A nonperiodic Chebyshev spectral method avoiding penalization techniques for a class of nonlinear peridynamic models

In the framework of elastodynamics, peridynamics is a nonlocal theory able to capture singularities without using partial derivatives. The governing equation is a second order in time partial integro-differential equation. In this article, we focus on a one-dimensional nonlinear model of peridynamics and propose a spectral method based on the Chebyshev polynomials to discretize in space. The main capability of the method is that it avoids the assumption of periodic boundary condition in the solution and can benefit of the use of the fast Fourier transform. We show its convergence and find that the method results to be very efficient in terms of accuracy and execution time with respect to spectral methods based on the Fourier trigonometric polynomials associated to a volume penalization technique.