TY - JOUR

T1 - A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

AU - Chen, Hao

AU - Qiu, Wenlin

AU - Zaky, Mahmoud A.

AU - Hendy, Ahmed S.

N1 - The authors are grateful for helpful comments and suggestions from the reviewers. This work was supported by Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20220454). Ahmed S. Hendy wishes to acknowledge the support of the RSF grant, project 22-21-00075.

PY - 2023

Y1 - 2023

N2 - 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 L2-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. © 2023, The Author(s) under exclusive licence to Istituto di Informatica e Telematica (IIT).

AB - 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 L2-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. © 2023, The Author(s) under exclusive licence to Istituto di Informatica e Telematica (IIT).

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U2 - 10.1007/s10092-023-00508-6

DO - 10.1007/s10092-023-00508-6

M3 - Article

VL - 60

JO - Calcolo

JF - Calcolo

SN - 0008-0624

IS - 1

M1 - 13

ER -