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Flux-Corrected Transport pp 145–192Cite as

Algebraic Flux Correction I

Scalar Conservation Laws

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Part of the book series: Scientific Computation ((SCIENTCOMP))

Abstract

This chapter is concerned with the design of high-resolution finite element schemes satisfying the discrete maximum principle. The presented algebraic flux correction paradigm is a generalization of the flux-corrected transport (FCT) methodology. Given the standard Galerkin discretization of a scalar transport equation, we decompose the antidiffusive part of the discrete operator into numerical fluxes and limit these fluxes in a conservative way. The purpose of this manipulation is to make the antidiffusive term local extremum diminishing. The available limiting techniques include a family of implicit FCT schemes and a new linearity-preserving limiter which provides a unified treatment of stationary and time-dependent problems. The use of Anderson acceleration makes it possible to design a simple and efficient quasi-Newton solver for the constrained Galerkin scheme. We also present a linearized FCT method for computations with small time steps. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations.

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  1. Applied Mathematics III, University Erlangen-Nuremberg, Cauerstr. 11, 91058, Erlangen, Germany

    Dmitri Kuzmin

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  1. , Chair of Applied Mathematics III, University of Erlangen-Nuremberg, Haberstr. 2, Erlangen, 91058, Germany

    Dmitri Kuzmin

  2. , School of Computational Sciences, George Mason University, 4400 University Drive MS 6A2, Fairfax, 22030, Virginia, USA

    Rainald Löhner

  3. Institute of Applied Mathematics, LS III, Dortmund University of Technology, Vogelpothsweg 87, Dortmund, 44227, Germany

    Stefan Turek

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Kuzmin, D. (2012). Algebraic Flux Correction I. In: Kuzmin, D., Löhner, R., Turek, S. (eds) Flux-Corrected Transport. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4038-9_6

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