斜压流体动力学哈密顿公式的探讨
Exploration of Hamiltonian formulations of baroclinic hydrodynamics
摘要：
在致密双星中的中子星，在内旋阶段被正压流很好地近似。在合并阶段，潮汐扰动和激波加热发生，斜压的描述，而不是需要。在正压的情况下，一个哈密顿公式可能提供独特的好处，数值相对论模拟的inspiral阶段，包括高度精确的守恒的流通和超收敛的流体变量，并正在积极探索。在这项工作中，我们调查的可行性的哈密顿制定斜压的情况下。与正压情况不同的是，这个公式是非保守的，但它可以用近似黎曼解算器算法很好地处理，因为非保守项在真正的非线性场中消失了。尽管如此，使用数值1维激波管测试，我们发现，弱解的哈密顿系统不同的标准通过强制执行守恒的静止质量密度，动量密度，和能量密度跨越间断。我们还表明，正压哈密顿公式可以承认冲击波在流体真空界面，这可能与在过去的数值试验中观察到的恒星表面的不稳定行为。鉴于非物理的弱解，我们预计，在未来的数值相对论中的流体动力学的哈密顿公式的实现，将有必要使用显式正压配方在螺旋阶段，然后切换到一个强大的斜压配方合并之前。
英文摘要：
A neutron star in a compact binary is expected to be well-approximated by a barotropic flow during the inspiral phase. During the merger phase, where tidal disruption and shock-heating occur, a baroclinic description is needed instead. In the barotropic case, a Hamiltonian formulation potentially offers unique benefits for numerical relativity simulations of the inspiral phase, including highly accurate conservation of circulation and superconvergence of the fluid variables, and is actively being explored. In this work, we investigate the viability of a Hamiltonian formulation in the baroclinic case. At odds with the barotropic case, this formulation is non-conservative, yet it can be treated well with approximate Riemann solver algorithms since the non-conservative terms vanish across genuinely nonlinear fields. Nonetheless, using numerical 1-dimensional shock tube tests we find that the weak solutions of the Hamiltonian system differ from the standard ones obtained by enforcing conservation of rest mass density, momentum density, and energy density across discontinuities. We also show that barotropic Hamiltonian formulations can admit shockwaves at fluid-vacuum interfaces, which may be related to the unstable behavior of stellar surfaces observed in past numerical tests. In light of the unphysical weak solutions, we expect that in future implementations of the Hamiltonian formulation of hydrodynamics in numerical relativity it will be necessary to use an explicitly barotropic formulation during the inspiral phase, and then switch to a robust baroclinic formulation prior to merger.
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