Convergence analysis of a second-order finite difference scheme on Shishkin-type meshes for a singularly perturbed Volterra integro-differential equation

Jiwen Chen; Li-Bin Liu; Guangqing Long; Zaitang Huang; Jiwen Chen; Li-Bin Liu; Guangqing Long; Zaitang Huang

doi:10.3934/nhm.2025057

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1333-1345. doi: 10.3934/nhm.2025057

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Convergence analysis of a second-order finite difference scheme on Shishkin-type meshes for a singularly perturbed Volterra integro-differential equation

Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, China

Received: 25 August 2025 Revised: 26 November 2025 Accepted: 02 December 2025 Published: 08 December 2025

In this paper, a novel second-order numerical method on a Shishkin mesh is constructed to solve a singularly perturbed Volterra integro-differential equation. The proposed numerical scheme employs a second-order backward differentiation formula (BDF2) for discretizing the first-order derivative term, while utilizing the trapezoidal rule to approximate the integral term. Specifically, at the grid transition point, a first-order finite difference approximation is implemented to handle the first-order derivative computation. Subsequently, comprehensive truncation error estimations and rigorous convergence analyses are systematically conducted. Finally, two numerical examples are performed to verify the theoretical findings.
- Volterra integro-differential equation,
- singularly perturbed,
- Shishkin mesh,
- BDF2
Citation: Jiwen Chen, Li-Bin Liu, Guangqing Long, Zaitang Huang. Convergence analysis of a second-order finite difference scheme on Shishkin-type meshes for a singularly perturbed Volterra integro-differential equation[J]. Networks and Heterogeneous Media, 2025, 20(4): 1333-1345. doi: 10.3934/nhm.2025057

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Abstract

In this paper, a novel second-order numerical method on a Shishkin mesh is constructed to solve a singularly perturbed Volterra integro-differential equation. The proposed numerical scheme employs a second-order backward differentiation formula (BDF2) for discretizing the first-order derivative term, while utilizing the trapezoidal rule to approximate the integral term. Specifically, at the grid transition point, a first-order finite difference approximation is implemented to handle the first-order derivative computation. Subsequently, comprehensive truncation error estimations and rigorous convergence analyses are systematically conducted. Finally, two numerical examples are performed to verify the theoretical findings.

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