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High Order Positivity- and Bound-Preserving Hybrid Compact-WENO Finite Difference Scheme for the Compressible Euler Equations

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Abstract

Based on the same hybridization framework of Don et al. (SIAM J Sci Comput 38:A691–A711 2016), an improved hybrid scheme employing the nonlinear 5th-order characteristic-wise WENO-Z5 finite difference scheme for capturing high gradients and discontinuities in an essentially non-oscillatory manner and the linear 5th-order conservative compact upwind (CUW5) scheme for resolving the fine scale structures in the smooth regions of the solution in an efficient and accurate manner is developed. By replacing the 6th-order non-dissipative compact central scheme (CCD6) with the CUW5 scheme, which has a build-in dissipation, there is no need to employ an extra high order smoothing procedure to mitigate any numerical oscillations that might appear in an hybrid scheme. The high order multi-resolution algorithm of Harten is employed to detect the smoothness of the solution. To handle the problems with extreme conditions, such as high pressure and density ratios and near vacuum states, and detonation diffraction problems, we design a positivity- and bound-preserving limiter by extending the one developed in Hu et al. (J Comput Phys 242, 2013) for solving the high Mach number jet flows, detonation diffraction problems and detonation passing multiple obstacles problems. Extensive one- and two-dimensional shocked flow problems demonstrate that the new hybrid scheme is less dispersive and less dissipative, and allows a potential speedup up to a factor of more than one and half times faster than the WENO-Z5 scheme.

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References

  1. Adams, N., Shariff, K.: High-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. J. Comput. Phys. 127, 27–51 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3101–3211 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Costa, B., Don, W.S.: High order hybrid central-WENO finite difference scheme for conservation laws. J. Comput. Appl. Math. 204(2), 209–218 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Don, W.S., Gao, Z., Li, P., Wen, X.: Hybrid compact-WENO finite difference scheme with conjugate Fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38(2), A691–A711 (2016)

    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  MATH  Google Scholar 

  7. Guo, Y., Xiong, T., Shi, Y.: A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations. J. Comput. Phys. 274, 505–523 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu, X.Y., Adams, N.A., Shu, C.-W.: Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys. 242, 169–180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 50, 97–127 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jia, F., Gao, Z., Don, W.S.: A spectral study on the dissipation and dispersion of the WENO schemes. J. Sci. Comput. 63, 69–77 (2015)

    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. Lele, S.A.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Niu, Y., Gao, Z., Don, W.S., Xie, S.S., Li, P.: Hybrid compact-WENO finite difference scheme for detonation waves simulations, spectral and high order methods for partial differential equations ICOSAHOM 2014. In: Kirby, R.M., Berzins, M., Hesthaven, J.S. (eds.) Lecture Notes on Computer Science Engineering, vol. 106. Springer, New York (2015)

    Google Scholar 

  15. Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J. Comput. Phys. 178(1), 81–117 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ren, Y.X., Liu, M., Zhang, H.: A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 192(2), 365–386 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shu, C.-W.: High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tolstykh, A.I., Lipavskii, M.V.: On performance of methods with third- and fifth-order compact upwind differencing. J. Comput. Phys. 140(2), 205–232 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vasilyev, O., Lund, T., Moin, P.: A general class of commutative filters for LES in complex geometries. J. Comput. Phys. 146(1), 82–104 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, C., Zhang, X.X., Shu, C.-W., Ning, J.: Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. J. Comput. Phys. 231, 653–665 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, X.X., Shu, C.-W.: On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, X.X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 2752–2776 (2011)

    MATH  Google Scholar 

  24. Zhang, X.X., Shu, C.-W.: Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. J. Comput. Phys. 230, 1238–1248 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, X.X., Shu, C.-W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245–2258 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhuang, M., Cheng, R.F.: Optimized upwind dispersion-relation-preserving finite difference schemes for computational aeroacoustics. AIAA J. 36(11), 2146–2148 (1998)

    Article  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the funding support of this research by National Natural Science Foundation of China (11201441, 11325209, 11521062), Science Challenge Project (TZ2016001), Natural Science Foundation of Shandong Province (ZR2012AQ003) and Fundamental Research Funds for the Central Universities (201562012). The author (Don) also likes to thank the Ocean University of China for providing the startup fund (201712011) that is used in supporting this work.

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Authors and Affiliations

  1. State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, China

    Peng Li & Cheng Wang

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, China

    Wai-Sun Don & Zhen Gao

Authors
  1. Peng Li
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  2. Wai-Sun Don
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  3. Cheng Wang
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  4. Zhen Gao
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Corresponding author

Correspondence to Cheng Wang.

Appendix: Two-Dimensional Flags

Appendix: Two-Dimensional Flags

Here, we show the WENO flags in the x- and y-directions where the nonlinear WENO-Z5 scheme is employed for solving the PDEs instead of the compact upwind (CUW5) scheme for examples presented at the final time (Figs. 17, 18, 19, 20, 21).

Fig. 17
figure 17

The shock-turbulence problem

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Fig. 18
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The double Mach reflection (DMR) problem

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Fig. 19
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The Mach 2000 jet problem

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Fig. 20
figure 20

The detonation diffraction over a backward facing step and a plate problems

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Fig. 21
figure 21

The detonation diffraction over multiple square obstacles problem

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Li, P., Don, WS., Wang, C. et al. High Order Positivity- and Bound-Preserving Hybrid Compact-WENO Finite Difference Scheme for the Compressible Euler Equations. J Sci Comput 74, 640–666 (2018). https://doi.org/10.1007/s10915-017-0452-5

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  • DOI: https://doi.org/10.1007/s10915-017-0452-5

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