Convergence analysis of an efficient multistep pseudo-spectral continuous Galerkin approach for solving Volterra integro-differential equations

In this research article, we apply the multistep pseudo-spectral continuous Galerkin approach for solving the first-order Volterra integro-differential equations by the aid of the orthogonal Legendre polynomials. This approach is a recursive scheme that the accuracy of the numerical solution at the present subinterval depends on the numerical solutions at the previous subintervals. Convergence analysis of the suggested numerical approach is discussed via using an important auxiliary problem. Extensive numerical test problems with high oscillating analytical solutions, exact solutions with steep gradients, and long time computational intervals are considered and both of the h-version and p-version convergence rates are examined experimentally. Finally, the conclusions regarding the presented approach and the considered model are provided and we point out to some other models that can be solved numerically via this efficient and robust method.