Accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are but tiny perturbations on top of a close balance between gravity and 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 are applied in the implicit, finite-volume code SLH 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 $\alpha$-$\beta$ 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 time scales. 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.

中文翻译：

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在低马赫数下的天体流体动力学模拟中重力的均衡处理

准确模拟恒星内部的流动对于增进我们对恒星结构和演化的理解至关重要。由于通常缓慢的流动只是重力和压力梯度之间紧密平衡之上的微小扰动，因此此类模拟对数值流体力学方案提出了很高的要求。我们证明了合理大小的网格上的离散误差如何导致比物理流快几个数量级的虚假流。均衡的数值方案可以解决这个问题。结合低马赫数数值通量函数，在隐式有限体积代码SLH中应用了三种这样的方案。我们比较了该方案在四个数值实验中的性能，以解决恒星流体力学中典型问题带来的一些挑战。我们发现，只要重力势能包含在总能量平衡中，$ \ alpha $-$ \ beta $和“偏差”平衡方法可以准确地维持静水力解决方案。它们精确地保存了等熵分层中平移的微小熵波动，这使方法能够随着加热速率再现对流流速的预期比例。偏差方法还大大提高了在长时间尺度上保持开普勒圆盘中平稳轨道运动的准确性。Cargo-LeRoux方法在我们的测试中的表现要差得多，尽管它的简单性在某些情况下仍具有一些优点。全面的，