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

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 t=0$$ t equal to 0 $$. Under the suitable assumptions, the stability and convergence are established by the energy argument, and the error is of order k2$$ {k} circumflex 2 $$, where k$$ k $$ is the parameter for the time grids. Numerical experiments validate the theoretical estimate.

Automated summary

This paper proposes a second-order accurate scheme for the temporal discretization of abstract Volterra integrodifferential equation with a weakly singular kernel. The scheme is constructed using the Crank-Nicolson method for the time derivative approximation and the product integration rule for the integral term approximation. The proposed scheme employs a graded mesh for time to compensate for the singular behavior of the exact solution at t=0$$ t equal to 0 $$. Stability and convergence are established by the energy argument, and numerical experiments validate the theoretical estimate.