Upwind Schemes for Scalar Advection-Dominated Problems in the Discrete Exterior Calculus

  • Michael GriebelEmail author
  • Christian Rieger
  • Alexander Schier
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)


We present the discrete exterior calculus (DEC) to solve discrete partial differential equations on discrete objects such as cell complexes. To cope with advection-dominated problems, we introduce a novel stabilization technique to the DEC. To this end, we use the fact that the DEC coincides in special situations with known discretization schemes such as finite volumes or finite differences. Thus, we can carry over well-established upwind stabilization methods introduced for these classical schemes to the DEC. This leads in particular to a stable discretization of the Lie-derivative. We present the numerical features of this new discretization technique and study its numerical properties for simple model problems and for advection-diffusion processes on simple surfaces.



M. Griebel and A. Schier thank the DFG for the financial support through the Priority Programme 1506: Transport Processes at Fluidic Interfaces (SPP 1506). The authors would also like to thank B. Zwicknagl for reading parts of the manuscript and for inspiring discussions.


  1. 1.
    Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auchmann, B., Kurz, S.: A geometrically defined discrete Hodge operator on simplicial cells. IEEE Trans. Magn. 42(4), 643 (2006)CrossRefGoogle Scholar
  3. 3.
    Aurenhammer, F.: Voronoi diagrams – a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)CrossRefGoogle Scholar
  4. 4.
    Baker, B.S., Grosse, E., Rafferty, C.S.: Nonobtuse triangulation of polygons. Discret. Comput. Geom. 3(2), 147–168 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bell, N., Hirani, A.N.: PyDEC: software and algorithms for discretization of exterior calculus. ACM Trans. Math. Softw. (TOMS) 39(1), (2012). doi:10.1145/2382585.2382588Google Scholar
  6. 6.
    Bey, J.: Finite–Volumen–und Mehrgitterverfahren für elliptische Randwertprobleme. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    Croce, R., Griebel, M., Schweitzer, M.A.: Numerical simulation of bubble and droplet-deformation by a level set approach with surface tension in three dimensions. Int. J. Numer. Methods Fluids 62(9), 963–993 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus. Preprint (2005). Google Scholar
  9. 9.
    Edelsbrunner, H.: Shape reconstruction with Delaunay complex. In: Lucchesi, C.L., Moura, A.V. (eds.) LATIN’98: Theoretical Informatics: Third Latin American Symposium Campinas, Brazil, 20–24 Apr 1998 Proceedings, pp. 119–132. Springer, Berlin/Heidelberg (1998)Google Scholar
  10. 10.
    Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Edelsbrunner, H.: Roots of Geometry and Topology. Springer International Publishing, Cham (2014)CrossRefzbMATHGoogle Scholar
  12. 12.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence, RI (2010)zbMATHGoogle Scholar
  13. 13.
    Griebel, M., Dornseifer, T., Neunhoeffer, T.: Numerical Simulation in Fluid Dynamics: A Practical Introduction. Mathematical Modeling and Simulation, vol. 3. SIAM, Philadelphia, PA (1997)Google Scholar
  14. 14.
    Heumann, H.: Eulerian and semi-Lagrangian methods for advection-diffusion of differential forms, Ph.D. thesis, Dissertation, Eidgenössische Technische Hochschule ETH Zürich, Nr. 19608 (2011)Google Scholar
  15. 15.
    Hirani, A.N.: Discrete exterior calculus, Ph.D. thesis, California Institute of Technology (2003).
  16. 16.
    Hirani, A.N., Kalyanaraman, K., VanderZee, E.B.: Delaunay Hodge star. Comput. Aided Des. 45(2), 540–544 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hirani, A.N., Nakshatrala, K.B., Chaudhry, J.H.: Numerical method for Darcy flow derived using discrete exterior calculus. Int. J. Comput. Methods Eng. Sci. Mech. 16(3), 151–169 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Texts in Applied Mathematics, vol. 44. Springer, New York (2003)Google Scholar
  19. 19.
    Laadhari, A., Saramito, P., Misbah, C.: Improving the mass conservation of the level set method in a finite element context. C.R. Math. 348(9), 535–540 (2010)Google Scholar
  20. 20.
    Mohamed, M.S., Hirani, A.N., Ravi, S.: Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes. J. Comput. Phys. 312, 175–191 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mullen, P., McKenzie, A., Pavlov, D., Durant, L., Tong, Y., Kanso, E., Marsden, J.E., Desbrun, M.: Discrete Lie advection of differential forms. Found. Comput. Math. 11(2), 131–149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nestler, M., Nitschke, I., Praetorius, S., Voigt, A.: Orientational order on surfaces – the coupling of topology, geometry and dynamics. ArXiv e-prints (2016).
  23. 23.
    Nitschke, I., Reuther, S., Voigt, A.: Discrete exterior calculus (DEC) for the surface Navier-Stokes equation. ArXiv e-prints (2016).
  24. 24.
    VanderZee, E., Hirani, A.N., Guoy, D., Zharnitsky, V., Ramos, E.: Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions. Comput. Geom. Theory Appl. 46(6), 700–724 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Michael Griebel
    • 1
    Email author
  • Christian Rieger
    • 1
  • Alexander Schier
    • 1
  1. 1.Institute for Numerical SimulationUniversity of BonnBonnGermany

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