A Hybrid PSTD/DG Method to Solve the Linearized Euler Equations: Optimization and Accuracy

Abstract

The wave-based Fourier Pseudospectral time-domain (Fourier PSTD) method was shown to be an effective way of modelling acoustic propagation problems as described by the linearized Euler equations (LEE), but is limited to real-valued frequency independent boundary conditions and predominantly staircase-like boundary shapes. This paper presents a hybrid approach to solve the LEE, coupling Fourier PSTD with a nodal Discontinuous Galerkin (DG) method. DG exhibits almost no restrictions with respect to geometrical complexity or boundary conditions. The aim of this novel method is to allow the computation of arbitrary boundary conditions and complex geometries by using the benefits of DG, while keeping Fourier PSTD in the bulk of the domain. The hybridization approach is based on conformal meshes to avoid spatial interpolation of the DG solutions when transferring values from DG to Fourier PSTD and a Gaussian window function to enforce periodicity, while the data transfer from Fourier PSTD to DG is done utilizing spectral interpolation of the Fourier PSTD solutions. Furthermore, the coupling algorithm includes a low-pass filtering approach to suppress instabilities arising from the Fourier PSTD solver. In this paper, the influence of the main parameters of the data-exchange areas and the Gaussian window function on the hybrid method results, is investigated. The precision of the hybrid approach is presented and compared with the precision of the Fourier PSTD standalone solver, showing no significant additional error from the hybrid approach for a suitable selection of parameters.