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Dmitri Kuzmin, Technical University of Dortmund Institute of Applied Mathematics
Flux-limiting approaches to enforcing discrete maximum principles and entropy stability in finite element schemes
This talk presents a new kind of flux limiters for high-order discretizations of nonlinear hyperbolic problems. The proposed methodology is based on an algebraic splitting of a high-order semi-discrete scheme into a low-order approximation of local Lax-Friedrichs type and an antidiffusive correction term. The latter is decomposed into numerical fluxes, which are limited to preserve relevant properties of the low-order scheme. Preservation of local and/or global bounds for functions of conserved variables is enforced using monolithic convex limiting, a flux correction procedure based on representation of spatial semi-discretizations in terms of admissible intermediate states. Additional limiting is performed to satisfy Tadmor's entropy stability conditions and inequality constraints for fully discrete schemes. The benefit of each correction step is illustrated by numerical examples for scalar equations and hyperbolic systems. The main focus of this presentation is on continuous P1/Q1 Lagrange approximations but extensions to discontinuous Galerkin methods and high-order finite elements are available as well.