A new fourth-order difference scheme with a temporal graded mesh for a nonlinear partial integrodifferential equation with the multi-term weakly singular kernels

Abstract In this paper, we consider a new fourth-order difference scheme for solving a nonlinear partial integrodifferential equation with the multi-term weakly singular kernels. First, we obtain a temporal semi-discrete scheme by using the Crank-Nicolson (C-N) method, product-integration (P-I) rule and linear interpolation. Then, we employ the reduced order method and two compact operators to discretize the spatial direction, and finally obtain a linearized fully discrete difference scheme on a graded mesh with the convergence of second-order in time and fourth-order in space. The main contribution of this paper is improve the theoretical and numerical results in [1, 2] and only the linear systems are solved. We prove the theoretical properties of our established scheme, where the existence, uniqueness and convergence are obtained. At the end of the paper, two examples are given, where the numerical results are in line with our theoretical expectations.