In this paper, two important aspects of a numerical method for the simulation of sound propagation in complex flows are addressed. This method is based on an explicit sixth-order accurate approximation of spatial derivatives, high-order Adams-Bashforth time integration and an explicit linear filtering step. The first aspect is an analysis of the global properties of this method, including spatial filtering and nonreflecting exit boundary conditions, using a stability analysis and an eigenmodes decomposition. The described procedure is shown to be stable and accurate. Secondly, wave propagation on adjacent zones with multi-size mesh is performed. A high-order interpolation technique is proposed and assessed at the coupling interface between two adjacent domains. The propagation of an initial pulse through the interface is simulated. The proposed method may bring a significant reduction of the computational cost in the simulation of the sound propagation through complex flows.