  We develop a semi-implicit algorithm for time-accurate simulation of the
compressible Navier-Stokes equations, with special reference to wall-bounded
flows. The method is based on linearization of the partial convective fluxes
associated with acoustic waves, in such a way to suppress, or at least mitigate
the acoustic time step limitation. Together with replacement of the total
energy equation with the entropy transport equation, this approach avoids the
inversion of block-banded matrices involved in classical methods, which is
replaced by less demanding inversion of standard banded matrices. The method is
extended to deal with implicit integration of viscous terms and to multiple
space dimensions through approximate factorization, and used as a building
block of third-order Runge-Kutta time stepping scheme. Numerical experiments
are carried out for isotropic turbulence, plane channel flow, and flow in a
square duct. All available data support higher computational efficiency than
existing methods, and saving of resources ranging from $85\%$ under
low-subsonic flow conditions, to about $50\%$ in supersonic flow.
