Optimal error estimate of accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels

TL;DR

This paper develops a second-order accurate, unconditionally stable numerical scheme for Volterra integrodifferential equations with tempered multi-term kernels, using Crank-Nicolson and product integration methods, validated by numerical tests.

Contribution

It introduces a novel second-order scheme employing Crank-Nicolson and product integration for tempered kernels, with proven stability and convergence, addressing solution singularities at zero time.

Findings

The scheme achieves second-order temporal accuracy in L2-norm.

Numerical examples confirm the scheme's stability and effectiveness.

The method handles singular behavior at t=0 successfully.

Abstract

In this paper, we investigate and analyze numerical solutions for the Volterra integrodifferential equations with tempered multi-term kernels. Firstly we derive some regularity estimates of the exact solution. Then a temporal-discrete scheme is established by employing Crank-Nicolson technique and product integration (PI) rule for discretizations of the time derivative and tempered-type fractional integral terms, respectively, from which, nonuniform meshes are applied to overcome the singular behavior of the exact solution at $t=0$. Based on deduced regularity conditions, we prove that the proposed scheme is unconditionally stable, and possesses accurately temporal second-order convergence in $L_2$-norm. Numerical examples confirm the effectiveness of the proposed method.