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

作     者：Chen, Hao Zaky, Mahmoud A. Hendy, Ahmed S. Qiu, Wenlin 

作者机构：MOE-LCSM School of Mathematics and Statistics Hunan Normal University Hunan Changsha410081 China Department of Applied Mathematics National Research Centre Cairo Dokki12622 Egypt Department of Computational Mathematics and Computer Science Institute of Natural Sciences and Mathematics Ural Federal University 19 Mira St. Yekaterinburg620002 Russia Department of Mathematics Faculty of Science Benha University Benha13511 Egypt 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2022年

核心收录：

主　　题：Nonlinear equations 

摘      要：In this paper, a two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel is proposed to reduce the computation time and improve the accuracy of the scheme developed by Xu et al. (Applied Numerical Mathematics 152 (2020) 169-184). The proposed scheme consists of three steps: First, a small nonlinear system is solved on the coarse grid using fix-point iteration. Second, the Lagrange s linear interpolation formula is used to arrive at some auxiliary values for 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 Crank-Nicolson technique and product integral rule, respectively. With the help of the discrete energy method, the stability and space-time second-order convergence of the proposed approach are obtained in L2-norm. Finally, the numerical results agree with the theoretical analysis and verify the effectiveness of the algorithm. Copyright © 2022, The Authors. All rights reserved.