The flux-corrected-transport paradigm is generalized to implicit finite element schemes for hyperbolic systems. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. A mathematical theory for positivity-preserving schemes is
reviewed. A nonoscillatory low-order method is constructed by elimination of negative off-diagonal entries of the discrete transport operator. Zalesak`s multi-dimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof
of positivity is provided. A feasible generalization of the scalar methodology for the construction of low-order methods is elucidated. An efficient edge-based algorithm for the matrix assembly for nonlinear systems is devised. Scalar dissipation proportional to the spectral radius of the Roe matrix is used to construct the low-order method for hyperbolic systems. A block-diagonal preconditioner is utilized to work out an efficient defect correction procedure for coupled systems. Several 2D examples for both stationary and highly dynamic flow in a wide range of Mach numbers are presented
to demonstrate the potential of the new methodology.