On the Discrete Maximum Principle for Algebraic Flux Correction Schemes with Limiters of Upwind Type

  • Petr Knobloch
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 120)


Algebraic flux correction (AFC) schemes are applied to the numerical solution of scalar steady-state convection-diffusion-reaction equations. A general result on the discrete maximum principle (DMP) is established under a weak assumption on the limiters and used for proving the DMP for a particular limiter of upwind type under an assumption that may hold also on non-Delaunay meshes. Moreover, a simple modification of this limiter is proposed that guarantees the validity of the DMP on arbitrary simplicial meshes. Furthermore, it is shown that AFC schemes do not provide sharp approximations of boundary layers if meshes do not respect the convection direction in an appropriate way.



This work has been supported through the grant No. 16-03230S of the Czech Science Foundation.


  1. 1.
    Barrenechea, G.R., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427–2451 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barrenechea, G.R., John, V., Knobloch, P.: An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes. Math. Models Methods Appl. Sci. 27(3), 525–548 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Kuzmin, D.: Algebraic flux correction for finite element discretizations of coupled systems. In: Papadrakakis, M., Oñate, E., Schrefler, B. (eds.) Proceedings of the International Conference on Computational Methods for Coupled Problems in Science and Engineering, pp. 1–5. CIMNE, Barcelona (2007)Google Scholar
  4. 4.
    Kuzmin, D.: Explicit and implicit FEM-FCT algorithms with flux linearization. J. Comput. Phys. 228, 2517–2534 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Kuzmin, D.: Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. 236, 2317–2337 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kuzmin, D., Möller, M.: Algebraic flux correction I. Scalar conservation laws. In: Kuzmin, D., Löhner, R., Turek, S. (eds.) Flux-Corrected Transport. Principles, Algorithms, and Applications, pp. 155–206. Springer, Berlin (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Department of Numerical MathematicsCharles UniversityPraha 8Czech Republic

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