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Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations. Solving such equations numerically is both complicated and computationally intensive, and numerical methods often prove to be unstable. Collocation schemes are easier to implement than full finite element formulations, but little appears to be known about their stability and convergence. Here we derive and analyse some new stable collocation schemes for the single layer equation for transient acoustic scattering, and use (spatial) Fourier and (temporal) Laplace transform techniques to demonstrate that such stable schemes are second order convergent. 1 2. Preliminaries. We now describe the notation and some basic results used in the manuscript. The stability and convergence analysis in § §4-5 is for the scalar RPIE (1.1) posed on an infinite flat surface, i.e. for R 2 u(x , t−|x −x|) |x −x| dx = a(x, t) on R 2 × (0, T ) , (2.1)where u and a satisfy (1.2).

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