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High-Order Positivity-Preserving Entropy Stable Schemes for the 3-D Compressible Navier–Stokes Equations

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Abstract

This paper extends a new family of high-order positivity-preserving, entropy stable spectral collocation schemes developed for the one-dimensional compressible Navier–Stokes equations in Upperman and Yamaleev (J Comput Phys 466, 2022; J Comput Phys 466, 2022) to three spatial dimensions. The proposed schemes are constructed by using a flux-limiting technique that combines a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable finite volume-type scheme discretized on the same Legendre–Gauss–Lobatto grid points used for constructing the high-order discrete operators. The positivity-preserving and excellent discontinuity-capturing properties are achieved by adding an artificial dissipation in the form of the low- and high-order Brenner–Navier–Stokes diffusion operators. Furthermore, the new schemes are entropy conservative for smooth inviscid flows and freestream preserving. To our knowledge, this is the first family of schemes of arbitrary order of accuracy that provably guarantee both the pointwise positivity of thermodynamic variables and \(L_2\) stability for the 3-D compressible Navier–Stokes equations. Numerical results demonstrating accuracy and positivity-preserving properties of the new schemes are presented for 2-D and 3-D viscous and inviscid flows with strong discontinuities.

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Data Availability

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

The first author acknowledges the support from Army Research Office through grant W911NF-17-0443. The second author was supported by the Virginia Space Grant Consortium Graduate STEM Research Fellowship and the Department fo Defense Science, Mathematics and Research for Transformation (SMART) Scholarship.

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  1. Johnathon Upperman

    Present address: AFRL, Kirtland AFB, Albuquerque, NM, 87117, USA

Authors and Affiliations

  1. Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, 23529, USA

    Nail K. Yamaleev & Johnathon Upperman

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  1. Nail K. Yamaleev
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  2. Johnathon Upperman
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Correspondence to Nail K. Yamaleev.

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Yamaleev, N.K., Upperman, J. High-Order Positivity-Preserving Entropy Stable Schemes for the 3-D Compressible Navier–Stokes Equations. J Sci Comput 95, 11 (2023). https://doi.org/10.1007/s10915-023-02136-2

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