Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of Any Order for the Compressible Navier-Stokes equations

M. Parsani, M.H. Carpenter, T.C. Fisher, E.J. Nielsen
SIAM Journal on Scientific Computing, 38(5), A3129-A3162, (2016)

Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of Any Order for the Compressible Navier-Stokes equations

Keywords

High-order accurate discontinuous methods, Entropy stability, SBP-SAT, Compressible Navier-Stokes, staggered grid, conservation

Abstract

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 ows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are signi cantly 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 di erence, or ux reconstruction operators).

Code

DOI: 10.1137/15M1043510

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