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On the Accuracy of WENO and CWENO Reconstructions of Third Order on Nonuniform Meshes

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Third order WENO and CWENO reconstruction are widespread high order reconstruction techniques for numerical schemes for hyperbolic conservation and balance laws. In their definition, there appears a small positive parameter, usually called \(\epsilon \), that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. Recently, two detailed studies of the role of this parameter, in the case of uniform meshes, were published. In this paper we extend their results to the case of finite volume schemes on non-uniform meshes, which is very important for h-adaptive schemes, showing the benefits of choosing \(\epsilon \) as a function of the local mesh size \(h_j\). In particular we show that choosing \(\epsilon =h_j^2\) or \(\epsilon =h_j\) is beneficial for the error and convergence order, studying on several non-uniform grids the effect of this choice on the reconstruction error, on fully discrete schemes for the linear transport equation, on the stability of the numerical schemes. Finally we compare the different choices for \(\epsilon \) in the case of a well-balanced scheme for the Saint-Venant system for shallow water flows and in the case of an h-adaptive scheme for nonlinear systems of conservation laws and show numerical tests for a two-dimensional generalisation of the CWENO reconstruction on locally adapted meshes.

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  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. Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comp. 25, 2050–2065 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berger, M.J., LeVeque, R.J.: Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35, 2298–2316 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977–3014 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Constantinescu, E., Sandu, A.: Multirate timestepping methods for hyperbolic conservation laws. J. Sci. Comput. 33(3), 239–278 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Don, W.S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013)

    Article  MathSciNet  Google Scholar 

  7. Feng, H., Huang, C., Wang, R.: An improved mapped weighted essentially non-oscillatory scheme. Appl. Math. Comput. 232, 453–468 (2014)

    MathSciNet  Google Scholar 

  8. Gerolymos, G.: Representation of the lagrange reconstructing polynomial by combination of substencils. J. Comput. Appl. Math. 236, 2763–2794 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)

    Article  MATH  Google Scholar 

  11. Hu, C., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150(1), 97–127 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Kirby, R.: On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws. Math. Comp. 72(243), 1239–1250 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Lahooti, M., Pishevar, A.: A new fourth order central WENO method for 3D hyperbolic conservation laws. Appl. Math. Comput. 218, 10258–10270 (2012)

    MathSciNet  MATH  Google Scholar 

  16. LeVeque, R.J.: Numerical Methods for Conservation Laws. Lecture Notes in Math. Bikhäuser, Basel (1999)

    Google Scholar 

  17. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. M2AN. Math. Model. Numer. Anal. 33, 547–571 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22(2), 656–672 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Levy, D., Puppo, G., Russo, G.: A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws. SIAM. J. Sci. Comput. 24, 480–506 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213(2), 474–499 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Puppo, G., Semplice, M.: Numerical entropy and adaptivity for finite volume schemes. Commun. Comput. Phys. 10(5), 1132–1160 (2011)

    MathSciNet  Google Scholar 

  22. Puppo, G., Semplice, M.: Well-balanced high order 1D schemes on non-uniform grids and entropy residuals. J. Sci. Comput. (2015). doi:10.1007/s10915-015-0056-x

  23. Semplice, M., Coco, A., Russo, G.: Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction. J. Sci. Comput. (2015). doi:10.1007/s10915-015-0038-z

  24. 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), Lecture Notes in Math., vol. 1697, pp. 325–432. Springer, Berlin (1998)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Xing, Y., Shu, C.W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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The authors wish to thank the anonymous referees for for their punctual remarks and interesting remarks comments that helped to improve this paper.

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Cravero, I., Semplice, M. On the Accuracy of WENO and CWENO Reconstructions of Third Order on Nonuniform Meshes. J Sci Comput 67, 1219–1246 (2016).

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