Numerical analysis of temporal second‐order accurate scheme for the abstract Volterra integrodifferential equation

Abstract

In this paper, a second‐order accurate scheme is considered for the temporal discretization of the abstract Volterra integrodifferential equation with a weakly singular kernel. The time discrete scheme is constructed by the Crank–Nicolson method for approximating the time derivative and product integration (PI) rule for approximating the integral term. The proposed scheme employs a graded mesh for time to compensate for the singular behavior of the exact solution at . Under the suitable assumptions, the stability and convergence are established by the energy argument, and the error is of order , where is the parameter for the time grids. Numerical experiments validate the theoretical estimate.