Fully Well-Balanced Methods for Schwarzschild-Euler Equation in Gullstrand-Painlevé Coordinates

Area of Science:

• Astrophysics
• General Relativity
• Computational Fluid Dynamics

Background:

• Simulating fluid dynamics near black holes is challenging due to the extreme gravitational environment and the event horizon.
• Existing models often struggle with mathematical singularities at the event horizon.

Purpose of the Study:

• To formulate the general-relativistic Euler equations in coordinates that avoid singularities at the event horizon.
• To develop accurate and robust numerical schemes for simulating relativistic fluid flows in curved spacetimes.

Main Methods:

• Formulation of the general-relativistic Euler equations in Gullstrand-Painlevé coordinates.
• Mathematical analysis of the system's properties (well-posedness, Riemann invariants, wave speeds).
• Design and construction of high-order, exactly well-balanced numerical schemes (first- and second-order).

Main Results:

• The formulation provides a regular description of fluid dynamics across the Schwarzschild event horizon.
• Stationary solutions were obtained and analyzed for a representative equation of state.
• Numerical schemes demonstrated accuracy, robustness, and well-balanced behavior in extensive experiments.

Conclusions:

• The study offers theoretical insights into relativistic fluid dynamics near black holes.
• The developed numerical schemes provide practical tools for reliable long-term simulations in curved spacetimes.