低马赫数时天体物理学流体动力学模拟中重力的平衡处理
Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers
摘要：
恒星内部流动的精确模拟对于提高我们对恒星结构和演化的理解至关重要。由于典型的慢流仅仅是重力和压力梯度之间的紧密平衡之上的微小扰动，因此这种模拟对数值流体动力学方案提出了很高的要求。我们演示了如何在合理大小的网格上的离散化误差可以导致比物理流快几个数量级的虚假流。良好平衡的数值格式可以处理这个问题。在隐式、有限体积的Seven-League Hydro（SLH）代码中，结合低马赫数数值通量函数，应用了三种这样的格式。我们比较了如何执行的计划，在四个数值实验解决一些典型的恒星流体力学问题所带来的挑战。我们发现，$α$ - $β$和偏差井平衡方法可以准确地保持流体静力学的解决方案，重力势能是包括在总能量平衡。他们准确地保存微小的熵波动平流在等熵层结，这使得该方法能够重现预期的缩放的对流流速与加热速率。偏差方法也大大提高了在长时间尺度上保持开普勒盘中静止轨道运动的准确性。在我们的测试中，Cargo-LeRoux方法的表现要差得多，尽管它的简单性在某些情况下仍然有一些优点。总的来说，我们找到了良好的平衡处理重力结合低马赫数通量功能至关重要的复制正确的物理解决方案，具有挑战性的恒星慢流问题负担得起的同位网格。
英文摘要：
Accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are merely tiny perturbations on top of a close balance between gravity and the pressure gradient, such simulations place heavy demands on numerical hydrodynamics schemes. We demonstrate how discretization errors on grids of reasonable size can lead to spurious flows orders of magnitude faster than the physical flow. Well-balanced numerical schemes can deal with this problem. Three such schemes were applied in the implicit, finite-volume Seven-League Hydro (SLH) code in combination with a low-Mach-number numerical flux function. We compare how the schemes perform in four numerical experiments addressing some of the challenges imposed by typical problems in stellar hydrodynamics. We find that the $α$-$β$ and deviation well-balancing methods can accurately maintain hydrostatic solutions provided that gravitational potential energy is included in the total energy balance. They accurately conserve minuscule entropy fluctuations advected in an isentropic stratification, which enables the methods to reproduce the expected scaling of convective flow speed with the heating rate. The deviation method also substantially increases accuracy of maintaining stationary orbital motions in a Keplerian disk on long timescales. The Cargo-LeRoux method fares substantially worse in our tests, although its simplicity may still offer some merits in certain situations. Overall, we find the well-balanced treatment of gravity in combination with low Mach number flux functions essential to reproducing correct physical solutions to challenging stellar slow-flow problems on affordable collocated grids.
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