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Algebraic Flux Correction I. Scalar Conservation Laws

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Book cover Flux-Corrected Transport

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

Summary

An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context. A centered space discretization of unstable convective terms is rendered local extremum diminishing by a conservative elimination of negative off-diagonal coefficients from the discrete transport operator. This modification leads to an upwind-biased low-order scheme which is nonoscillatory but overly diffusive. In order to reduce the incurred error, a limited amount of compensating antidiffusion is added in regions where the solution is sufficiently smooth. Two closely related flux correction strategies are presented. The first one is based on a multidimensional generalization of total variation diminishing (TVD) schemes, whereas the second one represents an extension of the FEM-FCT paradigm to implicit time-stepping. Nonlinear algebraic systems are solved by an iterative defect correction scheme preconditioned by the low-order evolution operator which enjoys the M-matrix property. The dffusive and antidiffusive terms are represented as a sum of antisymmetric internodal fluxes which are constructed edge-by-edge and inserted into the global defect vector. The new methodology is applied to scalar transport equations discretized in space by the Galerkin method. Its performance is illustrated by numerical examples for 2D benchmark problems.

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Author information

Authors and Affiliations

  1. Institute of Applied Mathematics (LS III), University of Dortmund, Vogelpothsweg 87, D-44227, Dortmund, Germany

    Dmitri Kuzmin

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  1. Dmitri Kuzmin
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  2. Matthias Möller
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Editors and Affiliations

  1. Department of Mathematics, University of Dortmund, LS III, Vogelpothsweg 87, 44227, Dortmund, Germany

    Dmitri Kuzmin  & Stefan Turek  & 

  2. School of Computational Sciences, George Mason University, Chain Bridge Road, 103 4260, Fairfax, Virginia, 22030-4444, USA

    Rainald Löhner

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Kuzmin, D., Möller, M. (2005). Algebraic Flux Correction I. Scalar Conservation Laws. In: Kuzmin, D., Löhner, R., Turek, S. (eds) Flux-Corrected Transport. Scientific Computation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27206-2_6

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