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Speaker
Nail Yamaleev, Associate Professor Department of Mathematics & Statistics Old Dominion University
Title
High-order Positivity-Preserving Entropy Stable Spectral Collocation Schemes for the 3-D Navier-Stokes Equations
Abstract
In this talk, we present a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 3-D compressible Navier-Stokes equations. The key distinctive feature of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables 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 collocation points used for the high-order counterpart. To provide the positivity-preserving and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding an artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high-and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves design-order of accuracy for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorousL2-stability proof for the symmetric form of the 3-D discretized Navier-Stokes equations. Numerical results demonstrating accuracy and positivity-preserving properties of the new schemes will be presented for viscous flows with nearly vacuum regions and very strong shocks and contact discontinuities.