Hyperboloidal evolution for scalar scattering in Minkowski space

Abstract

We develop a time-domain numerical framework for global scalar wave scattering in Minkowski spacetime. The main contribution is an exact conformal matching of three compactified regions: a past hyperboloidal domain attached to $\mathscr I^-$, a Penrose domain covering a neighborhood of spatial infinity $i^0$, and a future hyperboloidal domain attached to $\mathscr I^+$. The matching surfaces are identical conformal hypersurfaces in the adjacent charts. This yields a global evolution scheme connecting $\mathscr I^-$, the neighborhood of $i^0$, and $\mathscr I^+$ without artificial timelike outer boundaries and without interpolation between scri-fixing gauges. We implement the construction for spherically symmetric scalar waves, including free propagation, localized linear scattering potentials such as the Pöschl--Teller potential, and semilinear wave equations with cubic, quintic, and septic nonlinearities. The numerical experiments demonstrate stable propagation across the matching interfaces, direct extraction of radiation at $\mathscr I^+$, and fourth-order convergence for the free and linear-potential tests. The quintic and septic nonlinear tests exhibit approximately fourth-order convergence and recover the expected late-time tail rates. The cubic case, by contrast, shows only first-order convergence, revealing a limitation of our treatment near compactified boundaries when the conformally rescaled nonlinear source remains non-vanishing. These results validate the conformal matching strategy for long-time simulations, while identifying the boundary regularity issues that must be addressed using a more robust treatment of spatial infinity.