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

TL;DR

This paper introduces a two-grid, second-order temporal scheme for efficiently solving two-dimensional nonlinear Volterra integro-differential equations with weakly singular kernels, enhancing accuracy and reducing computation time.

Contribution

It presents a novel two-grid method combining fix-point iteration, interpolation, and linearized Crank-Nicolson techniques for improved efficiency and accuracy in solving complex integro-differential equations.

Findings

The scheme achieves second-order convergence in space and time.

Numerical results confirm stability and theoretical accuracy.

The method reduces computational effort compared to existing approaches.

Abstract

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 $L^2$-norm. Finally, the numerical results agree with the theoretical analysis and verify the effectiveness of the algorithm.