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

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aràndiga, F., Martí, M.C., Mulet, P.: Weights design for maximal order WENO schemes. J. Sci. Comput. 60(3), 641–659 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Artebrant, R., Schroll, H.J.: Conservative logarithmic reconstructions and finite volume methods. SIAM J. Sci. Comput. 27(1), 294–314 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Čada, M., Torrilhon, M.: Compact third order limiter functions for finite volume methods. J. Comput. Phys. 228(11), 4118–4145 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  8. Colella, P., Woodward, P.R.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dubois F.: Nonlinear interpolation and total variation diminishing schemes. Technical report, (1990)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Keppens, R., Porth, O.: Scalar hyperbolic PDE simulations and coupling strategies. J. Comput. Appl. Math. 266, 87–101 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. LeVeque, R.J.: Numerical methods for conservation laws, 2nd edn. Birkhäuser, Basel (1992)

  17. LeVeque, R.J.: Finite volume methods for hyperbolic problems, 1st edn. Cambridge University Press, Cambridge (2002)

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Marquina, A.: Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws. SIAM J. Sci. Comput. 15, 892–915 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  22. Roe, P.L.: Some contributions to the modelling of discontinuous flows. Lect. Notes Appl. Math. 22, 163–193 (1985)

    MathSciNet  Google Scholar 

  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. 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 

  25. 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 

  26. Suresh, A., Huynh, H.T.: Accurate monotonicity-preserving schemes with Runge-Kutta time stepping. J. Comput. Phys. 136, 83–99 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yamaleev, N.K., Carpenter, M.H.: Third-order energy stable WENO scheme. J. Comput. Phys. 228(8), 3025–3047 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Center for Computational Engineering Science, RWTH Aachen University, Schinkelstr. 2, 52062, Aachen, Germany

    B. Schmidtmann & M. Torrilhon

  2. Department of Mathematics, Temple University, 1801 North Broad Street, Philadelphia, PA, 19122, USA

    B. Seibold

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  1. B. Schmidtmann
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  2. B. Seibold
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  3. M. Torrilhon
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Correspondence to B. Schmidtmann.

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Schmidtmann, B., Seibold, B. & Torrilhon, M. Relations Between WENO3 and Third-Order Limiting in Finite Volume Methods. J Sci Comput 68, 624–652 (2016). https://doi.org/10.1007/s10915-015-0151-z

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  • DOI: https://doi.org/10.1007/s10915-015-0151-z

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