Convolution spline approximations for time domain boundary integral equations

We introduce a new "convolution spline" temporal approximation of time domain boundary integral equations (TDBIEs). It shares some properties of convolution quadrature (CQ), but instead of being based on an underlying ODE solver the approximation is explicitly constructed in terms of compactly supported basis functions. This results in sparse system matrices and makes it computationally more efficient than using the linear multistep version of CQ for TDBIE time-stepping. We use a Volterra integral equation (VIE) to illustrate the derivation of this new approach: at time step \(t_n = n h\) the VIE solution is approximated in a backwards-in-time manner in terms of basis functions \(\phi_j\) by \(u(t_n-t) \approx \sum_{j=0}^n u_{n-j}\,\phi_j(t/h)\) for \(t \in [0,t_n]\). We show that using isogeometric B-splines of degree \(m\ge 1\) on \([0,\infty)\) in this framework gives a second order accurate scheme, but cubic splines with the parabolic runout conditions at \(t=0\) are fourth order accurate. We establish a methodology for the stability analysis of VIEs and demonstrate that the new methods are stable for non-smooth kernels which are related to convergence analysis for TDBIEs, including the case of a Bessel function kernel oscillating at frequency \(O(1/h)\). Numerical results for VIEs and for TDBIE problems on both open and closed surfaces confirm the theoretical predictions.