  Flux-corrected transport (FCT) is one of the flux limiter methods. Unlike the
total variation diminishing methods, obtaining the known FCT formulas for
computing flux limiters is not quite transparent, and their transformation is
not obvious when the original differential operator changes. We propose a novel
formal mathematical approach to design flux correction for weighted hybrid
difference schemes by using linear programming. The hybrid scheme is a linear
combination of a monotone scheme and a high order scheme. The determination of
maximal antidiffusive fluxes is treated as an optimization problem with a
linear objective function. To obtain constraints for the optimization problem,
inequalities that are valid for the monotone difference scheme are applied to
the hybrid difference scheme. The numerical solution of the nonlinear
optimization problem is reduced to the iterative solution of linear programming
problems. A nontrivial approximate solution of the corresponding linear
programming problem can be treated as the required flux limiters. We present
flux correction formulas for scalar hyperbolic conservation laws and
convection-diffusion equations. The designed flux-corrected transport for
scalar hyperbolic conservation laws yields entropy solutions. Numerical results
are presented.
