First-Order Positivity-Preserving Entropy Stable Scheme for the 3-D Compressible Navier–Stokes Equations

• Published in: Journal of scientific computing 2023, Vol.94 (1), p.18, Article 18
• Main Authors: Upperman, Johnathon, Yamaleev, Nail K.
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Boundary conditions; Collocation methods; Compressibility; Computational Mathematics and Numerical Analysis; Discretization; Entropy; Fluid flow; Inequality; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Navier-Stokes equations; Theoretical; Three dimensional flow; Viscosity; Viscous flow
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-02062-9

In this paper, we extend the positivity-preserving, entropy stable first-order scheme developed for the one-dimensional compressible Navier–Stokes equations in Upperman et al. (J Comput Phys 466, 2022) to three spatial dimensions. The new first-order scheme is provably entropy stable, design-order accurate for smooth solutions, and guarantees the pointwise positivity of thermodynamic variables for 3-D compressible viscous flows. Similar to the 1-D counterpart, the proposed scheme for the 3-D Navier–Stokes equations is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The positivity of density is achieved by adding an artificial dissipation in the form of the first-order Brenner–Navier–Stokes diffusion operator. Another distinctive feature of the proposed scheme is that the Navier–Stokes viscous terms are discretized by high-order spectral collocation summation-by-parts operators. To eliminate time step stiffness caused by the high-order approximation of the viscous terms and the temperature positivity constraint, the velocity and temperature limiters developed for the 1-D compressible Navier–Stokes equations in Upperman et al. (J. Comput. Phys., 466, 2022) are generalized to three spatial dimensions. These limiters bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results are presented to demonstrate design-order accuracy and positivity-preserving properties of the new first-order scheme for 2-D and 3-D inviscid and viscous flows with strong shocks and contact discontinuities.