Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Lax–Friedrichs Multigrid Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

An Erratum to this article was published on 12 April 2015

Abstract

Fast sweeping methods are efficient Gauss–Seidel iterative numerical schemes originally designed for solving static Hamilton–Jacobi equations. Recently, these methods have been applied to solve hyperbolic conservation laws with source terms. In this paper, we propose Lax–Friedrichs fast sweeping multigrid methods which allow even more efficient calculations of viscosity solutions of stationary hyperbolic problems. Due to the choice of Lax–Friedrichs numerical fluxes, general problems can be solved without difficult inversion. High order discretization, e.g., WENO finite difference method, can be incorporated to achieve high order accuracy. On the other hand, multigrid methods, which have been widely used to solve elliptic equations, can speed up the computation by smoothing errors of low frequencies on coarse meshes. We modify the classical multigrid method with regard to properties of viscous solutions to hyperbolic conservation equations by introducing WENO interpolation between levels of mesh grids. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving singularities of the viscosity solutions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  1. Abgrall, R.: Toward the ultimate conservative scheme: following the quest. J. Comput. Phys. 167, 277–315 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abgrall, R., Barth, T.: Residual distribution schemes for conservation laws via adaptive quadrature. SIAM J. Sci. Comput. 24, 732–769 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abgrall, R., Mezine, M.: Construction of second order accurate monotone and stable residual distribution schemes for steady problems. J. Comput. Phys. 195, 474–507 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abgrall, R., Roe, P.L.: High order fluctuation scheme on triangular meshes. J. Sci. Comput. 19, 3–36 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Amarala, S., Wan, J.: Multigrid methods for systems of hyperbolic conservation laws. Multiscale Model. Simul. 11(2), 586–614 (2013)

  6. Buckley, Se E., Leverett, MCi: Mechanism of fluid displacement in sands. Trans. AIME 146(107), 1–7 (1942)

    Google Scholar 

  7. Chen, W., Chou, C.-S., Kao, C.-Y.: Lax-Friedrichs fast sweeping methods for steady state problems for hyperbolic conservation laws. J. Comput. Phys. 234, 452–471 (2013)

  8. Chou, C.-S., Shu, C.-W.: High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes. J. Comput. Phys. 214, 698–724 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deconinck, H., Struijs, R., Bourgeois, G., Roe, P.: Compact advection schemes on unstructured meshes, computational fluid dynamics. Computational Fluid Dynamics, VKI Lecture Series 1993–04, 1993

  10. Embid, P., Goodman, J., Majda, A.: Multiple steady states for 1-d transonic flow. SIAM J. Sci. Stat. Comput. 5, 21–41 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goedbloed, J., Poedts, S.: Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press, (2004)

  12. Harten, A., Hyman, J., Lax, P., Keyfitz, B.: On finite-difference approximations and entropy conditions for shocks. Commun. Pure Appl. Math. 29(3), 297–322 (1976)

  13. Hemker, P.W., Spekreijse, S.P.: Multigrid solution of the steady euler equations. In Dietrich B., Wolfgang H., and Ulrich T., (eds.), Advances in Multi-Grid Methods, of Notes on Numerical Fluid Mechanics, vol. 11, pp. 33–44. Vieweg+Teubner Verlag, (1985)

  14. Jespersen, D.: Design and implementation of a multigrid code for the euler equations. Appl. Math. Comput. 13(3–4), 357–374 (1983)

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Koren, B., Hemker, P.W.: Damped, direction-dependent multigrid for hypersonic flow computations. Appl. Numer. Math. 7(4), 309–328 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13(2), 217–237 (1960)

  18. Leclercq, M.P., Stoufflet, B.: Characteristic multigrid method application to solve the euler equations with unstructured and unnested grids. J. Comput. Phys. 104(2), 329–346 (1993)

    Article  MATH  Google Scholar 

  19. Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  20. Liepmann, H.W., Roshko, A.: Elements of Gas Dynamics. Wiley, New York (1957)

    Google Scholar 

  21. Liu, T.P.: Hyperbolic and viscous conservation laws. CBMS-NSF regional conference series in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000

  22. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially nonoscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Roberts, T., Sidilkover, D., Swanson, R.C.: Textbook multigrid efficiency for the steady Euler equations. Aiaa Pap. 97, 1949 (1997)

  24. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Roe, P.L., Sidilkover, D.: Optimum positive linear schemes for advection in two or three dimensions. SIAM J. Numer. Anal. 29, 1542–1588 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Quarteroni, A. (ed.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer (1998)

  27. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sidikover, D., Brandt, A.: Multigrid solution to steady-state two-dimensional conservation laws. SIAM J. Numer. Anal. 30(1), 249–274 (1993)

  30. Struijs, R., Deconinck, H., Roe, P.L.: Fluctuation splitting schemes for the 2d euler equations. In: In its Computational Fluid Dynamics 94 p (SEE N91-32426 24–34), vol. 1, 1991

  31. Tsai, Y.-H.R., Cheng, L.-T., Osher, S., Zhao, H.-K.: Fast sweeping algorithms for a class of Hamilton–Jacobi equations. SIAM J. Numer. Anal. 41, 673–694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vincenti, W.G., Kruger, C.H.: Introduction to Physical Gas Dynamics. Wiley, New York (1965)

    Google Scholar 

  33. Wan, J.W.L., Jameson, A.: Monotonicity preserving multigrid time stepping schemes for conservation laws. Comput. Vis. Sci. 11(1), 41–58 (2008)

    Article  MathSciNet  Google Scholar 

  34. Wesseling, P.: Principles of Computational Fluid Dynamics. Springer, Secaucus, NJ (2000)

    MATH  Google Scholar 

  35. Zhang, S., Jiang, S., Shu, C.-W.: Improvement of convergence to steady state solutions of Euler equations with the WENO schemes. J Sci. Comput. 47(2), 216–238 (2011)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92617, USA

    Weitao Chen

  2. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Ching-Shan Chou

  3. Department of Mathematical Sciences, Claremont McKenna College, Claremont, CA, 91711, USA

    Chiu-Yen Kao

Authors
  1. Weitao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ching-Shan Chou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chiu-Yen Kao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ching-Shan Chou.

Additional information

C.-S. Chou: This author is supported by NSF DMS1253481.

 C.-Y. Kao: This author is partially supported by NSF DMS1318364.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Chou, CS. & Kao, CY. Lax–Friedrichs Multigrid Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws. J Sci Comput 64, 591–618 (2015). https://doi.org/10.1007/s10915-015-0006-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0006-7

Keywords

Search

Navigation