High-order Positivity-preserving L2-stable Spectral Collocation Schemes for the 3-D compressible Navier-Stokes equations

TL;DR

This paper develops high-order, positivity-preserving spectral collocation schemes for 3-D compressible Navier-Stokes equations that are entropy stable, L2 stable, and guarantee positive thermodynamic variables, extending previous 1-D methods.

Contribution

It introduces the first family of arbitrary-order, positivity-preserving, entropy stable spectral collocation schemes for 3-D compressible Navier-Stokes equations, with proven stability and accuracy.

Findings

Schemes are provably L2 stable and high-order accurate for smooth solutions.

Guarantee pointwise positivity of thermodynamic variables in 3-D flows.

Demonstrate effective discontinuity capturing with artificial dissipation.

Abstract

This paper extends a new class of positivity-preserving, entropy stable spectral collocation schemes developed for the one-dimensional compressible Navier-Stokes equations in [1,2] to three spatial dimensions. The new high-order schemes are provably L2 stable, design-order accurate for smooth solutions, and guarantee the pointwise positivity of thermodynamic variables for 3-D compressible viscous flows. Similar to the 1-D counterpart, the proposed schemes for the 3-D Navier-Stokes equations 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 preservation 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. To our knowledge, this is the first family of positivity-preserving, entropy stable schemes of arbitrary order of accuracy for the 3-D compressible Navier-Stokes equations.