Accelerated sound propagation simulations using an error-free Fourier method coupled with the spectral element method

Simulating acoustics efficiently and accurately using numerical methods has been an active research area for the last decades and has applications in computer games, VR/AR, and architectural design.
However, their extensive computation time makes these methods challenging for large scenes and broad frequency ranges. This work attempts to accelerate the simulations using rectangular decomposition, enabling error-free propagation in the bulk of the domain consisting of air. We exploit the analytical solution to the wave equation in rectangular domains calculated using the Fast Fourier Transform with near-optimal spatial discretization satisfying the Nyquist criterium. Coupling with the spectral element method near the boundaries results in a method capable of handling complex
geometries with realistic boundaries, though with the caveat that additional errors and computational overhead may result from the interface. This paper will investigate the accuracy and efficiency of the proposed domain decomposition method compared to a spectral element implementation running in the entire domain and the results in 1D indicate an 18 times speedup factor for relative errors below 9%.