Convolution-in-Time Approximations of Time Domain Boundary Integral Equations

We present a new temporal approximation scheme for the boundary integral formulation of time-dependent scattering problems which can be combined with either collocation or Galerkin approximation in space. It uses the backward-in-time framework introduced in [P. J. Davies and D. B. Duncan, Convolution Spline Approximations of Volterra Integral Equations, www.mathstat. strath.ac.uk/research/reports/2012 (2012)] with new temporal basis functions which share some properties with radial basis function multiquadrics. We analyze the stability and convergence properties of the new scheme for associated Volterra integral equations and perform extensive numerical tests for scattering from flat polygonal plates and open and closed cubes and spheres, which demonstrate effectiveness of this approach.