Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels

The cubic “convolution spline” method for first kind Volterra convolution integral equations was introduced in [Convolution spline approximations of Volterra integral equations, J. Integral Equations Appl., 26:369–410, 2014 ]. Here we analyse its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function, and a new error bound obtained from a particular B-spline quasi-interpolant.