The flux-corrected transport (FCT) methodology is generalized to implicit
finite element schemes and applied to the Euler equations of gas dynamics.
For scalar equations, a local extremum diminishing scheme is
constructed by
adding artificial diffusion so as to eliminate negative off-diagonal
entries
from the high-order transport operator. To obtain a nonoscillatory
low-order
method in the case of hyperbolic systems, the artificial viscosity
tensor is
designed so that all off-diagonal blocks of the discrete Jacobians are
rendered positive semi-definite. Compensating antidiffusion is applied
within a fixed-point defect correction loop so as to recover the high
accuracy of the Galerkin discretization in regions of smooth
solutions. All
conservative matrix manipulations are performed edge-by-edge which
leads to
an efficient algorithm for the matrix assembly.