A second-order accurate Crank-Nicolson finite difference method on uniform meshes for nonlinear partial integro-differential equations with weakly singular kernels?

A second-order accurate Crank-Nicolson finite difference method on uniform meshes is proposed and analyzed for a class of nonlinear partial integro-differential equations with weakly singular kernels. The first-order time derivative is approximated by using a Crank-Nicolson time-stepping technique, and the singular integral term is treated by a product averaged integration rule, which preserves the positive semi-definite property of the singular integral operator. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative, and a suitable second-order discretization is adopted for the nonlinear convection term. The solvability, stability and convergence of the method are rigorously proved by the discrete energy method, the positive semi-definite property of the associated quadratic form of the method and a perturbation technique. The error estimation shows that the method has the optimal second-order convergence in time and space for non-smooth solutions. Newton's iterative method and its algorithm implementation are presented to solve the resulting nonlinear system. Numerical results confirm the theoretical convergence result and show the effectiveness of the method. (c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

This paper proposes and analyzes a second-order accurate Crank-Nicolson finite difference method for a class of nonlinear partial integro-differential equations with weakly singular kernels. The method approximates the first-order time derivative using a Crank-Nicolson time-stepping technique and treats the singular integral term with a product averaged integration rule. The method is proven to be solvable, stable, and convergent through the discrete energy method, positive semi-definite property, and perturbation technique. Numerical results confirm the theoretical convergence and demonstrate the effectiveness of the method.