  Schemes with the second-order approximation in time are considered for
numerical solving the Cauchy problem for an evolutionary equation of first
order with a self-adjoint operator. The implicit two-level scheme based on the
Pad\'{e} polynomial approximation is unconditionally stable. It demonstrates
good asymptotic properties in time and provides an adequate evolution in time
for individual harmonics of the solution (has spectral mimetic stability). In
fact, the only drawback of this scheme is the necessity to solve an equation
with an operator polynomial of second degree at each time level. We consider
modifications of these schemes, which are based on solving equations with
operator polynomials of first degree. Such computational implementations occur,
for example, if we apply the fully implicit two-level scheme (the backward
Euler scheme). A three-level modification of the SM-stable scheme is proposed.
Its unconditional stability is established in the corresponding norms. The
emphasis is on the scheme, where the numerical algorithm involves two stages,
namely, the backward Euler scheme of first order at the first (prediction)
stage and the following correction of the approximate solution using a
factorized operator. The SM-stability is established for the proposed scheme.
To illustrate the theoretical results of the work, a model problem is solved
numerically.
