Skip to main content
SpringerLink
Log in

High-Order Upwind Methods for Wave Equations on Curvilinear and Overlapping Grids

  • Conference paper
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 106))

  • 1328 Accesses

Abstract

In this work we discuss a newly developed class of robust and high-order accurate upwind schemes for wave equations in second-order form on curvilinear and overlapping grids. The schemes are based on embedding d’Alembert’s exact solution for a local Riemann-type problem directly into the discretization (Banks and Henshaw, J Comput Phys 231(17):5854–5889, 2012). High-order accuracy is obtained using a single-step space-time scheme. Overlapping grids are used to represent geometric complexity. The method of manufactured solutions is used to demonstrate that the dissipation introduced through upwinding is sufficient to stabilize the wave equation in the presence of overlapping grid interpolation.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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. D. Appelö, J.W. Banks, W.D. Henshaw, D.W. Schwendeman, Numerical methods for solid mechanics on overlapping grids: linear elasticity. J. Comput. Phys. 231 6012–6050 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. J.W. Banks, W.D. Henshaw, Upwind schemes for the wave equation in second-order form. J. Comput. Phys. 231(17), 5854–5889 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. J.W. Banks, D.W. Schwendeman, A.K. Kapila, W.D. Henshaw, A high-resolution Godunov method for compressible multi-material flow on overlapping grids. J. Comput. Phys. 223, 262–297 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. J.W. Banks, W.D. Henshaw, J.N. Shadid, An evaluation of the FCT method for high-speed flows on structured overlapping grids. J. Comput. Phys. 228(15), 5349–5369 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  6. G. Chesshire, W. Henshaw, Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys. 90, 1–64 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. B. Cockburn, C.W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite-element method for conservation-laws 2: general framework. Math. Comput. 52, 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  8. P. Colella, P.R. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54(1), 174–201 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Courant, E. Isaacson, M. Rees, On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure. Appl. Math. 5, 243–255 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Fornberg, A practical Guide to Pseudospectral Methods (Cambridge University Press, Cambridge, 1996)

    MATH  Google Scholar 

  11. B. Gustafsson, H.-O. Kreiss, J. Oliger, Time Dependent Problems and Difference Methods (Wiley, New York, 1995)

    MATH  Google Scholar 

  12. A. Harten, B. Engquist, S. Osher, S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. W.D. Henshaw, A high-order accurate parallel solver for Maxwell’s equations on overlapping grids. SIAM J. Sci. Comput. 28(5), 1730–1765 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. W.D. Henshaw, D.W. Schwendeman, Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow. J. Comput. Phys. 216(2), 744–779 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, New York, NY, USA

    J. W. Banks & W. D. Henshaw

Authors
  1. J. W. Banks
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. D. Henshaw
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. W. Banks .

Editor information

Editors and Affiliations

  1. School of Computing , University of Utah, Salt Lake City, Utah, USA

    Robert M. Kirby

  2. School of Computing, University of Utah, Salt Lake City, Utah, USA

    Martin Berzins

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Banks, J.W., Henshaw, W.D. (2015). High-Order Upwind Methods for Wave Equations on Curvilinear and Overlapping Grids. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_10

Download citation

Publish with us

Policies and ethics

Search

Navigation