The flux-corrected-transport (FCT) paradigm is generalized to
implicit finite element schemes for hyperbolic systems and applied
to the Euler equations of gas dynamics. First,
the scalar version of the algorithm is presented. Building on the
mathematical theory of positivity-preserving schemes, a nonoscillatory
low-order method is constructed by elimination of negative off-diagonal
entries of the discrete transport operator. The difference between the
linear discretizations of high and low order is decomposed into a sum
of antidiffusive fluxes and embedded into an outer iteration loop of
defect correction type. The monotone low-order operator
enjoys the M-matrix property and constitutes an excellent preconditioner.
Zalesak`s multidimensional limiter is generalized to implicit
time-stepping and used as a starting point to derive a superior
iterative FCT limiter independent of the time step. It is proved that
the fully implicit FEM-FCT scheme is unconditionally positivity-preserving,
and a computable upper bound is derived for the admissible time step under
a (semi-) explicit time discretization. The methodology is extended to
problems with nonlinear source/sink terms by linearizing them in a
positivity-preserving fashion. Last but not least, the origins of
ripples produced by standard FCT methods in some cases are elucidated
and effective remedies are proposed.
In the second part of the talk, the scalar FEM-FCT methodology is carried
over to nonlinear hyperbolic systems. An efficient edge-based algorithm for
the matrix assembly is introduced. The `discrete upwinding` is performed
as in the scalar case by rendering local Jacobians positive definite.
This leads to a multidimensional extension of Roe`s approximate Riemann
solver. For practical purposes, the low-order method can be constructed
by adding scalar dissipation proportional to the spectral radius of the
Roe matrix. The coupled system is decomposed into scalar subproblems
for individual variables by resorting to a block-diagonal low-order
preconditioner for the outer defect correction loop. Synchronized
correction factors are applied to the conservative fluxes. Limiting
in terms of arbitrary `indicator` variables is shown to be feasible.
The performance of the algorithm is illustrated by numerical examples
for the compressible Euler equations in a wide range of Mach numbers.
Both stationary and time-dependent results are presented for standard
two-dimensional benchmark problems.