Discretization of the Time Domain CFIE for Acoustic Scattering Problems Using Convolution Quadrature

Abstract

We consider the problem of approximating time domain acoustic scattering by a bounded sound-soft obstacle in three dimensional space via a time domain combined field integral equation (CFIE). Convolution quadrature is used for time discretization of the retarded operators, using an underlying A-stable time-stepping method. We prove operator bounds that imply convergence of a semidiscrete approximation of the time domain CFIE in three cases: (1) a standard regularized time domain CFIE, (2) a new regularized CFIE, and (3) the standard unregularized CFIE. To analyze the first case we use a special mixed formulation of the problem, in the second case we use a perturbation argument, while in the third case a Rellich argument gives the necessary estimates. We conclude with fully discrete error estimates in the case of the new regularized CFIE.