README.md

2026CSY

Codex/ChatGPT (July 2026)

Summary

This paper proves stability and convergence results for discontinuous Galerkin discretizations of spherically symmetric Einstein-scalar equations in Bondi coordinates. The setting is chosen as a mathematically tractable nonlinear-relativity model that still includes large initial data leading to black-hole formation. The authors formulate the DG scheme, prove $L^2$ stability for arbitrary $L^2$ initial data, and establish a priori estimates for a broad class of exact solutions. Those estimates include behavior up to $r=0$ and provide event-horizon formation information. For polynomial degree $k \ge 1$, the paper proves an optimal $O(h^{k+1})$ error estimate, and it separately treats the $P^0$ case. Numerical experiments confirm that the theoretical convergence rates are sharp. The analysis gives rigorous support for high-order DG methods in a nonlinear general-relativistic model where much prior numerical-relativity stability work has been empirical or linearized.

Contributions

1. Formulated a DG method for spherically symmetric Einstein-scalar equations in Bondi coordinates.
2. Proved $L^2$ stability for arbitrary $L^2$ initial data.
3. Established a priori estimates for large-data solutions that form black holes.
4. Proved optimal convergence for high-order DG approximations and handled the $P^0$ case separately.
5. Confirmed the sharpness of the error estimates through numerical experiments.