A Tutorial on Non-Reflective Characteristic Boundary Conditions for Direct and Adjoint Time-Domain Acoustic Simulations

Accurate free-field acoustic simulations require non-reflective boundary conditions to suppress spurious reflections at the computational domain boundaries. Although several characteristic-based formulations for direct (forward) simulations have been proposed in recent decades, adjoint formulations of such characteristic-based boundary conditions (CBCs) have received limited attention in the literature and lack a comprehensive analysis.

[Formula: see text]This paper presents the derivation and evaluation of adjoint CBCs complementing existing direct CBCs. Both direct and adjoint CBCs are applied to the nonlinear Euler equations and linear acoustic equations in time-domain simulations. The resulting boundary treatments are compared in terms of accuracy, reflection behavior, and direct-adjoint consistency.

[Formula: see text]The CBCs are implemented using a single-point approach with both static and dynamic wave-vector estimates and are optionally combined with a sponge layer. Over the evaluated frequency range of 350–4000[Formula: see text]Hz, spurious reflections are effectively attenuated in both direct and adjoint simulations. The adjoint CBCs show reflection behavior consistent with their direct counterparts, demonstrating that the proposed formulation provides a consistent open-boundary treatment for adjoint time-domain acoustics.