On hybrid convolution quadrature approaches for modeling time-domain wave problems with broadband frequency content

Abstract

We propose two hybrid convolution quadrature based discretizations of the wave equation on interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time-domain wave problem into a series of Helmholtz problems with complex-valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the (Formula presented.) -transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies, we employ the boundary element method, while for more oscillatory problems, we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach, we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann-to-Dirichlet map for plane waves to the given boundary data. Finally, we investigate the effectiveness of both hybrid approaches across a range of numerical experiments.