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.
翻译:最近已经证明,动态低位算法可以对一系列运动式方程式提供强大和高效的近似近似。 特别是如果解决方案接近某种无药可治的极限, 已知解决方案是低位的。 一个特别有趣的案例是,在小Knudsen 数的限度内通常获得的流体动态限制。 但是, 在本案中, 描述相应均衡分布的Maxwellian 并不一定是低位的; 因此, 文献中知道的方法只能适用于薄弱的压缩案例。 在本文中, 我们建议一个高效的动态低位混凝物体, 它可以捕捉到博尔茨曼- BGK 模型的流体极限 -- -- 纳维尔- 斯托克斯方程式 -- -- 即使在可调制内也是如此。 然而, 完成这一点的办法是将解决方案写成$=mg$=mg$, 其中美元是Maxwellian, 低位的近似偏差只适用于$g$。 为了在低位框架内高效地实施这一变异性配置, 我们建议一个高效的低级缩缩缩缩配置器需要, 在等热案中, 使用某种精确的计算方法来大幅度地显示我们所了解的演算, 以快速变动的变的计算。