Performance Analysis of the Crank–Nicolson Scheme for Volterra Integro-Differential Equations

Source abstract

In this study, the Crank–Nicolson (CN) method is applied to solve higher-order Volterra integro-differential equations. The original integro-differential equation is first transformed into an equivalent system of first-order equations. The CN method is then used to discretize the differential terms, while the integral terms are approximated using the composite trapezoidal rule. To handle the nonlinear terms, a semi-implicit predictor–corrector scheme is employed. For the stability analysis, perturbation arguments and Grönwall's inequality are used to show that the numerical errors remain bounded under suitable conditions. Numerical experiments on both linear and nonlinear Volterra integro-differential equations demonstrate that the proposed method produces highly accurate solutions and achieves second-order convergence for linear problems. For nonlinear problems, however, a reduction in the convergence rate is observed.