Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

High Order Residual Distribution for Steady State Problems for Hyperbolic Conservation Laws

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

Abstract

In this paper, we propose a high order residual distribution conservative finite difference scheme for solving steady state conservation laws. A new type of WENO (weighted essentially non-oscillatory) termed as WENO-ZQ integration is used to compute the numerical fluxes and source term based on the point values of the solution, and the principles of residual distribution schemes are adapted to obtain steady state solutions. 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 shocks of the proposed methods.

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

Similar content being viewed by others

References

  1. Abgrall, R.: Toward the ultimate conservative scheme: following the quest. J. Comput. Phys. 167(2), 277–315 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Abgrall, R., Mezine, M.: Construction of second-order accurate monotone and stable residual distribution schemes for steady problems. J. Comput. Phys. 195(2), 474–507 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Abgrall, R., Roe, P.L.: High order fluctuation schemes on triangular meshes. J. Sci. Comput. 19(1–3), 3–36 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Abgrall, R., Mer, K., Nkonga, B.: A Lax–Wendroff type theorem for residual schemes. In: Innovative Methods for Numerical Solutions of Partial Differential Equations (Arcachon, 1998), pp. 243–266. World Sci. Publ., River Edge, NJ (2002)

  5. Balsara, D.S., Rumpf, T., Dumbser, M., Munz, C.-D.: Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J. Comput. Phys. 228(7), 2480–2516 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Cai, W., Gottlieb, D., Shu, C.-W.: Essentially nonoscillatory spectral Fourier methods for shock wave calculations. Math. Comput. 52(186), 389–410 (1989)

    MathSciNet  MATH  Google Scholar 

  8. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    MathSciNet  MATH  Google Scholar 

  9. 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(2), 698–724 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Deconinck, H., Struijs, R., Bourgeois, G., Roe, P.: Compact advection schemes on unstructured meshes. Comput. Fluid Dyn., VKI Lecture Series 1993–04 (1993)

  11. Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221(2), 693–723 (2007)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Engquist, B., Osher, S.: Stable and entropy satisfying approximations for transonic flow calculations. Math. Comput. 34(149), 45–75 (1980)

    MathSciNet  MATH  Google Scholar 

  14. Harten, A., Lax, P.D.: A random choice finite difference scheme for hyperbolic conservation laws. SIAM J. Numer. Anal. 18(2), 289–315 (1981)

    MathSciNet  MATH  Google Scholar 

  15. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys. 71(2), 231–303 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Hughes, T.J.R., Franca, L.P., Mallet, M.: A new finite element formulation for computational fluid dynamics. I. Symmetric forms of the compressible Euler and Navier–Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54(2), 223–234 (1986)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Johnson, C., Saranen, J.: Streamline diffusion methods for the incompressible Euler and Navier–Stokes equations. Math. Comp. 47(175), 1–18 (1986)

    MathSciNet  MATH  Google Scholar 

  20. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Osher, S., Chakravarthy, S.: High resolution schemes and the entropy condition. SIAM J. Numer. Anal. 21(5), 955–984 (1984)

    MathSciNet  MATH  Google Scholar 

  22. Osher, S., Chakravarthy, S.: Very high order accurate TVD schemes. In: Oscillation Theory, Computation, and Methods of Compensated Compactness (Minneapolis, Minn., 1985), Volume 2 of IMA Vol. Math. Appl., pp. 229–274. Springer, New York (1986)

  23. Osher, S., Solomon, F.: Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comput. 38(158), 339–374 (1982)

    MathSciNet  MATH  Google Scholar 

  24. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981)

    MathSciNet  MATH  Google Scholar 

  25. Roe, P.L.: Upwind differencing schemes for hyperbolic conservation laws with source terms. In: Nonlinear Hyperbolic Problems (St. Etienne, 1986), Volume 1270 of Lecture Notes in Mathematics, pp. 41–51. Springer, Berlin (1987)

  26. Roe, P.L., Sidilkover, D.: Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal. 29(6), 1542–1568 (1992)

    MathSciNet  MATH  Google Scholar 

  27. Salas, M.D., Abarbanel, S., Gottlieb, D.: Multiple steady states for characteristic initial value problems. Appl. Numer. Math. 2(3–5), 193–210 (1986)

    MathSciNet  MATH  Google Scholar 

  28. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997), Volume 1697 of Lecture Notes in Mathematics, pp. 325–432. Springer, Berlin (1998)

  29. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988a)

    MathSciNet  MATH  Google Scholar 

  31. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988b)

    MathSciNet  MATH  Google Scholar 

  32. Struijs, R., Deconinck, H., Roe, P.: Fluctuation splitting for the 2D Euler equations. Comput. Fluid Dyn. –1, 01 (1991)

    Google Scholar 

  33. Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21(5), 995–1011 (1984)

    MathSciNet  MATH  Google Scholar 

  34. Yee, H.C.: Linearized form of implicit TVD schemes for the multidimensional Euler and Navier–Stokes. Comput. Math. Appl. Part A 12(4–5), 413–432 (1986)

    MathSciNet  MATH  Google Scholar 

  35. Zhong, X.H., Shu, C.-W.: A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 232, 397–415 (2013)

    MathSciNet  Google Scholar 

  36. Zhu, J., Qiu, J.-X.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Zhu, J., Qiu, J.-X.: A new type of finite volume WENO schemes for hyperbolic conservation laws. J. Sci. Comput. 73(2–3), 1338–1359 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

J. Lin and J. Qiu are partly supported by NSFC Grant 11571290 and NSAF Grant U1630247, J. Lin also is supported by the China Scholarship Council and SNF Grant FZEB-0-166980. This work was performed while the first author was visiting the Institute of Mathematics, University of Zurich.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jianfang Lin

  2. Institute of Mathematics, University of Zurich, 8057, Zurich, Switzerland

    Rémi Abgrall

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jianxian Qiu

Authors
  1. Jianfang Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rémi Abgrall
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxian Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rémi Abgrall.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, J., Abgrall, R. & Qiu, J. High Order Residual Distribution for Steady State Problems for Hyperbolic Conservation Laws. J Sci Comput 79, 891–913 (2019). https://doi.org/10.1007/s10915-018-0878-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0878-4

Keywords

Search

Navigation