Advertisement

Journal of Scientific Computing

, Volume 68, Issue 2, pp 624–652 | Cite as

Relations Between WENO3 and Third-Order Limiting in Finite Volume Methods

  • B. SchmidtmannEmail author
  • B. Seibold
  • M. Torrilhon
Article

Abstract

Weighted essentially non-oscillatory (WENO) and finite volume (FV) methods employ different philosophies in their way to perform limiting. We show that a generalized view on limiter functions, which considers a two-dimensional, rather than a one-dimensional dependence on the slopes in neighboring cells, allows to write WENO3 and 3rd-order FV schemes in the same fashion. Within this framework, it becomes apparent that the classical approach of FV limiters to only consider ratios of the slopes in neighboring cells, is overly restrictive. The hope of this new perspective is to establish new connections between WENO3 and FV limiter functions, which may give rise to improvements for the limiting behavior in both approaches.

Keywords

Non-linear and non-polynomial limiter WENO Finite volume Hyperbolic conservation laws Shock capturing 

References

  1. 1.
    Aràndiga, F., Baeza, A., Belda, A.M., Mulet, P.: Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49(2), 893–915 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aràndiga, F., Martí, M.C., Mulet, P.: Weights design for maximal order WENO schemes. J. Sci. Comput. 60(3), 641–659 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Artebrant, R., Schroll, H.J.: Conservative logarithmic reconstructions and finite volume methods. SIAM J. Sci. Comput. 27(1), 294–314 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Čada, M., Torrilhon, M.: Compact third order limiter functions for finite volume methods. J. Comput. Phys. 228(11), 4118–4145 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chidyagwai, P., Nave, J.-C., Rosales, R.R., Seibold, B.: A comparative study of the efficiency of jet schemes. Int. J. Numer. Anal. 3(3), 297–306 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cockburn, B., Shu, C.-W.: The local Discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Colella, P., Woodward, P.R.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dubois F.: Nonlinear interpolation and total variation diminishing schemes. Technical report, (1990)Google Scholar
  10. 10.
    Godunov, S.K.: A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations. Math. Sbornik 47, 271–306 (1959)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order accurate essentially non-oscillatory schemes. III. J. Comput. Phys. 71(2), 231–303 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kemm, F.: A comparative study of TVD-limiters—well-known limiters and an introduction of new ones. Int. J. Numer. Meth. Fluids 67(4), 404–440 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Keppens, R., Porth, O.: Scalar hyperbolic PDE simulations and coupling strategies. J. Comput. Appl. Math. 266, 87–101 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kolb, O.: On the full and global accuracy of a compact third order WENO scheme. SIAM J. Numer. Anal. 52(5), 2335–2355 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    LeVeque, R.J.: Numerical methods for conservation laws, 2nd edn. Birkhäuser, Basel (1992)Google Scholar
  17. 17.
    LeVeque, R.J.: Finite volume methods for hyperbolic problems, 1st edn. Cambridge University Press, Cambridge (2002)Google Scholar
  18. 18.
    Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Marquina, A.: Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws. SIAM J. Sci. Comput. 15, 892–915 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mignone, A., Tzeferacos, P., Bodo, G.: High-order conservative finite difference GLM-MHD schemes for cell-centered MHD. J. Comput. Phys. 229(17), 5896–5920 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, (1973)Google Scholar
  22. 22.
    Roe, P.L.: Some contributions to the modelling of discontinuous flows. Lect. Notes Appl. Math. 22, 163–193 (1985)MathSciNetGoogle Scholar
  23. 23.
    Schmidtmann, B., Abgrall, R., Torrilhon, M.: On third-order limiter functions for finite volume methods. In Proceedings of the XV International Conference on Hyperbolic Problems (HYP2014), Bulletin of the Brazilian Math, 2014. http://arxiv.org/abs/1411.0868
  24. 24.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. II. J. Comput. Phys. 83, 32–78 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Suresh, A., Huynh, H.T.: Accurate monotonicity-preserving schemes with Runge-Kutta time stepping. J. Comput. Phys. 136, 83–99 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    van Leer, B.: Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme. J. Comput. Phys. 14, 361–370 (1974)CrossRefzbMATHGoogle Scholar
  28. 28.
    van Leer, B.: Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)CrossRefGoogle Scholar
  29. 29.
    Yamaleev, N.K., Carpenter, M.H.: A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228(11), 4248–4272 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Yamaleev, N.K., Carpenter, M.H.: Third-order energy stable WENO scheme. J. Comput. Phys. 228(8), 3025–3047 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Center for Computational Engineering ScienceRWTH Aachen UniversityAachenGermany
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA

Personalised recommendations