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Developing shock-capturing difference methods

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Abstract

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.

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References

  1. Harten A. High resolution schemes for hypersonic conservation laws[J]. Journal of Computational Physics, 1983, 49(2):367–393.

    Google Scholar 

  2. Roe P L. Approximate Riemann solvers, parameter vectors and difference schemes[J]. Journal of Computational Physics, 1981, 43(2):357–372.

    Article  MATH  MathSciNet  Google Scholar 

  3. Liou M S. A sequel to AUSM: AUSM+[J]. Journal of Computational Physics, 1996, 129(2):364–382.

    Article  MATH  MathSciNet  Google Scholar 

  4. Harten A, Osher S. Uniformly high-order accurate non-oscillatory schemes I[J]. SIAM Journal on Numerical Analysis. 1987, 24(2):279–309.

    Article  MATH  MathSciNet  Google Scholar 

  5. Shu C W, Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes[J]. Journal of Computational Physics, 1988, 77(2):439–471.

    Article  MATH  MathSciNet  Google Scholar 

  6. Jiang G S, Shu C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126(1):202–228.

    Article  MATH  MathSciNet  Google Scholar 

  7. Daru V, Tenaud C. High order one step monotonicity-preserving schemes for unsteady compressible flow calculations[J]. Journal of Computational Physics, 2004, 193(2):563–594.

    Article  MATH  MathSciNet  Google Scholar 

  8. Tu Guohua, Yuan Xiangjiang, Xia Zhiqiang, et al. A class of compact upwind TVD difference schemes[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(6):765–772.

    MathSciNet  Google Scholar 

  9. Ravichandran K S. Higher order KFVS algorithms using compact upwind difference operators[J]. Journal of Computational Physics, 1997, 230(2):161–173.

    Article  Google Scholar 

  10. Tu Guohua, Luo Junrong. Improve compact schemes by limiting flux method[J]. Chinese Journal of Computational Physics, 2005, 22(4):40–47 (in Chinese).

    Google Scholar 

  11. Ren Y X, Liu M, Zhang H. A characteristic-wise hybrid compact-WENO schemes for solving hyperbolic conservations[J]. Journal of Computational Physics, 2003, 192(2):365–386.

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. China Aerodynamics Research and Development Center, Mianyang, 621000, Sichuan Province, P. R. China

    Tu Guo-hua Doctor  (涂国华) & Yuan Xiang-jiang  (袁湘江)

  2. Tianjin Key Laboratory of Modern Engineering Mechanics, Tianjin, 300072, P. R. China

    Tu Guo-hua Doctor  (涂国华)

  3. National CFD Laboratory, Beihang University, Beijing, 100083, P. R. China

    Tu Guo-hua Doctor  (涂国华) & Yuan Xiang-jiang  (袁湘江)

  4. School of Jet Propulsion, Beihang University, Beijing, 100083, P. R. China

    Lu Li-peng  (陆利篷)

Authors
  1. Tu Guo-hua Doctor  (涂国华)
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  2. Yuan Xiang-jiang  (袁湘江)
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  3. Lu Li-peng  (陆利篷)
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Corresponding author

Correspondence to Tu Guo-hua Doctor  (涂国华).

Additional information

Communicated by LU Chuan-jing

Project supported by the National Natural Science Foundation of China (Nos.10321002 and 10672012)

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Tu, Gh., Yuan, Xj. & Lu, Lp. Developing shock-capturing difference methods. Appl Math Mech 28, 477–486 (2007). https://doi.org/10.1007/s10483-007-0408-z

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  • DOI: https://doi.org/10.1007/s10483-007-0408-z

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