Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: First-order approximation

• Published in: Journal of computational physics 2022-10, Vol.466, p.111355, Article 111355
• Main Authors: Upperman, Johnathon, Yamaleev, Nail K.
• Format: Article
• Language: English
• Subjects: Approximation; Artificial dissipation; Brenner regularization; Collocation methods; Compressibility; Computational physics; Discontinuity; Entropy; Entropy stability; Finite volume method; Fluid flow; Internal energy; Inviscid flow; Mathematical analysis; Navier-Stokes equations; Positivity-preserving methods; Stiffness; Summation-by-parts (SBP) operators; The Navier-Stokes equations; Viscous flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111355

This is the first of two companion papers that introduce a new class of positivity-preserving, entropy stable schemes for the 1-D compressible Navier-Stokes equations. In this paper, we develop and analyze a novel first–order finite volume scheme that is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The scheme is constructed by regularizing the Navier-Stokes equations by adding an artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The key distinctive property of the proposed scheme is that it is proven to satisfy the discrete entropy inequality and to guarantee the pointwise positivity of density, temperature, pressure, and internal energy for 1–D compressible viscous flows, while using high–order spectral collocation summation-by-parts operators for discretizing the Navier–Stokes viscous terms. To eliminate time step stiffness caused by the high-order approximation of the viscous terms, we develop two new limiters that bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results demonstrating positivity-preserving and discontinuity-capturing capabilities of the new scheme are presented for viscous and inviscid flows with nearly vacuum regions and very strong shocks and contact discontinuities. •Novel first-order artificial dissipation based on the Brenner diffusion operator.•New 1st-order positivity-preserving L2 stable scheme for 1D Navier-Stokes equations.•New positivity-preserving entropy stable velocity and temperature limiters.