Stability and Instability of Time‐Domain Boundary Element Methods for the Acoustic Neumann Problem

ABSTRACT This work presents a stable time‐domain boundary element method for the acoustic wave equation in three‐dimensional unbounded domains. Other formulations of time‐domain boundary element methods based on retarded potential operators are known to exhibit stability issues, which often hinder their use in industrial contexts. We have investigated the stability properties of a Galerkin first‐kind boundary integral formulation for sound emission problems, where well‐posedness can be established in both the continuous and the discrete setting. Numerical experiments confirm the accuracy and convergence of the method. We assess long‐time stability through extensive simulations focusing on fine temporal resolutions and large time ranges. The proposed formulation is compared with two alternative approaches used in practice: a space‐time single‐layer potential approach and a semi‐discretized collocation method.