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

We present a comprehensive formulation of the general-relativistic Euler equations in Gullstrand-Painlevé coordinates, providing a description of fluid dynamics that remains regular across the Schwarzschild event horizon. After establishing the system and examining its mathematical properties including well-posedness behaviour, Riemann invariants, and characteristic wave speeds, we obtain its stationary solutions in implicit form and analyze them in detail for a representative equation of state. Building on these results, we design high-order exactly well-balanced numerical schemes capable of preserving all stationary states of the system. Both first- and second-order methods are constructed. Extensive numerical experiments confirm the accuracy, robustness, and well-balanced behavior of the proposed schemes. The study offers both theoretical insight into relativistic fluid flows near black holes and practical tools for reliable long-term simulations in curved spacetimes.