  Recently, there has been much interest in black hole echoes, based on the
idea that there may be some mechanism (e.g., from quantum gravity) that
waves/fields falling into a black hole could partially reflect off of an
interface before reaching the horizon. There does not seem to be a good
understanding of how to properly model a reflecting surface in numerical
relativity, as the vast majority of the literature avoids the implementation of
artificial boundaries, or applies transmitting boundary conditions. Here, we
present a framework for reflecting a scalar field in a fully dynamical
spherically symmetric spacetime, and implement it numerically. We study the
evolution of a wave packet in this situation and its numerical convergence,
including when the location of a reflecting boundary is very close to the
horizon of a black hole. This opens the door to model exotic near-horizon
physics within full numerical relativity.

General Relativity and Quantum Cosmology