Skip to main content
SpringerLink
Log in

Time-Dependent Conservation Laws on Cut Cell Meshes and the Small Cell Problem

  • Conference paper
  • First Online:
Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples (FVCA 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 323))

Included in the following conference series:

Abstract

When solving time-dependent conservation laws on cut cell meshes, one has to face the small cell problem: standard explicit schemes are not stable if the time step is chosen based on the size of the background cells. Therefore, special schemes must be developed. The first part of this contribution discusses the small cell problem in detail and summarizes several existing solution approaches in the context of both finite volume (FV) schemes and discontinuous Galerkin (DG) schemes. In the second part, we present our two fundamentally different solution approaches for overcoming the small cell problem: the FV based mixed explicit implicit scheme, developed in collaboration with Berger (J. Sci. Comput. 71, pp. 919–943, 2017), and the DG based Domain-of-Dependence (DoD) stabilization, joint work with Engwer, Nüßing, and Streitbürger (ArXiv:1906.05642).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Berger, M., Helzel, C.: A simplified h-box method for embedded boundary grids. SIAM J. Sci. Comput. 34(2), A861–A888 (2012)

    Article  MathSciNet  Google Scholar 

  2. Berger, M.J., Helzel, C., LeVeque, R.J.: H-Box method for the approximation of hyperbolic conservation laws on irregular grids. SIAM J. Numer. Anal. 41, 893–918 (2003)

    Article  MathSciNet  Google Scholar 

  3. Burman, E.: Ghost penalty. C. R. Math. 348(21), 1217–1220 (2010)

    Article  MathSciNet  Google Scholar 

  4. Chern, I.L., Colella, P.: A conservative front tracking method for hyperbolic conservation laws. Tech. rep., Lawrence Livermore National Laboratory, Livermore, CA (1987). Preprint UCRL-97200

    Google Scholar 

  5. Colella, P.: A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Stat. Comput. 6, 104–117 (1985)

    Article  MathSciNet  Google Scholar 

  6. Colella, P., Graves, D.T., Keen, B.J., Modiano, D.: A cartesian grid embedded boundary method for hyperbolic conservation laws. J. Comput. Phys. 211, 347–366 (2006)

    Article  MathSciNet  Google Scholar 

  7. Engwer, C., May, S., Nüßing, C., Streitbürger, F.: A stabilized discontinuous Galerkin cut cell method for discretizing the linear transport equation (2019). ArXiv:1906.05642

  8. Gokhale, N., Nikiforakis, N., Klein, R.: A dimensionally split cartesian cut cell method for hyperbolic conservation laws. J. Comput. Phys. 364, 186–208 (2018)

    Article  MathSciNet  Google Scholar 

  9. Gottlieb, S., Shu, C.W.: Total-variation-diminishing Runge-Kutta schemes. Math. Comp. 67, 73–85 (1998)

    Article  MathSciNet  Google Scholar 

  10. Gürkan, C., Massing, A.: A stabilized cut discontinuous Galerkin framework: II. Hyperbolic problems. ArXiv:1807.05634

  11. Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comp. 29, 396–406 (1975)

    Article  MathSciNet  Google Scholar 

  12. Hartmann, D., Meinke, M., Schröder, W.: An adaptive multilevel multigrid formulation for cartesian hierarchical grid methods. Comput. Fluids 37, 1103–1125 (2008)

    Article  MathSciNet  Google Scholar 

  13. Helzel, C., Berger, M.J., LeVeque, R.: A high-resolution rotated grid method for conservation laws with embedded geometries. SIAM J. Sci. Comput. 26, 785–809 (2005)

    Article  MathSciNet  Google Scholar 

  14. Hunt, J.D.: An adaptive 3D cartesian approach for the parallel computation of inviscid flow about static and dynamic configurations. Ph.D. thesis, University of Michigan (2004)

    Google Scholar 

  15. Klein, R., Bates, K.R., Nikiforakis, N.: Well-balanced compressible cut-cell simulation of atmospheric flow. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367, 4559–4575 (2009)

    Google Scholar 

  16. Krivodonova, L., Qin, R.: A discontinuous Galerkin method for solutions of the Euler equations on cartesian grids with embedded geometries. J. Comput. Sci-neth. 4, 24–35 (2013)

    Article  Google Scholar 

  17. May, S., Berger, M., Laakmann, F.: Accuracy considerations of mixed explicit implicit schemes for embedded boundary meshes. In: Eberhardsteiner, J., Schöberl, M. (eds.) PAMM. Proceedings in Applied Mathematics and Mechanics, Pamm.201900411, Wiley (2019)

    Google Scholar 

  18. May, S., Berger, M.: A mixed explicit implicit time stepping scheme for cartesian embedded boundary meshes. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds.) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, pp. 393–400. Springer International Publishing (2014)

    Google Scholar 

  19. May, S.: Embedded boundary methods for flow in complex geometries. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University (2013)

    Google Scholar 

  20. May, S., Berger, M.J.: An explicit implicit scheme for cut cells in embedded boundary meshes. J. Sci. Comput. 71, 919–943 (2017)

    Article  MathSciNet  Google Scholar 

  21. Müller, B., Krämer-Eis, S., Kummer, F., Oberlack, M.: A high-order discontinuous Galerkin method for compressible flows with immersed boundaries. Int. J. Numer. Meth, Eng (2016)

    Google Scholar 

  22. Pember, R., Bell, J.B., Colella, P., Crutchfield, W., Welcome, M.L.: An adaptive cartesian grid method for unsteady compressible flow in irregular regions. J. Comput. Phys. 120, 278–304 (1995)

    Article  MathSciNet  Google Scholar 

  23. Quirk, J.J.: An alternative to unstructured grids for computing gas dynamic flows around arbitrarily complex two-dimensional bodies. Comput. Fluids 23(1), 125–142 (1994)

    Article  Google Scholar 

  24. Sticko, S., Kreiss, G.: Higher order cut finite elements for the wave equation. J. Sci. Comput. 80, 1867–1887 (2019)

    Article  MathSciNet  Google Scholar 

  25. Streitbürger, F., Engwer, C., May, S., Nüßing, C.: Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equation (2019). ArXiv:1912.11933

  26. van Leer, B.: Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s methods. J. Comput. Phys. 32, 101–136 (1979)

    Google Scholar 

  27. Wendroff, B., White, A.B.: A supraconvergent scheme for nonlinear hyperbolic systems. Comput. Math. Appl 18(8), 761–767 (1989)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technical University of Munich, Boltzmannstraße 3, Garching Close to Munich, 85748, Germany

    Sandra May

Authors
  1. Sandra May
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sandra May .

Editor information

Editors and Affiliations

  1. NORCE Norwegian Research Centre AS, Bergen, Norway

    Robert Klöfkorn

  2. Department of Mathematics, University of Bergen, Bergen, Norway

    Eirik Keilegavlen

  3. Department of Mathematics, University of Bergen, Bergen, Norway

    Florin A. Radu

  4. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

Rights and permissions

Reprints and permissions

Copyright information

© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

May, S. (2020). Time-Dependent Conservation Laws on Cut Cell Meshes and the Small Cell Problem. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation