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Improvement of convergence to steady state solutions of Euler equations with weighted compact nonlinear schemes

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Abstract

The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) [Deng X. and Zhang H. (2000), J. Comput. Phys. 165, 22–44 and Zhang S., Jiang S. and Shu C.-W. (2008), J. Comput. Phys. 227, 7294–7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. (2007), J. Sci. Comput. 31, 273–305 and Zhang S., Jiang S and Shu. C.-W. (2011), J. Sci. Comput. 47, 216–238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.

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Authors and Affiliations

  1. State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, 621000, China

    Shu-hai Zhang & **ao-gang Deng

  2. China Aerodynamics Research and Development Center, Mianyang, 621000, China

    Mei-liang Mao

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

    Chi-Wang Shu

Authors
  1. Shu-hai Zhang
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  2. **ao-gang Deng
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  3. Mei-liang Mao
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  4. Chi-Wang Shu
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Corresponding author

Correspondence to Shu-hai Zhang.

Additional information

Supported by the National Natural Science Foundation of China (Grants 11172317, 91016001) and 973 Program 2009CB724104.

Supported by 973 program 2009CB723800.

Supported by AFOSR Grant FA9550-09-1-0126 and NSF grants DMS-0809086 and DMS-1112700.

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Zhang, Sh., Deng, Xg., Mao, Ml. et al. Improvement of convergence to steady state solutions of Euler equations with weighted compact nonlinear schemes. Acta Math. Appl. Sin. Engl. Ser. 29, 449–464 (2013). https://doi.org/10.1007/s10255-013-0230-6

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  • DOI: https://doi.org/10.1007/s10255-013-0230-6

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