Optimal convergence of an Euler and finite difference method for nonlinear partial integro-differential equations

Fully discretized Euler method in time and finite difference method in space are constructed and analyzed for a class of nonlinear partial integro-differential equations emerging from practical applications of a wide range, such as the modeling of physical phenomena associated with non-Newtonian fluids. Though first-order and second-order time discretizations (based on truncation errors) have been investigated recently, due to lack of the smoothness of the exact solutions, the overall numerical procedures do not achieve the optimal convergence rates in time. In this paper, however, by using the energy method, we prove that it is possible for the scheme to obtain the optimal convergence rate O(τ). Numerical demonstrations are given to illustrate our result.