  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.
