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Numerical study on the convergence to steady-state solutions of a new class of finite volume WENO schemes: triangular meshes

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Abstract

In this paper, we continue our research on the numerical study of convergence to steady-state solutions for a new class of finite volume weighted essentially non-oscillatory (WENO) schemes in Zhu and Shu (J Comput Phys 349:80–96, 2017), from tensor product meshes to triangular meshes. For the case of triangular meshes, this new class of finite volume WENO schemes was designed for time-dependent conservation laws in Zhu and Qiu (SIAM J Sci Comput 40(2):A903–A928, 2018) for the third- and fourth-order versions. In this paper, we extend the design to a new fifth-order version in the same framework to keep the essentially non-oscillatory property near discontinuities. Similar to the case of tensor product meshes in Zhu and Shu  (2017), by performing such spatial reconstruction procedures together with a TVD Runge–Kutta time discretization, these WENO schemes do not suffer from slight post-shock oscillations that are responsible for the phenomenon wherein the residues of classical WENO schemes hang at a truncation error level instead of converging to machine zero. The third-, fourth-, and fifth-order finite volume WENO schemes in this paper can suppress the slight post-shock oscillations and have their residues settling down to a tiny number close to machine zero in steady-state simulations in our extensive numerical experiments.

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References

  1. Zhu, J., Shu, C.-W.: Numerical study on the convergence to steady state solutions of a new class of high order WENO schemes. J. Comput. Phys. 349, 80–96 (2017). https://doi.org/10.1016/j.jcp.2017.08.012

    Article  MathSciNet  MATH  Google Scholar 

  2. Zhu, J., Qiu, J.: New finite volume weighted essentially non-oscillatory schemes on triangular meshes. SIAM J. Sci. Comput. 40(2), A903–A928 (2018). https://doi.org/10.1137/17M1112790

    Article  MATH  Google Scholar 

  3. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994). https://doi.org/10.1006/jcph.1994.1187

    Article  MathSciNet  MATH  Google Scholar 

  4. Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996). https://doi.org/10.1006/jcph.1996.0130

    Article  MathSciNet  MATH  Google Scholar 

  5. Friedrich, O.: Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys. 144, 194–212 (1998). https://doi.org/10.1006/jcph.1998.5988

    Article  MathSciNet  MATH  Google Scholar 

  6. Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999). https://doi.org/10.1006/jcph.1998.6165

    Article  MathSciNet  MATH  Google Scholar 

  7. Titarev, V.A., Toro, E.F.: Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201, 238–260 (2004). https://doi.org/10.1016/j.jcp.2004.05.015

    Article  MathSciNet  MATH  Google Scholar 

  8. Titarev, V.A., Tsoutsanis, P., Drikakis, D.: WENO schemes for mixed-element unstructured meshes. Commun. Comput. Phys. 8, 585–609 (2010). https://doi.org/10.4208/cicp.040909.080110a

    MathSciNet  MATH  Google Scholar 

  9. Zhang, Y.T., Shu, C.-W.: Third order WENO scheme on three dimensional tetrahedral meshes. Commun. Comput. Phys. 5, 836–848 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114, 45–58 (1994). https://doi.org/10.1006/jcph.1994.1148

    Article  MathSciNet  MATH  Google Scholar 

  11. Shi, J., Hu, C., Shu, C.-W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002). https://doi.org/10.1006/jcph.2001.6892

    Article  MATH  Google Scholar 

  12. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E., Quarteroni, A. (eds.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998). https://doi.org/10.1007/bfb0096355

    Chapter  Google Scholar 

  13. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001). https://doi.org/10.1137/s003614450036757X

    Article  MathSciNet  MATH  Google Scholar 

  14. Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988). https://doi.org/10.1137/0909073

    Article  MathSciNet  MATH  Google Scholar 

  15. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988). https://doi.org/10.1016/0021-9991(88)90177-5

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016). https://doi.org/10.1016/j.jcp.2016.05.010

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhu, J., Qiu, J.: A new type of finite volume WENO schemes for hyperbolic conservation laws. J. Sci. Comput. 73, 1338–1359 (2017). https://doi.org/10.1007/s10915-017-0486-8

    Article  MathSciNet  MATH  Google Scholar 

  18. Balsara, D.S., Garain, S., Shu, C.-W.: An efficient class of WENO schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016). https://doi.org/10.1016/j.jcp.2016.09.009

    Article  MathSciNet  MATH  Google Scholar 

  19. Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977–3014 (2008). https://doi.org/10.1016/j.jcp.2007.11.029

    Article  MathSciNet  MATH  Google Scholar 

  20. 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). https://doi.org/10.1007/s10915-015-0123-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Fu, L., Hu, X.Y.Y., Adams, N.A.: A family of high-order targeted ENO schemes for compressible-fluid simulations. J. Comput. Phys. 305, 333–359 (2016). https://doi.org/10.1016/j.jcp.2015.10.037

    Article  MathSciNet  MATH  Google Scholar 

  22. Kolb, O.: On the full and global accuracy of a compact third order WENO scheme. SIAM J. Numer. Anal. 52, 2335–2355 (2014). https://doi.org/10.1137/130947568

    Article  MathSciNet  MATH  Google Scholar 

  23. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. Math. Model. Numer. Anal. 33, 547–571 (1999). https://doi.org/10.1051/m2an:1999152

    Article  MathSciNet  MATH  Google Scholar 

  24. Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22, 656–672 (2000). https://doi.org/10.1137/s1064827599359461

    Article  MathSciNet  MATH  Google Scholar 

  25. Semplice, M., Coco, A., Russo, G.: Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction. J. Sci. Comput. 66, 692–724 (2016). https://doi.org/10.1007/s10915-015-0038-z

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, Y., Zhang, Y.T.: A robust reconstruction for unstructured WENO schemes. J. Sci. Comput. 54, 603–621 (2013). https://doi.org/10.1007/s10915-012-9598-3

    Article  MathSciNet  MATH  Google Scholar 

  27. Harten, A., Chakravarthy, S.R.: Multi-dimensional ENO schemes for general geometries. ICASE Report, 91-76, NASA Contractor Report 187637 (1991)

  28. Sonar, T.: On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection. Comput. Methods Appl. Mech. Eng. 140, 157–181 (1997). https://doi.org/10.1016/S0045-7825(96)01060-2

    Article  MathSciNet  MATH  Google Scholar 

  29. Vankeirsbilck, P.: Algorithmic developments for the solution of hyperbolic conservation laws on adaptive unstructured grids. Ph.D. Thesis, Katholieke Universiteit Leuven, Faculteit Toegepaste Wetenschappen, Afdeling Numerieke Analyse en Toegepaste Wiskunde, Celestijnenlaan 2OOA, 3001 Leuven (Heverlee) (1993)

  30. Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221, 693–723 (2007). https://doi.org/10.1016/j.jcp.2006.06.043

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhong, X., Shu, C.-W.: A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 232, 397–415 (2013). https://doi.org/10.1016/j.jcp.2012.08.028

    Article  MathSciNet  Google Scholar 

  32. Zhu, J., Zhong, X., Shu, C.-W., Qiu, J.: Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes. J. Comput. Phys. 248, 200–220 (2013). https://doi.org/10.1016/j.jcp.2013.04.012

    Article  MathSciNet  MATH  Google Scholar 

  33. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008). https://doi.org/10.1016/j.jcp.2007.11.038

    Article  MathSciNet  MATH  Google Scholar 

  34. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011). https://doi.org/10.1016/j.jcp.2010.11.028

    Article  MathSciNet  MATH  Google Scholar 

  35. Don, W.S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013). https://doi.org/10.1016/j.jcp.2013.05.018

    Article  MathSciNet  MATH  Google Scholar 

  36. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981). https://doi.org/10.1016/0021-9991(81)90128-5

    Article  MathSciNet  MATH  Google Scholar 

  37. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984). https://doi.org/10.1016/0021-9991(84)90142-6

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, S., Shu, C.-W.: A new smoothness indicator for WENO schemes and its effect on the convergence to steady state solutions. J. Sci. Comput. 31, 273–305 (2007). https://doi.org/10.1007/s10915-006-9111-y

    Article  MathSciNet  MATH  Google Scholar 

  39. Luo, H., Baum, J.D., Löhner, R.: A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys. 225, 686–713 (2007). https://doi.org/10.1016/j.jcp.2006.12.017

    Article  MathSciNet  MATH  Google Scholar 

  40. Luo, H., Baum, J.D., Löhner, R.: On the computation of steady-state compressible flows using a discontinuous Galerkin method. Int. J. Numer. Methods Eng. 73, 597–623 (2008). https://doi.org/10.1002/nme.2081

    Article  MathSciNet  MATH  Google Scholar 

  41. Shida, Y., Kuwahara, K., Ono, K., Takami, H.: Computation of dynamic stall of a NACA-0012 airfoil. AIAA J. 25, 408–413 (1987). https://doi.org/10.2514/3.9638

    Article  Google Scholar 

  42. Cummings, R.M., Mason, W.H., Morton, S.A., McDaniel, D.R.: Applied Computational Aerodynamics: A Modern Engineering Approach. Cambridge University Press, Cambridge (2015)

    Book  MATH  Google Scholar 

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Acknowledgements

J. Zhu: Research is supported by NSFC Grant 11372005 and the state scholarship fund of China for studying abroad. C.-W. Shu: Research is supported by ARO Grant W911NF-15-1-0226 and NSF Grant DMS-1719410.

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  1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, People’s Republic of China

    J. Zhu

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    C.-W. Shu

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Correspondence to C.-W. Shu.

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Communicated by D. Zeidan and H. D. Ng.

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Zhu, J., Shu, CW. Numerical study on the convergence to steady-state solutions of a new class of finite volume WENO schemes: triangular meshes. Shock Waves 29, 3–25 (2019). https://doi.org/10.1007/s00193-018-0833-1

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