计算欧拉流体力学与伽利略不变性
Computational Eulerian Hydrodynamics and Galilean Invariance
摘要：
欧拉流体动力学模拟是天体物理系统中流体建模的一种强大而流行的工具。在这项工作中，我们严格审查最近声称，这些方法违反伽利略不变性的欧拉方程。我们证明了欧拉流体力学方法收敛到伽利略不变的解决方案，提供了一个定义良好的收敛解存在。具体而言，我们表明，数值扩散，导致扩散类项的离散流体动力学方程的欧拉方法解决，占以前确定的影响，这些方法的伽利略非不变性的证据。当模拟的空间分辨率保持固定时，这些速度相关的扩散项导致不同的体积速度的不同结果，但是随着模拟的分辨率增加以获得收敛的解，它们的影响变得可以忽略不计。特别是，我们发现，开尔文-亥姆霍兹不稳定性发展正常，在现实的欧拉计算，无论体积速度提供的问题是模拟足够的分辨率（一个因素的2-4增加相比，没有体积流量的情况下，现实的速度）。我们的研究结果重申，高分辨率欧拉方法可以很好地执行，并获得收敛的解决方案，即使在存在的高度超音速散装流。
英文摘要：
Eulerian hydrodynamical simulations are a powerful and popular tool for modeling fluids in astrophysical systems. In this work, we critically examine recent claims that these methods violate Galilean invariance of the Euler equations. We demonstrate that Eulerian hydrodynamics methods do converge to a Galilean-invariant solution, provided a well-defined convergent solution exists. Specifically, we show that numerical diffusion, resulting from diffusion-like terms in the discretized hydrodynamical equations solved by Eulerian methods, accounts for the effects previously identified as evidence for the Galilean non-invariance of these methods. These velocity-dependent diffusive terms lead to different results for different bulk velocities when the spatial resolution of the simulation is kept fixed, but their effect becomes negligible as the resolution of the simulation is increased to obtain a converged solution. In particular, we find that Kelvin-Helmholtz instabilities develop properly in realistic Eulerian calculations regardless of the bulk velocity provided the problem is simulated with sufficient resolution (a factor of 2-4 increase compared to the case without bulk flows for realistic velocities). Our results reiterate that high-resolution Eulerian methods can perform well and obtain a convergent solution, even in the presence of highly supersonic bulk flows.
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