Space-time discretized retarded potential boundary integral operators: Quadrature for collocation methods

We propose a novel discretization method for time domain boundary integral equations of the three-dimensional wave equation. The basic idea of the discussed approach is to treat time as if it were an additional spatial coordinate. This leads to a discretization of the boundary integral operators in 3+1 dimensional space-time by use of basis functions which do not separate space and time variables. These functions are based on tetrahedral meshes of the lateral boundary of the space-time cylinder. We discuss an explicit representation of the integral operators of the wave equation, so-called retarded potential integral operators, which genuinely conforms to the space-time setting. The majority of this work is concerned with the numerical evaluation of these integrals. An accurate and robust Gaussian quadrature scheme is proposed and verified by means of numerical experiments.