Numerical Stability of Collocation Schemes for Time Domain Boundary Integral Equations

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations (RPIEs). Collocation schemes for RPIEs are often unstable, having errors which oscillate and grow exponentially with time. We describe how Fourier analysis can be used to analyse the stability of uniform grid schemes and to show that the instabilities are often very different from those observed in PDE approximations. We also present a new stable collocation scheme for a scalar RPIE, and show that it converges.