The following second-order methods are all closely related: 
The leapfrog method for the linear advection equation,
the Verlet and Gautschi methods for the wave equation,
the second-order Richardson and SOR methods for linear systems,
and the second-order Chebyshev iterative method for finding eigenvalues.
The linear stability analysis for all of these methods reduces to the classical equation generating the Chebyshev polynomials.

For many of these methods, it is desirable to use a step-size very close to the stability boundary.  Although formally stable, the resultant numerical solution can grow by several orders of magnitude.  A Chebyshev analysis accurately describes this growth. The analysis also suggests how to eliminate such growth even at the stability boundary.
Growth may be eliminated, or controlled, by (a) choosing suitable initial data, (b) choosing a suitable filter or finishing method at the end of the computation, or (c) by filtering mid-calculation and re-starting using the filtered values.