TY - JOUR

T1 - Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier--Stokes Equations

AU - Parsani, Matteo

AU - Carpenter, Mark H.

AU - Fisher, Travis C.

AU - Nielsen, Eric J.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This work was partially supported by King Abdullah University of Science & Technology (KAUST) in Thuwal, Saudi Arabia

PY - 2016/10/4

Y1 - 2016/10/4

N2 - Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier--Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, M. Parsani, M. H. Carpenter, and E. J. Nielsen, J. Comput. Phys., 292 (2015), pp. 88--113], extends the applicable set of points from tensor product, Legendre--Gauss--Lobatto (LGL), to a combination of tensor product Legendre--Gauss (LG) and LGL points. The new semidiscrete operators discretely conserve mass, momentum, energy, and satisfy a mathematical entropy inequality for the compressible Navier--Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly from a theoretical point of view. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinear stability proof for the compressible Navier--Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators).

AB - Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier--Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, M. Parsani, M. H. Carpenter, and E. J. Nielsen, J. Comput. Phys., 292 (2015), pp. 88--113], extends the applicable set of points from tensor product, Legendre--Gauss--Lobatto (LGL), to a combination of tensor product Legendre--Gauss (LG) and LGL points. The new semidiscrete operators discretely conserve mass, momentum, energy, and satisfy a mathematical entropy inequality for the compressible Navier--Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly from a theoretical point of view. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinear stability proof for the compressible Navier--Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators).

UR - http://hdl.handle.net/10754/621846

UR - http://epubs.siam.org/doi/10.1137/15M1043510

UR - http://www.scopus.com/inward/record.url?scp=84994164571&partnerID=8YFLogxK

U2 - 10.1137/15M1043510

DO - 10.1137/15M1043510

M3 - Article

VL - 38

SP - A3129-A3162

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 5

ER -