<p style='text-indent:20px;'>We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The Legendre-Petrov-Galerkin method is applied to time discretization and the nonlinear term is dealt with Chebyshev spectral collocation method. The scheme results in a simple algebraic system by choosing appropriate basis functions. Optimal error estimates in <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm for the single and multiple interval methods are given. As an application of the method, we construct the space-time spectral schemes for solving some nonlinear time-dependent partial differential equations (PDEs). Numerical experiments suggest the efficiency of the methods.</p>