Convergence analysis of the multistep Legendre pseudo-spectral method for Volterra integral equations with vanishing delays

In this paper, we extend the single-step pseudo-spectral method for second kind Volterra integral equations with vanishing variable delays to the multistep pseudo-spectral method. We also analyze the convergence of the hp-version of the multistep pseudo-spectral method under the L2-norm, and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. In addition, it is very suitable for long time calculations and large step size situations. Numerical results show good agreement with the theoretical analysis.