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In this paper, we extend the single-step pseudo-spectral method for nonlinear Volterra integral equations of the second kind to the multistep pseudo-spectral method. We also analyze the convergence of the  hp -version of the multistep pseudo-spectral method under the  L  2 -norm, and the result shows 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 experiments confirm the theoretical expectations.

A Multistep Legendre Pseudo-Spectral Method for Nonlinear Volterra Integral Equations