Convolution spline approximations of Volterra integral equations

We derive a new “convolution spline” approximation method for convolution Volterra integral equations. This shares some properties of convolution quadrature, but instead of being based on an underlying ODE solver is explicitly constructed in terms of basis functions which have compact support. At time step tn = nh > 0, the solution is approximated in a “backward time” manner in terms of basis functions φj by u(tn− t) ≈ ∑n j=0 un−jφj(t/h) for t ∈ [0, tn]. We carry out a detailed analysis for B-spline basis functions, but note that the framework is more general than this. For B-splines of degree m ≥ 1 we show that the schemes converge at the rate O(h) when the kernel is sufficiently smooth. We also establish a methodology for their stability analysis and obtain new stability results for several non-smooth kernels, including the case of a highly oscillatory Bessel function kernel (in which the oscillation frequency can be O(1/h)). This is related to convergence analysis for approximation of time domain boundary integral equations (TDBIEs), and provides evidence that the new convolution spline approach could provide a useful time-stepping mechanism for TDBIE problems. In particular, using compactly supported basis functions would give sparse system matrices.