Applied Math Seminar: Algebraic flux correction for finite element discretizations of hyperbolic conservation laws
Seminar | October 31 | 11 a.m.-12 p.m. | 891 Evans Hall
Dmitri Kuzmin, TU Dortmund University
In this talk, we review some recent advances in the analysis and design of algebraic flux correction (AFC) schemes for hyperbolic problems. In contrast to most variational stabilization techniques, AFC approaches modify the standard Galerkin discretization in a way which provably guarantees the validity of discrete maximum principles for scalar conservation laws and invariant domain preservation for hyperbolic systems. The corresponding inequality constraints are enforced by adding diffusive fluxes, and bound-preserving antidiffusive corrections are performed to obtain nonlinear high-order approximations. After introducing the AFC methodology and the underlying theoretical framework in the context of continuous piecewise-linear finite element discretizations, we present some of the limiting techniques that we use in high-resolution AFC schemes. As an alternative to flux-corrected transport (FCT) algorithms which apply limited antidiffusive corrections to bound-preserving low-order solutions, we propose a new limiting strategy based on representation of these solutions as convex combinations of "bar states" satisfying physical and numerical admissibility conditions. Each antidiffusive flux is limited so as to guarantee that the associated bar state remains in the convex invariant set and preserves appropriate local bounds. There is no free parameter, and the nonlinear discrete problem is well-defined even in the steady-state limit. In the case study for the Euler equations of gas dynamics, we enforce local maximum principles for the density, velocity, and specific total energy in addition to positivity preservation for the density and pressure. The results of numerical studies for standard test problems illustrate the ability of the methods under investigation to resolve steep gradients without generating spurious oscillations. In the last part of this talk, we discuss the design of AFC schemes for high-order finite elements. The approaches to be explored are based on the use of Bernstein basis functions and partitioned finite element spaces.