Fully Well-Balanced Methods for Schwarzschild–Euler Equation in Gullstrand–Painlevé Coordinates
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We present a comprehensive formulation of the general-relativistic Euler equations in Gullstrand-Painlevé coordinates (Poisson, Will, 2014), 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 will be shown to 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. The use of Gullstrand-Painlevé coordinates overcomes the difficulties that arise when using Schwarzschild coordinates near the event horizon as it was showned in (LeFloch, Parés, Pimentel-García, 2021).
