A new approach to the design of flux-corrected transport (FCT)
  algorithms for linear/bilinear finite element approximations
  of convection-dominated transport problems is pursued. The
  raw antidiffusive fluxes are linearized about an intermediate
  solution computed by a positivity-preserving low-order scheme.
  By virtue of this linearization, the costly evaluation of
  correction factors needs to be performed just once per time
  step, and no nonlinear algebraic systems need to be solved if
  the governing equation is linear. Furthermore, no questionable
  `prelimiting` of antidiffusive fluxes is required, which
  eliminates the danger of artificial steepening. Three FEM-FCT
  algorithms based on the Runge-Kutta, Crank-Nicolson, and
  backward Euler time-stepping are proposed. Numerical results
  are presented for the linear convection equation as well as
  for the shock tube problem of gas dynamics.