Artificial boundary conditions for the numerical simulation of unsteady acoustic waves

Abstract

We construct non-local artificial boundary conditions (ABCs) for the numerical simulation of genuinely time-dependent acoustic waves that propagate from a compact source in an unbounded unobstructed space. The key property used for obtaining the ABCs is the presence of lacunae, i.e., sharp aft fronts of the waves, in wave-type solutions in odd-dimension spaces. This property can be considered a manifestation of the Huygens’ principle. The ABCs are obtained directly for the discrete formulation of the problem. They truncate the original unbounded domain and guarantee the complete transparency of the new outer boundary for all the outgoing waves. A central feature of the proposed ABCs is that the extent of their temporal non-locality is fixed and limited, and it does not come at the expense of simplifying the original model. It is rather a natural consequence of the existence of lacunae, which is a fundamental property of the corresponding solutions. The proposed ABCs can be built for any consistent and stable finite-difference scheme. Their accuracy can always be made as high as that of the interior approximation, and it will not deteriorate even when integrating over long time intervals. Besides, the ABCs are most flexible from the standpoint of geometry and can handle irregular boundaries on regular grids with no fitting/adaptation needed and no accuracy loss induced. Finally, they allow for a wide range of model settings. In particular, not only one can analyze the simplest advective acoustics case with the uniform background flow, but also the case when the waves’ source (or scatterer) is engaged in an accelerated motion.