  We study continuous finite element dicretizations for one dimensional
hyperbolic partial differential equations. The main contribution of the paper
is to provide a fully discrete spectral analysis, which is used to suggest
optimal values of the CFL number and of the stabilization parameters involved
in different types of stabilization operators. In particular, we analyze the
streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the
continuous interior penalty (CIP) stabilization method and the local projection
stabilization (LPS). Three different choices for the continuous finite element
space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced
nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last
choice, we only consider inexact quadrature based on the formulas corresponding
to the degrees of freedom of the element, which allows to obtain a fully
diagonal mass matrix. We also compare different time stepping strategies,
namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred
correction time integration methods. The latter allows to alleviate the
computational cost as the mass matrix inversion is replaced by the high order
correction iterations. To understand the effects of these choices, both
time-continuous and fully discrete Fourier analysis are performed. These allow
to compare all the different combinations in terms of accuracy and stability,
as well as to provide suggestions for optimal values discretization parameters
involved. The results are thoroughly verified numerically both on linear and
non-linear problems, and error-CPU time curves are provided. Our final
conclusions suggest that cubature elements combined with SSPRK and CIP or LPS
stabilization are the most promising combinations.
