The algebraic flux correction methodology [1], [2] is extended to systems of nonlinear conservation laws.
The group finite element formulation is employed for the treatment of the compressible Euler equations.
An efficient algorithm is presented 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 operator/preconditioner is constructed by adding scalar artificial viscosity
proportional to the spectral radius of the Roe matrix. The limiting of raw antidiffusive fluxes is performed
in terms of local characteristic variables. A node-oriented flux limiter of TVD type is applied to the vector
of fluxes in each coordinate direction. The nonlinear algebraic system is solved by a preconditioned defect
correction scheme. The use of a block-diagonal preconditioner with scalar dissipation makes it possible to
decouple the discretized Euler equations and solve them in a segregated fashion. As an alternative, a
coupled solution strategy (global BiCGSTAB method with a block-Gauß-Seidel preconditioner) is
introduced for applications which call for the use of large time steps.
The applicability of the algebraic flux correction paradigm on unstructured meshes suggests the use of
adaptive grid refinement techniques. The co-existence of two different discretizations can be utilized to
devise a usable error indicator based on the amount of rejected antidiffusion which can be evaluated on the
fly at no additional cost. Various algorithmic aspects including the choice of efficient data structures for
dynamically changing unstructured meshes and the implementation of characteristic boundary conditions
are addressed. Simulation results are presented for inviscid flows in a wide range of Mach numbers.