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Despite the numerous advances made in increasing the computational efficiency of time-domain integral equation (TDIE)-based solvers, the stability and accuracy of TDIE solvers remain problematic. This paper introduces a new numerical method for the accurate solution of TDIEs for scattering from arbitrary perfectly conducting surfaces. The work described in this paper uses the higher order divergence-conforming basis functions of Graglia et al. for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations. Since the basis functions used for the temporal representation are noncausal, an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time. Numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization.