Uniform convergence analysis of the BDF2 scheme on Bakhvalov-type meshes for a singularly perturbed Volterra integro-differential equation

Article Mathematics, Interdisciplinary Applications

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

Li-Bin Liu et al.

Summary: This paper considers a singularly perturbed Volterra integro-differential problem and approximates the first-order derivative term using the variable two-step backward differentiation formula, and discretizes the integral term using the trapezoidal formula. The stability and convergence analysis of the proposed numerical method are then proved. It is shown that the proposed scheme is uniformly second-order convergent with respect to the perturbation parameter e in the discrete maximum norm. Finally, a numerical experiment verifies the theoretical results.

NETWORKS AND HETEROGENEOUS MEDIA (2023)

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Article Mathematics, Applied

A novel parameter-uniform numerical method for a singularly perturbed Volterra integro-differential equation

Li-Bin Liu et al.

Summary: This study investigates a novel parameter-uniform finite difference scheme for a singularly perturbed Volterra integro-differential equation on a Shishkin-type mesh. The problem is discretized using the variable two-step backward differentiation formula (BDF2) for the first-order derivative term and the trapezoidal formula for the integral term. The stability of the proposed numerical method is analyzed, and the convergence analysis shows that the presented method is almost second-order uniformly convergent with respect to the perturbation parameter ? in the discrete maximum norm. Numerical results are provided to support the theoretical findings.

COMPUTATIONAL & APPLIED MATHEMATICS (2023)

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Article Mathematics, Applied

ANALYSIS OF ADAPTIVE BDF2 SCHEME FOR DIFFUSION EQUATIONS

Hong-lin Liao et al.

Summary: In this paper, the variable two-step backward differentiation formula (BDF2) is reexamined using a new theoretical framework, highlighting its stability and convergence properties for linear reaction-diffusion equations. The study shows that under specific conditions, the adaptive BDF2 time-stepping scheme is unconditionally stable and demonstrates (potentially first-order) convergence in the L-2 norm. Additionally, for linear dissipative diffusion problems, the stable BDF2 method is shown to preserve energy dissipation law in the H-1 seminorm and maintain monotonicity in the L-2 norm at discrete levels.

MATHEMATICS OF COMPUTATION (2021)

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Article Mathematics, Interdisciplinary Applications

Convergence analysis of the homogeneous second order difference method for a singularly perturbed Volterra delay-integro-differential equation

Omer Yapman et al.

Summary: The study examines a linear Volterra delay-integro-differential equation with a singular perturbation parameter ε, discretizing the problem using exponentially fitted schemes on Shishkin type meshes. It is proven that the numerical approximations generated by this method converge at a rate of O(N-2lnN) in the discrete maximum norm, where N is the mesh parameter. The numerical results exhibit good agreement with theoretical analysis, demonstrating the effectiveness of the method.

CHAOS SOLITONS & FRACTALS (2021)

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Article Mathematics, Applied

Richardson Extrapolation Method on an Adaptive Grid for Singularly Perturbed Volterra Integro-Differential Equations

Guangqing Long et al.

Summary: This paper examines an adaptive grid method for singularly perturbed Volterra integro-differential equations with a boundary layer. The method is shown to be first-order uniformly convergent with respect to the perturbation parameter and improved using Richardson extrapolation technique. Numerical results are provided to support the theoretical findings.

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION (2021)

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Article Mathematics, Applied