Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

A spatial two-grid compact difference method for the nonlinear Volterra integro-differential equations with the Abel kernel is proposed to reduce the computational cost and improve the accuracy of the scheme. The proposed scheme firstly solves a small nonlinear compact finite difference system on a coarse grid and then solves a large linear compact finite difference system on a fine grid based on the coarse-grid solution using Newton’s linearization and the higher-order mapping operator. Then, we combine the properties of the higher-order mapping operator with the two-grid analysis method as well as the singularity of the solutions to prove the stability and convergence of the proposed algorithm under the \(L^2\) $L^2$ -norm with the order \(O(\tau ^2+H^8+h^4)\) $O({\tau }^2+H^8+h^4)$ . Moreover, the constructed method is extended to the two-dimensional case. Several numerical experiments verify the effectiveness of the proposed method and show its competitiveness compared with the existing methods.