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Positivity-Preserving High Order Finite Volume HWENO Schemes for Compressible Euler Equations

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Abstract

In this paper, we present a positivity-preserving high order finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations based on the framework for constructing uniformly high order accurate positivity-preserving discontinuous Galerkin and finite volume schemes for Euler equations proposed in Zhang and Shu (J Comput Phys 230:1238–1248, 2011). The major advantages of the HWENO schemes is their compactness in the spacial field because the function and its first derivative are evolved in time and used in the reconstructions. On the other hand, the HWENO reconstruction tends to be more oscillatory than those of conventional WENO schemes. Thus positivity preserving techniques are more needed in HWENO schemes for the sake of stability. Numerical tests will be shown to demonstrate the robustness and high-resolution of the schemes.

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References

  1. Capdeville, G.: A Hermite upwind WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 227, 2430–2454 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin method for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ha, Y., Gardner, C.L.: Positive scheme numerical simulation of high mach number astrophysical jets. J. Sci. Comput. 34, 247–259 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ha, Y., Gardner, C., Gelb, A., Shu, C.-W.: Numerical simulation of high Mach number astrophysical jets with radiative cooling. J. Sci. Comput. 24, 597–612 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Korobeinikov, V.P.: Problems of Point-Blast Theory. American Institute of Physics, College Park (1991)

    Google Scholar 

  8. Lele, S.K.: Compact finite-difference schemes with spectra-like resolution. J. Comput. Phys. 103, 16–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, H., Qiu, J.: Finite difference Hermite WENO schemes for conservation laws. J. Sci. Comput. 63, 548–572 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Perthame, B., Shu, C.-W.: On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73, 119–130 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Qiu, J., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 193, 115–135 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Qiu, J., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method II: Two dimensional case. Comput. Fluids 34, 642–663 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, New York (1959)

    MATH  Google Scholar 

  14. 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 

  15. Van-Leer, B.: Towards the ultimate conservative difference scheme: III. A new approach to numerical convection. J. Comput. Phys. 23, 276–299 (1977)

    Article  MATH  Google Scholar 

  16. Wang, C., Zhang, 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 

  17. Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33, 1476–1493 (2010)

    Article  Google Scholar 

  18. Xing, Y., Zhang, X.: Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes. J. Sci. Comput. 57, 19–41 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, 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 

  21. Zhang, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, X., Xia, Y., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. J. Sci. Comput. 50, 29–62 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, 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 

  25. Zhang, X., Liu, Y.-Y., Shu, C.-W.: Maximum-principle-satisfying high order finite volume WENO schemes for convection–diffusion equations. SIAM J. Sci. Comput. 34, A627–A658 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, Y., Zhang, X., Shu, C.-W.: Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes. J. Comput. Phys. 234, 295–316 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, X., Shu, C.-W.: A minimum entropy principle of high order schemes for gas dynamics equations. Numer. Math. 121, 545–563 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhu, J., Qiu, J.: A class of fourth order finite volume hermite weighted essentially non-oscillatory schemes. Sci. China Ser. A Math. 51, 1549–1560 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Xiaofeng Cai

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-2067, USA

    Xiangxiong Zhang

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jianxian Qiu

Authors
  1. Xiaofeng Cai
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  2. Xiangxiong Zhang
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  3. Jianxian Qiu
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Corresponding author

Correspondence to Jianxian Qiu.

Additional information

Research was supported by NSFC Grants 91230110, 11328104, 11571290 and the NSF Grant DMS-1522593.

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Cai, X., Zhang, X. & Qiu, J. Positivity-Preserving High Order Finite Volume HWENO Schemes for Compressible Euler Equations. J Sci Comput 68, 464–483 (2016). https://doi.org/10.1007/s10915-015-0147-8

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  • DOI: https://doi.org/10.1007/s10915-015-0147-8

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