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Nonlinear Upwind-Biased Free-Stream-Preserving Schemes for Compressible Euler Equations

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Abstract

Concerning the satisfaction of free-stream preservation which reduces errors in the numerical evaluation of grid metrics, the nonlinear central schemes based on WENO interpolations are preferable because of their easy achievement of preservation even on distorted grids, whereas the preservation is hardly achieved by canonical WENO schemes due to their nonlinear upwind nature. However, the latter are typically robust in terms of capturing shocks on uniform or smooth grids, whereas the former are sometimes liable to numerical instability. Aiming at above predicaments, an upwind-biased approach in the presence of flux splitting is established by integrating two seemingly distinct techniques, WENO interpolation and reconstruction, and meanwhile free-stream preservation and increased robustness are acquired. Specifically, the proposed recipe consists of variable and corresponding flux at one midpoint as well as fluxes at sets of nodes as that in WENO; WENO interpolation is cast to derive the former and WENO reconstruction is implemented using the latter to incarnate their contribution; results by two nonlinearities are hybridized carefully and the target order can be achieved. The developed schemes include third-, fifth-, and seventh-order nonlinear methods. By the way, a WENO implementation with the free-stream preservation is obtained as a special case of the proposed method. Numerical examples are used to validate the third- and fifth-order schemes. The achievement of free-stream preservation property is testified (including the aforementioned WENO implementations). 1-D problems by Euler equations indicate the capability of proposed schemes to resolve shock discontinuities and their good resolution. In 2-D situation, vortex preservation and double Mach reflection problems on uniform and randomized grids are chosen, and the computation on the latter grids has been known to accomplish only if free-stream preservation is satisfied. The proposed schemes produce well results in both cases. Comparative studies demonstrate the accuracy and increased robustness of the newly developed schemes for solving flow problems under non-smooth grids.

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Acknowledgements

This study is sponsored by the project of National Numerical Wind-tunnel of China under the Grant No. NNW2019ZT4-B12, and is also co-supported by the National Natural Science Foundation of China under the Grant No. 11802324.

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Authors and Affiliations

  1. School of Aerospace Engineering, Xiamen University, Xiamen, 361102, Fujian, China

    Qin Li, Pan Yan & Xiao Huang

  2. State Key Laboratory of Aerodynamics, Mianyang, 621000, Sichuan, China

    Qin Li & Dong Sun

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  1. Qin Li
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  2. Dong Sun
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Correspondence to Qin Li.

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Li, Q., Sun, D., Yan, P. et al. Nonlinear Upwind-Biased Free-Stream-Preserving Schemes for Compressible Euler Equations. J Sci Comput 91, 72 (2022). https://doi.org/10.1007/s10915-022-01833-8

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  • DOI: https://doi.org/10.1007/s10915-022-01833-8

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