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.

磁通校正输运（FCT）是磁通限制器方法之一。与总方差减小方法不同，获得用于计算通量限制器的已知FCT公式并不十分透明，并且当原始微分算子更改时，它们的变换也不明显。我们提出了一种新颖的形式化数学方法，通过使用线性规划来设计用于加权混合差分方案的通量校正。混合方案是单调方案和高阶方案的线性组合。最大的确定抗扩散通量被视为具有线性目标函数的优化问题。为了获得优化问题的约束，将对单调差分方案有效的不等式应用于混合差分方案。非线性优化问题的数值解被简化为线性规划问题的迭代解。相应线性规划问题的非平凡近似解可以视为所需的磁通限制器。我们给出了标量双曲守恒定律和对流扩散方程的通量校正公式。为标量双曲守恒定律设计的经过通量校正的输运产生熵解。给出了数值结果。