  We investigate the stability of two families of three-level two-step schemes
that extend the classical second order BDF (BDF2) and second order
Adams-Moulton (AM2) schemes. For a free parameter restricted to an appropriate
range that covers the classical case, we show that both the generalized BDF2
and the generalized AM2 schemes are A-stable. We also introduce the concept of
uniform-in-time stability which characterizes a scheme's ability to inherit the
uniform boundedness over all time of the solution of damped and forced equation
with the force uniformly bounded in time. We then demonstrate that A-stability
and uniform-in-time stability are equivalent for three-level two-step schemes.
Next, these two families of schemes are utilized to construct efficient and
unconditionally stable IMEX schemes for systems that involve a damping term, a
skew symmetric term, and a forcing term. These novel IMEX schemes are shown to
be uniform-in-time energy stable in the sense that the norm of any numerical
solution is bounded uniformly over all time, provided that the forcing term is
uniformly bounded time, the skew symmetric term is dominated by the dissipative
term, together with a mild time-step restriction. Numerical experiments verify
our theoretical results. They also indicate that the generalized schemes could
be more accurate and/or more stable than the classical ones for suitable choice
of the parameter.
