Algebraic flux correction for hyperbolic systems: State of the art

  • Kuzmin, Dmitri (TU Dortmund University)

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Algebraic flux correction (AFC) schemes for finite element discretizations of hyperbolic equations modify a standard Galerkin discretization using graph viscosity operators and limiters. Modern extensions to nonlinear systems are based on convex decompositions that ensure preservation of invariant domains. In this talk, we review the current state of the art in the field of AFC. The algorithms to be discussed split a high-order baseline discretization into a low-order approximation of Lax--Friedrichs type and a sum of antidiffusive fluxes that may require limiting. Fractional-step approaches based on the flux-corrected transport (FCT) methodology apply limited fluxes to a low-order predictor. Monolithic alternatives enforce preservation of invariant domains and entropy stability at the level of spatial semi-discretization. Limiters for systems can be classified into synchronized, segregated, and sequential ones. We show how the AFC framework has evolved over the years, highlight interesting relationships between existing methods, and introduce some new approaches. The recent advances to be discussed include extensions of convex limiting to continuous and discontinuous Galerkin methods of very high order. A presentation of limiter-based entropy correction procedures is included as well.