Error estimate of BDF2 scheme on a Bakhvalov-type mesh for a singularly perturbed Volterra integro-differential equation

Li-Bin Liu; Yige Liao; Guangqing Long; Li-Bin Liu; Yige Liao; Guangqing Long

doi:10.3934/nhm.2023023

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 547-561. doi: 10.3934/nhm.2023023

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Error estimate of BDF2 scheme on a Bakhvalov-type mesh for a singularly perturbed Volterra integro-differential equation

School of Mathematics and Statistics, Nanning Normal University, Nanning 530100, China

Received: 26 November 2022 Revised: 11 January 2023 Accepted: 11 January 2023 Published: 31 January 2023

A singularly perturbed Volterra integro-differential problem is considered. The variable two-step backward differentiation formula is used to approximate the first-order derivative term and the trapezoidal formula is used to discretize the integral term. Then, the stability and convergence analysis of the proposed numerical method are proved. It is shown that the proposed scheme is second-order uniformly convergent with respect to perturbation parameter $ \varepsilon $ in the discrete maximum norm. Finally, a numerical experiment verifies the theoretical results.
- Volterra integro-differential equation,
- singularly perturbed,
- two-step backward differentiation formula,
- Bakhvalov mesh
Citation: Li-Bin Liu, Yige Liao, Guangqing Long. Error estimate of BDF2 scheme on a Bakhvalov-type mesh for a singularly perturbed Volterra integro-differential equation[J]. Networks and Heterogeneous Media, 2023, 18(2): 547-561. doi: 10.3934/nhm.2023023

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Abstract

A singularly perturbed Volterra integro-differential problem is considered. The variable two-step backward differentiation formula is used to approximate the first-order derivative term and the trapezoidal formula is used to discretize the integral term. Then, the stability and convergence analysis of the proposed numerical method are proved. It is shown that the proposed scheme is second-order uniformly convergent with respect to perturbation parameter $ \varepsilon $ in the discrete maximum norm. Finally, a numerical experiment verifies the theoretical results.

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