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

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

Abstract

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.

Notes

Acknowledgements

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.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

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

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