  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.
