Algebraic flux correction schemes of TVD and FCT type are extended to
systems of hyperbolic conservation laws.  The group finite element
formulation is employed for the treatment of the compressible Euler
equations. An efficient algorithm is proposed for the edge-by-edge
matrix assembly.  A generalization of Roe`s approximate Riemann solver
is derived by rendering all off-diagonal matrix blocks positive
semi-definite. Another usable low-order method is constructed by
adding scalar artificial viscosity proportional to the spectral radius
of the cumulative Roe matrix.  The limiting of antidiffusive fluxes is
performed using a transformation to the characteristic variables or a
suitable synchronization of correction factors for the conservative
ones. The outer defect correction loop is equipped with a
block-diagonal preconditioner so as to decouple the discretized Euler
equations and solve them in a segregated fashion. As an alternative, a
strongly coupled solution strategy (global BiCGSTAB method with a
block-Gau/ss-Seidel preconditioner) is introduced for applications
which call for the use of large time steps.  Various algorithmic
aspects including the implementation of characteristic boundary
conditions are addressed.  Simulation results are presented for
inviscid flows in a wide range of Mach numbers.