Algebraic Flux Correction I

Scalar Conservation Laws
Part of the Scientific Computation book series (SCIENTCOMP)


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.


Total Variation Diminish Galerkin Scheme Flux Limiter Slope Limiter Discrete Maximum Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Anderson, D.G.: Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Mach. 12, 547–560 (1965) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arioli, M., Loghin, D., Wathen, A.J.: Stopping criteria for iterations in finite element methods. Numer. Math. 99, 381–410 (2006) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arminjon, P., Dervieux, A.: Construction of TVD-like artificial viscosities on 2-dimensional arbitrary FEM grids. INRIA Research Report 1111 (1989) Google Scholar
  4. 4.
    Baum, J.D., Löhner, R.: Numerical simulation of pilot/seat ejection from an F-16. AIAA Paper, 93-0783 (1993) Google Scholar
  5. 5.
    Bochev, P., Ridzal, D., Scovazzi, G., Shashkov, M.: Constrained-optimization based data transfer: a new perspective on flux correction. Chap. 10 in this volume. doi: 10.1007/978-94-007-4038-9_10
  6. 6.
    Boris, J.P., Book, D.L.: Flux-Corrected Transport: I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11, 38–69 (1973) ADSMATHCrossRefGoogle Scholar
  7. 7.
    Book, D.L.: The conception, gestation, birth, and infancy of FCT. Chap. 1 in this volume. doi: 10.1007/978-94-007-4038-9_1
  8. 8.
    Book, D.L., Boris, J.P., Hain, K.: Flux-corrected transport: II. Generalizations of the method. J. Comput. Phys. 18, 248–283 (1975) ADSMATHCrossRefGoogle Scholar
  9. 9.
    Boris, J.P., Book, D.L.: Flux-Corrected Transport: III. Minimal-error FCT algorithms. J. Comput. Phys. 20, 397–431 (1976) ADSMATHCrossRefGoogle Scholar
  10. 10.
    Carette, J.-C., Deconinck, H., Paillère, H., Roe, P.L.: Multidimensional upwinding: its relation to finite elements. Int. J. Numer. Methods Fluids 20, 935–955 (1995) ADSMATHCrossRefGoogle Scholar
  11. 11.
    Catabriga, L., Coutinho, A.L.G.A.: Implicit SUPG solution of Euler equations using edge-based data structures. Comput. Methods Appl. Mech. Eng. 191, 3477–3490 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Ciarlet, P.G., Raviart, P.-A.: Maximum principle and convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2, 17–31 (1973) MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Dietachmayer, G.S.: A comparison and evaluation of some positive definite advection schemes. In: Noyle, J., May, R. (eds.) Computational Techniques and Applications, pp. 217–232. Elsevier, Amsterdam (1986) Google Scholar
  14. 14.
    DeVore, C.R.: An improved limiter for multidimensional flux-corrected transport. NASA Technical Report AD-A360122 (1998) Google Scholar
  15. 15.
    Donea, J., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, Chichester (2003) CrossRefGoogle Scholar
  16. 16.
    Donea, J., Giuliani, S., Laval, H., Quartapelle, L.: Time-accurate solution of advection-diffusion equations by finite elements. Comput. Methods Appl. Mech. Eng. 193, 123–145 (1984) MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Eyert, V.: A comparative study on methods for convergence acceleration of iterative vector sequences. J. Comput. Phys. 124, 271–285 (1996) MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Fang, H., Saad, Y.: Two classes of multisecant methods for nonlinear acceleration. Numer. Linear Algebra Appl. 16, 197–221 (2009) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Faragó, I., Horváth, R., Korotov, S.: Discrete maximum principle for linear parabolic problems solved on hybrid meshes. Appl. Numer. Math. 53, 249–264 (2005) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Fletcher, C.A.J.: The group finite element formulation. Comput. Methods Appl. Mech. Eng. 37, 225–243 (1983) MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Fletcher, C.A.J.: A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers’ equations. J. Comput. Phys. 51, 159–188 (1983) MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Godunov, S.K.: Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271–306 (1959) MathSciNetGoogle Scholar
  23. 23.
    Gottlieb, S., Shu, C.W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73–85 (1998) MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001) MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Hansbo, P.: Aspects of conservation in finite element flow computations. Comput. Methods Appl. Mech. Eng. 117, 423–437 (1994) MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983) MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Hubbard, M.E.: Non-oscillatory third order fluctuation splitting schemes for steady scalar conservation laws. J. Comput. Phys. 222, 740–768 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, Berlin (2003) MATHGoogle Scholar
  29. 29.
    Jameson, A.: Computational algorithms for aerodynamic analysis and design. Appl. Numer. Math. 13, 383–422 (1993) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Jameson, A.: Analysis and design of numerical schemes for gas dynamics 1. Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. Int. J. Comput. Fluid Dyn. 4, 171–218 (1995) CrossRefGoogle Scholar
  31. 31.
    Jemcov, A., Maruszewski, J.P.: Algorithm stabilization and acceleration in computational fluid dynamics: exploiting recursive properties of fixed point algorithms. In: Amano, R.S., Sundén, B. (eds.) Computational Fluid Dynamics and Heat Transfer. WIT Press, Southampton (2010) Google Scholar
  32. 32.
    John, V., Schmeyer, E.: On finite element methods for 3D time-dependent convection-diffusion-reaction equations with small diffusion. Comput. Methods Appl. Mech. Eng. 198, 475–494 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Karátson, J., Korotov, S.: Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 99, 669–698 (2005) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Karátson, J., Korotov, S., Křížek, M.: On discrete maximum principles for nonlinear elliptic problems. Math. Comput. Simul. 76, 99–108 (2007) MATHCrossRefGoogle Scholar
  35. 35.
    Kuzmin, D.: Positive finite element schemes based on the flux-corrected transport procedure. In: Bathe, K.J. (ed.) Computational Fluid and Solid Mechanics, pp. 887–888. Elsevier, Amsterdam (2001) CrossRefGoogle Scholar
  36. 36.
    Kuzmin, D.: On the design of general-purpose flux limiters for implicit FEM with a consistent mass matrix. I. Scalar convection. J. Comput. Phys. 219, 513–531 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  37. 37.
    Kuzmin, D.: Algebraic flux correction for finite element discretizations of coupled systems. In: Oñate, E., Papadrakakis, M., Schrefler, B. (eds.) Computational Methods for Coupled Problems in Science and Engineering II, CIMNE, Barcelona, pp. 653–656 (2007) Google Scholar
  38. 38.
    Kuzmin, D.: On the design of algebraic flux correction schemes for quadratic finite elements. J. Comput. Appl. Math. 218(1), 79–87 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    Kuzmin, D.: Explicit and implicit FEM-FCT algorithms with flux linearization. J. Comput. Phys. 228, 2517–2534 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  40. 40.
    Kuzmin, D.: A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods. J. Comput. Appl. Math. 233, 3077–3085 (2010) MathSciNetADSMATHCrossRefGoogle Scholar
  41. 41.
    Kuzmin, D.: A Guide to Numerical Methods for Transport Equations. University Erlangen-Nuremberg, Erlangen (2010). Google Scholar
  42. 42.
    Kuzmin, D.: Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. (2012, to appear) Google Scholar
  43. 43.
    Kuzmin, D., Möller, M.: Algebraic flux correction I. Scalar conservation laws. In: Kuzmin, D., et al. (eds.) Flux-Corrected Transport: Principles, Algorithms, and Applications, pp. 155–206. Springer, Berlin (2005) CrossRefGoogle Scholar
  44. 44.
    Kuzmin, D., Möller, M.: Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems. J. Comput. Appl. Math. 233, 3113–3120 (2010) MathSciNetADSMATHCrossRefGoogle Scholar
  45. 45.
    Kuzmin, D., Turek, S.: Flux correction tools for finite elements. J. Comput. Phys. 175, 525–558 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  46. 46.
    Kuzmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198, 131–158 (2004) MathSciNetADSMATHCrossRefGoogle Scholar
  47. 47.
    Kuzmin, D., Möller, M., Turek, S.: High-resolution FEM-FCT schemes for multidimensional conservation laws. Comput. Methods Appl. Mech. Eng. 193, 4915–4946 (2004) ADSMATHCrossRefGoogle Scholar
  48. 48.
    Kuzmin, D., Shashkov, M.J., Svyatskiy, D.: A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems. J. Comput. Phys. 228, 3448–3463 (2009) MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    LeVeque, R.J.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Yu.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 227, 492–512 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  51. 51.
    Löhner, R.: Applied CFD Techniques: An Introduction Based on Finite Element Methods, 2nd edn. Wiley, Chichester (2008) Google Scholar
  52. 52.
    Löhner, R.: Edges, stars, superedges and chains. Comput. Methods Appl. Mech. Eng. 111, 255–263 (1994) ADSMATHCrossRefGoogle Scholar
  53. 53.
    Löhner, R., Galle, M.: Minimization of indirect addressing for edge-based field solvers. Commun. Numer. Methods Eng. 18(5), 335–343 (2002) MATHCrossRefGoogle Scholar
  54. 54.
    Löhner, R., Morgan, K., Peraire, J., Vahdati, M.: Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 7, 1093–1109 (1987) MATHCrossRefGoogle Scholar
  55. 55.
    Löhner, R., Morgan, K., Vahdati, M., Boris, J.P., Book, D.L.: FEM-FCT: combining unstructured grids with high resolution. Commun. Appl. Numer. Methods 4, 717–729 (1988) MATHCrossRefGoogle Scholar
  56. 56.
    Luo, H., Baum, J.D., Löhner, R., Fast, A.: Matrix-free implicit method for compressible flows on unstructured grids. J. Comput. Phys. 146, 664–690 (1998) MathSciNetADSMATHCrossRefGoogle Scholar
  57. 57.
    Lyra, P.R.M.: Unstructured grid adaptive algorithms for fluid dynamics and heat conduction. PhD thesis, University of Wales, Swansea (1994) Google Scholar
  58. 58.
    Lyra, P.R.M., Willmersdorf, R.B., Martins, M.A.D., Coutinho, A.L.G.A.: Parallel implementation of edge-based finite element schemes for compressible flow on unstructured grids. In: Proceedings of the 3rd International Meeting on Vector and Parallel Processing, Faculdade de Engenharia da Universidade do Porto, Porto, Portugal, 21–23 Juni (1998) Google Scholar
  59. 59.
    Mer, K.: Variational analysis of a mixed element/volume scheme with fourth-order viscosity on general triangulations. Comput. Methods Appl. Mech. Eng. 153, 45–62 (1998) MathSciNetADSMATHCrossRefGoogle Scholar
  60. 60.
    Möller, M.: Hochauflösende FEM-FCT-Verfahren zur Diskretisierung von konvektionsdominanten Transportproblemen mit Anwendung auf die kompressiblen Eulergleichungen. Diploma thesis, University of Dortmund (2003) Google Scholar
  61. 61.
    Möller, M.: Efficient solution techniques for implicit finite element schemes with flux limiters. Int. J. Numer. Methods Fluids 55, 611–635 (2007) MATHCrossRefGoogle Scholar
  62. 62.
    Ni, P.: Anderson acceleration of fixed-point iteration with applications to electronic structure computations. PhD thesis, Worcester Polytechnic Institute (2009) Google Scholar
  63. 63.
    Schär, C., Smolarkiewicz, P.K.: A synchronous and iterative flux-correction formalism for coupled transport equations. J. Comput. Phys. 128, 101–120 (1996) MathSciNetADSMATHCrossRefGoogle Scholar
  64. 64.
    Oran, E.S., Boris, J.P.: Numerical Simulation of Reactive Flow, 2nd edn. Cambridge University Press, Cambridge (2001) MATHGoogle Scholar
  65. 65.
    Parrott, A.K., Christie, M.A.: FCT applied to the 2-D finite element solution of tracer transport by single phase flow in a porous medium. In: Numerical Methods for Fluid Dynamics, pp. 609–619. Oxford University Press, London (1986) Google Scholar
  66. 66.
    Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York (1980) MATHGoogle Scholar
  67. 67.
    Peraire, J., Vahdati, M., Peiro, J., Morgan, K.: The construction and behavior of some unstructured grid algorithms for compressible flows. In: Numerical Methods for Fluid Dynamics, IV, pp. 221–239. Oxford University Press, Oxford (1993) Google Scholar
  68. 68.
    Selmin, V.: Finite element solution of hyperbolic equations. I. One-dimensional case. INRIA Research Report 655 (1987) Google Scholar
  69. 69.
    Selmin, V.: Finite element solution of hyperbolic equations. II. Two-dimensional case. INRIA Research Report 708 (1987) Google Scholar
  70. 70.
    Selmin, V.: The node-centred finite volume approach: bridge between finite differences and finite elements. Comput. Methods Appl. Mech. Eng. 102, 107–138 (1993) ADSMATHCrossRefGoogle Scholar
  71. 71.
    Selmin, V., Formaggia, L.: Unified construction of finite element and finite volume discretizations for compressible flows. Int. J. Numer. Methods Eng. 39, 1–32 (1996) MATHCrossRefGoogle Scholar
  72. 72.
    Smith, D.A., Ford, W.F., Sidi, A.: Extrapolation methods for vector sequences. SIAM Rev. 29, 199–233 (1987) MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984) MathSciNetADSMATHCrossRefGoogle Scholar
  74. 74.
    van Slingerland, P.: An accurate and robust finite volume method for the advection diffusion equation. M.Sc. thesis, Delft University of Technology (June 2007) Google Scholar
  75. 75.
    van Slingerland, P., Borsboom, M., Vuik, C.: A local theta scheme for advection problems with strongly varying meshes and velocity profiles. Report 08-17, Department of Applied Mathematical Analysis, Delft University of Technology (June 2008) Google Scholar
  76. 76.
    Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1962) Google Scholar
  77. 77.
    Walker, H.W., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49, 1715–1735 (2011) MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31, 335–362 (1979) MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Applied Mathematics IIIUniversity Erlangen-NurembergErlangenGermany

Personalised recommendations