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 P.J. Davies and D.B. Duncan, Convolution spline approximations of Volterra integral equations, Journal of Integral Equations and Applications 26 (2014), 369-410. Here, we analyze 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.