Skip to main content
SpringerLink
Log in
Book cover

Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects pp 393–400Cite as

A Mixed Explicit Implicit Time Stepping Scheme for Cartesian Embedded Boundary Meshes

  • Conference paper
  • First Online:

  • 1272 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 77))

Abstract

We present a mixed explicit implicit time stepping scheme for solving the linear advection equationMay, Sandra on a Cartesian embedded boundary mesh. The scheme represents a new approach for overcoming the small cell problem—that standard finite volume schemes are not stable on the arbitrarily small cut cells. It uses implicit time stepping on cut cells for stability. On standard Cartesian cells, explicit time stepping is employed. This keeps the cost small and makes it possible to extend existing schemes from Cartesian meshes to Cartesian embedded boundary meshes. The coupling is done by flux bounding, for which we can prove a TVD result. We present numerical results in one and two dimensions showing second-order convergence in the \(L^1\) norm and between first- and second-order convergence in the \(L^{\infty }\) norm.

This is a preview of subscription content, 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
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

Learn about institutional subscriptions

References

  1. Aftosmis, M.J., Berger, M.J., Melton, J.E.: Robust and efficient Cartesian mesh generation for component-based geometry. AIAA J. 36(6), 952–960 (1998)

    Google Scholar 

  2. Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. J. Comput. Phys. 142, 1–46 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Almgren, A.S., Bell, J.B., Szymczak, W.G.: A numerical method for the incompressible Navier-Stokes equations based on an approximate projection. SIAM J. Sci. Comput. 17(2), 358–369 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85(2), 257–283 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bell, J.B., et al.: BoxLib user’s guide. Technical Report, CCSE, Lawrence Berkeley National Laboratory. https://ccse.lbl.gov/BoxLib/BoxLibUsersGuide.pdf (2012)

  6. Berger, M.J., Helzel, C., LeVeque, R.: H-box methods for the approximation of hyperbolic conservation laws on irregular grids. SIAM J. Numer. Anal. 41(3), 893–918 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boris, J.P., Book, D.L.: Flux corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11, 38–69 (1973)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  9. Colella, P.: Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87, 171–200 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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  MATH  MathSciNet  Google Scholar 

  11. 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  MATH  MathSciNet  Google Scholar 

  12. Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge-Kutta methods. Appl. Numer. Math. 59, 373–392 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  14. 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  MATH  MathSciNet  Google Scholar 

  15. Zalesak, S.T.: Fully multidimsional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31, 335–362 (1979)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Ann Almgren, John Bell, and Andy Nonaka from Lawrence Berkeley National Laboratory for providing and helping the authors with the software packages BoxLib and VarDen. This work was supported in part by the DOE office of Advanced Scientific Computing under grant DE-FG02-88ER25053 and by AFOSR grant FA9550-13-1-0052. S. M. was also supported by ERC STG. N 306279, SPARCLE.

Author information

Authors and Affiliations

  1. ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland

    Sandra May

  2. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Marsha Berger

Authors
  1. Sandra May
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marsha Berger
    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. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

  2. Institute Comp. Applied Mathematics, University of Münster Center for Nonlinear Sciences (CeNoS ), Münster, Germany

    Mario Ohlberger

  3. Inst. Appl. Analysis and Num. Simulation, University of Stuttgart, Stuttgart, Germany

    Christian Rohde

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

May, S., Berger, M. (2014). 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. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_38

Download citation

Publish with us

Policies and ethics

Search

Navigation