Abstract
This study presents a numerical investigation of parabolic Volterra integro-differential equations. The method of lines is employed, in which spatial discretization is performed first, and the spatial derivatives are approximated using second-order central finite differences. The time integration is then carried out using the classical fourth-order Runge–Kutta method. The study consists of two main stages. In the first stage, a semi-discrete formulation is considered, which allows for the analysis of the accuracy and stability properties of the fourth-order Runge–Kutta method with respect to time to be performed independently. In the second stage, a fully discrete scheme is developed, where both spatial and temporal discretizations are combined to construct the practical computational algorithm. For the fully discrete scheme, stability and convergence in the maximum norm are rigorously proven, and the consistency condition is verified. According to the Lax–Richtmyer equivalence theorem, stability together with consistency ensures the convergence of the numerical solution. Moreover, it is demonstrated that the CFL (Courant–Friedrichs–Lewy) condition preserves the stability of the numerical method and prevents solution blow-up. The efficiency and reliability of the proposed approach are confirmed through several numerical experiments and comparisons with other numerical methods.