Preserving equilibrium, capturing collapse: Well-balanced conservative finite-volume methods on dynamical spacetimes
Coupled hyperbolic systems on evolving geometries pose difficulty: the discretization must preserve nontrivial equilibria exactly yet remain conservative, admissibility-preserving, and stable through nonlinear transitions. We develop a strictly flux-conservative finite-volume framework for the two-way coupled Einstein-Euler system, built on a new first-order hyperbolic formulation of Einstein's equations and providing its first coupling to matter.
Both subsystems use LeVeque-type wave-propagation schemes, with different limiting strategies: the spacetime solver applies a symmetric second-order TVD limiter to the waves, whereas the fluid solver applies a higher-order monotonicity-preserving limiter to the characteristic variables. A fluctuation-based well-balanced formulation preserves Tolman-Oppenheimer-Volkoff equilibria to machine precision. The fluid reconstruction is symmetric and linearity-preserving, retaining constant and linear data when limiting is inactive and reducing order as required for admissibility. In vacuum, we rigorously prove preservation of strong hyperbolicity. In the coupled system, numerical positivity of the metric variables and satisfaction of the contracted Bianchi identities indicate preservation of hyperbolicity throughout evolution.
We demonstrate long-time neutron-star equilibrium, controlled collapse to a black hole, stable evolution after horizon formation, and possibility of extension to stellar-core collapse. The method provides a scalable route to multidimensional simulation of coupled PDEs on dynamical geometries.
