Convolution Quadrature Time-Domain Boundary Element Method for Viscoelastic Wave Scattering by Many Cavities in a 3D Infinite Space

This paper presents a time-domain boundary element method (BEM) for viscoelastic wave scattering by many cavities in 3D infinite space. The convolution quadrature method (CQM) is applied to the convolutions of the time-domain boundary integral equation in order to improve the numerical stability of BEM. Scattering of an incident wave by cavities in a viscoelastic solid is solved by the developed time-domain BEM. The incident wave propagating in a viscoelastic solid in time-domain is obtained using the inverse Fourier transform of that in frequency-domain. In addition, the hybrid parallelization using OpenMP and MPI is used to save the computational time and memory.

On superconvergence of Runge–Kutta convolution quadrature for the wave equation

The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge–Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of $$\left| s\right| $$ s (up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map.