Date of Award

Summer 2021

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematics & Statistics


Computational and Applied Mathematics

Committee Director

Nail Yamaleev

Committee Member

Mark Carpenter

Committee Member

Fang Hu

Committee Member

Fang Hu


High-order entropy stable schemes are a popular method used in simulations with the compressible Euler and Navier-Stokes equations. The strength of these methods is that they formally satisfy a discrete entropy inequality which can be used to guarantee L2 stability of the numerical solution. However, a fundamental assumption that is explicitly or implicitly used in all entropy stability proofs available in the literature for the compressible Euler and Navier-Stokes equations is that the thermodynamic variables (e.g., density and temperature) are strictly positive in the entire space{time domain considered. Without this assumption, any entropy stability proof for a numerical scheme solving the compressible Navier-Stokes equations is incomplete. Unfortunately, if the solution loses regularity the positivity assumption may fail to hold for a high-order entropy stable scheme unless special care is taken. To address this problem, we present a new class of positivity-preserving, entropy stable spectral collocation schemes for the 3-D compressible Navier-Stokes equations. The key distinctive property of our method is that it is proven to guarantee the pointwise positivity of density and temperature for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method discretized on the same Legendre-Gauss-Lobatto (LGL) collocation points used for the high-order counterpart. The proposed framework is general and can be directly extended to other SBP-SAT-type schemes. Numerical results demonstrating accuracy and positivity-preserving properties of the new spectral collocation schemes are presented for viscous and inviscid flows with nearly vacuum regions, very strong shocks, and contact discontinuities