The flux-corrected-transport paradigm is generalized to finite
element schemes based on arbitrary time-stepping. A conservative
flux decomposition procedure is proposed for both convective and
diffusive terms. Mathematical properties of positivity-preserving
schemes are reviewed. A nonoscillatory low-order method is
constructed by elimination of negative off-diagonal entries
of the discrete transport operator. The linearization of source
terms and extension to hyperbolic systems are discussed. Zalesak`s
multidimensional limiter is employed to switch between linear
discretizations
of high and low order. A rigorous proof of positivity is provided.
The treatment of nonlinearities and iterative solution of linear
systems are addressed.
The performance of the new algorithm
is illustrated by numerical examples for the shock tube problem
in 1D and scalar transport equations in 2D.