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Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

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Abstract

A stable high-order Runge-Kutta discontinuous Galerkin (RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional (2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.

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Acknowledgements

This work is supported by the National Supercomputer Center in Tianjin. All the numerical tests are run on the TH-I cluster.

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Authors and Affiliations

  1. School of Astronautics, Beihang University, Beijing, 100191, China

    Wei Su, Bijiao He & Guobiao Cai

  2. Beijing Institute of Spacecraft Environment Engineering, Beijing, 100094, China

    Zhenyu Tang

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  1. Wei Su
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  2. Zhenyu Tang
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  3. Bijiao He
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  4. Guobiao Cai
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Correspondence to Wei Su.

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Project supported by the National Natural Science Foundation of China (No. 11302017)

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Su, W., Tang, Z., He, B. et al. Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations. Appl. Math. Mech.-Engl. Ed. 38, 343–362 (2017). https://doi.org/10.1007/s10483-017-2177-8

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  • DOI: https://doi.org/10.1007/s10483-017-2177-8

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