Skip to main content
SpringerLink
Log in

An Example of High Order Residual Distribution Scheme Using non-Lagrange Elements

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

Abstract

We are interested in the numerical approximation of non-linear hyperbolic problems. The particular class of schemes we are interested in are the so-called Residual Distribution (RD) schemes. In their current form, they rely on the Lagrange interpolation of the point values of the approximated functions. This interpretation of the degrees of freedom as point values plays a fundamental role in the derivation of the schemes. The purpose of the present paper is to show that some non-Lagrange elements can also do the job, and maybe better. This opens the door to isogeometric analysis in the framework of RDS schemes.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1983)

  2. Ni, R.-H.: A multiple grid scheme for solving the Euler equations. AIAA J. 20(11), 1565–1571 (1981)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Hughes, T.J.R., Mallet, M.: A new finite element formulation for CFD, III: the generalized streamline operator for multidimensional advective-diffusive systems. Comput. Methods Appl. Mech. Eng. 58, 305–328 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  6. Struijs, R., Deconinck, H., Roe, P.L.: Fluctuation splitting schemes for the 2D Euler equations. Computational Fluid Dynamics, VKI-LS 1991-01 (1991)

  7. Paillère, H., Carette, J.-C., Deconinck, H.: Multidimentional upwind and supg methods for the solution of the compressible flow equations on unstructured grids. Computational Fluid Dynamics, VKI-LS 1994-05 (1994)

  8. van der Weide, E., Deconinck, H., Issmann, E., Degrez, G.: Fluctuation splitting schemes for multidimensional convection problems: an alternative to finite volume and finite element methods. Comput. Mech. 23(2), 199–208 (1999)

    Article  MATH  Google Scholar 

  9. Csík, Á., Ricchiuto, M., Deconinck, H.: A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws. J. Comput. Phys. 179(2), 286–312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. 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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Abgrall, R., Mezine, M.: Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems. J. Comput. Phys. 188(1), 16–55 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Abgrall, R., Shu, C.W.: Development of residual distribution schemes for the discontinuous Galerkin methods: the scalar case. Commun. Comput. Phys. 9, 376–390 (2009)

    MathSciNet  Google Scholar 

  15. Hubbard, M.E.: A framework for discontinuous fluctuation distribution. Int. J. Numer. Methods Fluids 56(8), 1305–1311 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ciarlet, P.G., Raviart, P.A.: General Lagrange and Hermite interpolation in R n with applications to finite element methods. Arch. Ration. Mech. Anal. 46, 177–199 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  18. de Boor, C.: A practical guide to splines. Applied Mathematical Sciences, vol. 27. Springer, Berlin (1978)

    MATH  Google Scholar 

  19. Abgrall, R., Larat, A., Ricchiuto, M., Tavé, C.: A simple construction of very high order non oscillatory compact schemes on unstructured meshes. Comput. Fluids 38(7), 1314–1323 (2009)

    Article  Google Scholar 

  20. Rogers, D. F.: An Introduction to NURBS: with Historical Perspectives. Morgan Kaufman, San Mateo (2001)

    Google Scholar 

  21. Guckenheimer, J.: Shocks and rarefactions in two space dimensions. Arch. Ration. Mech. Anal. 59, 281–291 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  22. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice–Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  23. Duruflé, M.: Intégration numérique et éléments finis d’ordre élevé appliqués aux équations de Maxwell en régime harmonique. PhD thesis, Université Paris Dauphine, february 2006

Download references

Author information

Authors and Affiliations

  1. Team Bacchus, INRIA Bordeaux Sud Ouest & Institut de Mathématiques de Bordeaux, 33 405, Talence, France

    R. Abgrall & J. Trefilík

  2. Institut Polytechnique de Bordeaux, 33 405, Talence, France

    R. Abgrall

  3. CTU Prague, 12 135, Praha 2, Czech Republic

    J. Trefilík

Authors
  1. R. Abgrall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Trefilík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Abgrall.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abgrall, R., Trefilík, J. An Example of High Order Residual Distribution Scheme Using non-Lagrange Elements. J Sci Comput 45, 3–25 (2010). https://doi.org/10.1007/s10915-010-9405-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9405-y

Keywords

Search

Navigation