmer;Jingwei Hu;Lexing Ying
最近已经证明，动态低秩算法可以提供强大的和有效的近似的范围内的动力学方程。这是真的，特别是如果解决方案是接近一些渐近极限，它是已知的解决方案是低秩。一个特别有趣的情况是流体动力学的限制，通常是在小努森数的限制。然而，在这种情况下，描述相应平衡分布的麦克斯韦不一定是低秩的;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，我们提出了一个有效的动态低秩积分，可以捕捉流体限制-Navier-Stokes方程-的Boltzmann-BGK模型，即使在可压缩的制度。这是通过将解写成$f=Mg$来实现的，其中$M$是麦克斯韦方程组，并且低秩近似仅适用于$g$。为了在低秩框架内有效地实现这种分解，在等温的情况下，需要使用卷积来评估某些系数，对于卷积，快速算法是已知的。使用建议的分解也有一个优点，即获得准确的结果所需的秩显着降低相比，以前的最先进的状态。我们证明了这一点，通过执行一些数值实验，也表明，我们的方法是能够捕捉尖锐的梯度/冲击波。
It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximation 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-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.
线性玻尔兹曼方程的动态低阶积分器：扩散极限中的误差分析
影响因子：
2.9
作者：
Ding, Zhiyan;Einkemmer, Lukas;Li, Qin
通讯作者：
Li, Qin
影响因子：
2.9
作者：
Kieri, Emil;Lubich, Christian;Walach, Hanna
通讯作者：
Walach, Hanna