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In this paper, a two-grid temporal second-order scheme for thetwo-dimensional nonlinear Volterra integro-differential equation with weaklysingular kernel is proposed to reduce the computation time and improve theaccuracy of the scheme developed by Xu et al. (Applied Numerical Mathematics152 (2020) 169-184). The proposed scheme consists of three steps: First, asmall nonlinear system is solved on the coarse grid using fix-point iteration.Second, the Lagrange's linear interpolation formula is used to arrive at someauxiliary values for analysis of the fine grid. Finally, a linearizedCrank-Nicolson finite difference system is solved on the fine grid. Moreover,the algorithm uses a central difference approximation for the spatialderivatives. In the time direction, the time derivative and integral term areapproximated by Crank-Nicolson technique and product integral rule,respectively. With the help of the discrete energy method, the stability andspace-time second-order convergence of the proposed approach are obtained in$L^2$-norm. Finally, the numerical results agree with the theoretical analysisand verify the effectiveness of the algorithm.

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