  The isentropic compressible Cahn-Hilliard-Navier-Stokes equations is a system
of fourth-order partial differential equations that model the evolution of some
binary fluids under convection.
  The purpose of this paper is the design of efficient numerical schemes to
approximate the solution of initial-boundary value problems with these
equations. The efficiency stems from the implicit treatment of the high-order
terms in the equations. Our proposal is a second-order linearly
implicit-explicit time stepping scheme applied in a method of lines approach,
in which the convective terms are treated explicitly and only linear systems
have to be solved.
  Some experiments are performed to assess the validity and efficiency of this
proposal.
