Skip to main content
SpringerLink
Log in

An Adaptive Artificial Viscosity Method for the Saint-Venant System

  • Chapter
Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

Part of the book series: Notes on Numerical Fluid Mechanics and Multidisciplinary Design ((NNFM,volume 120))

Abstract

We develop an adaptive artificial viscosity method for the one-dimensional Saint-Venant system of shallow water equations. The proposed method is a semi-discrete finite-volume method based on an appropriate numerical flux and a high-order piecewise polynomial reconstruction. The latter is utilized without any computationally expensive nonlinear limiters, which are typically needed to guarantee nonlinear stability of the scheme. Instead, we enforce stability by adding an adaptive artificial viscosity, whose coefficients are proportional to the size of the weak local residual. Our method is capable to preserve the “lake at rest” steady state and the positivity of water depth. We test the proposed scheme on a number of benchmarks. The obtained numerical results clearly demonstrate that our method is well-balanced, positivity preserving and highly accurate.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25(6), 2050–2065 (2004) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouchut, F.: Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004)

    Book  MATH  Google Scholar 

  3. Constantin, L.A., Kurganov, A.: Adaptive central-upwind schemes for hyperbolic systems of conservation laws. In: Hyperbolic Problems: Theory, Numerics, Applications (Osaka 2004), pp. 95–103. Yokohama Publishers (2006)

    Google Scholar 

  4. Gallouët, T., Hérard, J.M., Seguin, N.: Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. & Fluids 32(4), 479–513 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guermond, J.L., Pasquetti, R., Popov, B.: Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230(11), 4248–4267 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jin, S.: A steady-state capturing method for hyperbolic systems with geometrical source terms. M2AN Math. Model. Numer. Anal. 35(4), 631–645 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jin, S., Wen, X.: Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comput. 26(6), 2079–2101 (2005) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  9. Karni, S., Kurganov, A.: Local error analysis for approximate solutions of hyperbolic conservation laws. Adv. Comput. Math. 22, 79–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Karni, S., Kurganov, A., Petrova, G.: A smoothness indicator for adaptive algorithms for hyperbolic systems. J. Comput. Phys. 178, 323–341 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kurganov, A., Levy, D.: Central-upwind schemes for the saint-venant system. M2AN Math. Model. Numer. Anal. 36, 397–425 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kurganov, A., Lin, C.T.: On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2, 141–163 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Kurganov, A., Liu, Y.: New adaptive artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys. (submitted), http://www.math.tulane.edu/~kurganov/Kurganov-Liu.pdf

  14. Kurganov, A., Noelle, S., Petrova, G.: Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23, 707–740 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5(1), 133–160 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Kurganov, A., Tadmor, E.: New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146(1), 346–365 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  19. Lukácová-Medvidová, M., Noelle, S., Kraft, M.: Well-balanced finite volume evolution Galerkin methods for the shallow water equations. J. Comput. Phys. 221(1), 122–147 (2007)

    Article  MathSciNet  Google Scholar 

  20. Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213(2), 474–499 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Noelle, S., Xing, Y., Shu, C.W.: High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226(1), 29–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38(4), 201–231 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Russo, G.: Central schemes for balance laws. In: Hyperbolic Problems: Theory, Numerics, Applications: Proceedings of the Eighth International Conference in Magdeburg, February/March 2000, p. 821. Birkhauser (2002)

    Google Scholar 

  24. Russo, G.: Central schemes for conservation laws with application to shallow water equations. In: Trends and Applications of Mathematics to Mechanics, pp. 225–246. Springer Milan (2005)

    Google Scholar 

  25. de Saint-Venant, A.: Thèorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l’introduction des marèes dans leur lit. C.R. Acad. Sci. Paris 73, 147–154 (1871)

    MATH  Google Scholar 

  26. Vukovic, S., Sopta, L.: ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J. Comput. Phys. 179(2), 593–621 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xing, Y., Shu, C.W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208(1), 206–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xing, Y., Shu, C.W.: A new approach of high order well-balanced finite volume weno schemes and discontinuous galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys. 1, 100–134 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    Yunlong Chen, Alexander Kurganov, Minlan Lei & Yu Liu

Authors
  1. Yunlong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Kurganov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Minlan Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunlong Chen .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Rainer Ansorge

  2. , Department of Aerospace Engineering, Delft University of Technology, Delft, 2600, Netherlands

    Hester Bijl

  3. , Department of Mathematics and Natural Sc, University of Kassel, Heinrich-Plett-Str. 40, Kassel, 34132, Germany

    Andreas Meister

  4. Institute for Computational Mathematics, Technical University of Braunschweig, Pockelstr. 14, Braunschweig, 38106, Germany

    Thomas Sonar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Chen, Y., Kurganov, A., Lei, M., Liu, Y. (2013). An Adaptive Artificial Viscosity Method for the Saint-Venant System. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33221-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33221-0_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33220-3

  • Online ISBN: 978-3-642-33221-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation