Linear high-order methods are known to violate positivity constraints
and tend to produce spurious oscillations in the vicinity of steep
gradients. This can be overcome by applying variable artificial
diffusion so as to eliminate all negative off-diagonal entries of the
high-order operator. The excessive smearing engendered by the
resulting low-order scheme can be alleviated by using a nonlinear
switching between discretizations of high and low order. This idea can
be traced back to the flux-corrected-transport paradigm introduced by
Boris and Book [1] in the realm of finite differences and popularized
in the finite element context by L/``ohner {/it et al.} [2]. In a
series of recent publications [3], [4] the FEM-FCT methodology was
generalized to arbitrary time-stepping schemes, problems with
nonlinear source terms and systems of hyperbolic conservation laws.
In this talk an efficient edge-by-edge matrix assembly algorithm based
on the splitting of internal and boundary contributions is
proposed. The artificial diffusion is constructed in a tensorial form
so as to render the off-diagonal blocks positive definite. For
efficiency reasons scalar dissipation is adopted, its magnitude being
the same for all variables. For hyperbolic systems it is taken to be
proportional to the spectral radius of the Roe matrix. The
nonlinearities are handled within a fixed point defect correction
procedure. A block-diagonal preconditioner based on the low-order
evolution operator is utilized to decompose the entire system into a
sequence of well-behaved scalar subproblems. Synchronized correction
factors are applied to the conservative fluxes as proposed in [2].
Limiting in terms of nonconservative variables is shown to be
feasible. The new `add-ons` to the FEM-FCT formulation are applied to
reactive bubbly flows and the compressible Euler equations of gas
dynamics.