Towards an Entropy Stable Spectral Element Framework for Computational Fluid Dynamics

Mark H. Carpenter, Matteo Parsani, Eric J. Nielsen, Travis C. Fisher

Research output: Chapter in Book/Report/Conference proceedingConference contribution

13 Scopus citations

Abstract

Nonlinearly stable finite element methods of arbitrary type and order, are currently unavailable for discretizations of the compressible Navier-Stokes equations. Summation-by-parts (SBP) entropy stability analysis provides a means of constructing nonlinearly stable discrete operators of arbitrary order, but is currently limited to simple element types. Herein, recent progress is reported, on developing entropy-stable (SS) discontinuous spectral collocation formulations for hexahedral elements. Two complementary efforts are discussed. The first effort generalizes previous SS spectral collocation work to extend the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to tensor product Legendre-Gauss (LG) points. The LG and LGL point formulations are compared on a series of test problems. Both the LGL and LG operators are of comparable efficiency and robustness, as is demonstrated using test problems for which conventional FEM techniques suffer instability. The second effort extends previous work on entropy stability to include p-refinement at nonconforming interfaces. A generalization of existing entropy stability theory is required to accommodate the nuances of fully multidimensional SBP operators. The entropy stability of the compressible Euler equations on nonconforming interfaces is demonstrated using the newly developed LG operators and multidimensional interface interpolation operators. Preliminary studies suggest design order accuracy at nonconforming interfaces.
Original languageEnglish (US)
Title of host publication54th AIAA Aerospace Sciences Meeting
PublisherAmerican Institute of Aeronautics and Astronautics (AIAA)
ISBN (Print)9781624103933
DOIs
StatePublished - Jan 2 2016

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