A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Num...

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Bibliographic Details
Published in:Calcolo Vol. 60; no. 1; p. 13
Main Authors: Chen, Hao, Qiu, Wenlin, Zaky, Mahmoud A., Hendy, Ahmed S.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.03.2023
Springer Nature B.V
Subjects:
Algorithms
Approximation
Derivatives
Differential equations
Energy methods
Integrals
Mathematical analysis
Mathematics
Mathematics and Statistics
Methods
Nonlinear systems
Numerical Analysis
Theory of Computation
Viscoelasticity
Viscosity
Volterra integral equations
Accurate second order
Stability and convergence
65M06
Time two-grid algorithm
Nonlinear fractional evolution equation
45K05
Numerical experiments
65M22
65M12
ISSN:0008-0624, 1126-5434
Online Access:Get full text
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Summary:A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in L 2 -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm.
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ISSN:0008-0624
1126-5434
DOI:10.1007/s10092-023-00508-6