Axisymmetric hydrodynamics in numerical relativity: treating coordinate singularity, artificial heating and modeling MHD instabilities

TL;DR

Problem: axisymmetric 2D simulations of post-merger neutron-star remnants must avoid axis-related artifacts and drift in conserved quantities while capturing long-term evolution. Approach: introduce a flux-conservative 'modified conservative' scheme to enforce zero axis flux, implement entropy-density evolution to suppress numerical heating, and incorporate MRI-driven effective viscosity to model angular-momentum transport, with consideration of Tayler-Spruit instabilities. Contributions: demonstrate axis-smooth, conservation-compatible evolution on timescales of order $10^2$ ms, provide a practical entropy-transport framework, and compare schemes against alternative axis treatments. Significance: enables efficient, reliable 2D axisymmetric studies of post-merger remnants, complements 3D results, and helps constrain turbulence models, despite remaining uncertainties (neutrino treatment, nonaxisymmetric effects).

Abstract

Two-dimensional axisymmetric simulations of binary neutron star (BNS) merger remnant are a cheap alternative to 3D simulations. To maintain realism for secular timescales, simulations must avoid accumulated errors from drifts in conserved quantities and artificial heating, and they must model turbulent transport in a way that remains plausible throughout the evolution. It is also crucial to avoid numerical artifacts due to the polar coordinate axis singularity. Methods that behave well near the axis often break flux-conservative form of the hydrodynamic equations, resulting in significant drifts in conserved quantities. We present a flux-conservative scheme that maintains smoothness near the axis without sacrificing conservative formulation of the equations or incurring drifts in conserved global quantities. We compare the numerical performance of different treatments of the hydrodynamic equations when evolving a hypermassive neutron star resembling the remnant of a BNS merger. These simulations demonstrate that the new scheme combines the axis smoothness of non-conservative methods with the mass and angular momentum conservation of other conservative methods on $\sim$ $10^2$ ms timescales of viscous and neutrino-driven evolution. Because fluid profiles remain smooth in the remnant interior, it is possible to remove artificial heating by evolving the entropy density. We show how physical heating and cooling terms can be easily calculated from source terms of the conservative evolution variables and demonstrate our implementation. Finally, we discuss and implement improvements to the effective viscosity scheme to better model the effect of magnetohydrodynamic instabilities as the remnant evolves.

Figures (8)

• Figure 1: Angular velocity $\Omega$ profiles at different times as a function of cylindrical radius, $\varpi$, for different schemes plotted on the equator.
• Figure 2: Drift in baryonic mass(top) and angular momentum (bottom)for conservative and factored schemes plotted against time. The angular momentum drift is negative in sign for modified conservative and factored schemes.
• Figure 3: The rate of change of the relative error for total angular momentum (filled circles in black) and total baryonic mass (inverted triangles in green) for 5 different finite difference resolutions, $\Delta x = \{ 148\text{ m},99\text{ m},74\text{ m},59\text{ m},49\text{ m} \}$
• Figure 4: Schematic illustrating the modified conservative implementation. Quantities labeled by $A$ denote conservative variables; those labeled by $F$ denote the corresponding fluxes. Grid points marked with filled circles represent cell-center, and grid points with open circles represent cell-faces. The fluxes are altered for the first three layers of cell-faces that correspond to two layers of cell-centers
• Figure 5: Angular velocity plotted on the equator for the remnant at different times evolved for a longer time using the modified conservative scheme. The black-dashed line shows the velocity profile for a remnant evolved with no viscosity.