Global behavior of temporal discretizations for Volterra integrodifferential equations with certain nonsmooth kernels

In this work, the z-transform is presented to analyze two kinds of time-discrete techniques for Volterra integrodifferential equations in Hilbert space. These approaches are based on Crank-Nicolson method and second-order backward differentiation formula (BDF2) with certain nonsmooth kernels, respectively, combining with the corresponding quadrature rules for the Volterra integral term. Furthermore, for the temporal discretizations, the long-time global stability and convergence with formally and accurate second orders are proved under the appropriate assumptions.