The flux-corrected-transport (FCT) paradigm [1], [2] 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 [3]. 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.
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In the second part of the talk, the scalar FEM-FCT methodology is
carried over to nonlinear hyperbolic systems [4]. 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. Application of
synchronized correction factors to the conservative fluxes is shown
to preserve positivity. 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