Towards an Entropy Stable Spectral Element Framework for Computational Fluid Dynamics

Entropy stable (SS) discontinuous spectral collocation formulations of any order are developed for the compressible Navier-Stokes equations on hexahedral elements. Recent progress on two complementary efforts is presented. The first effort is a generalization of 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. Although being more costly to implement, it is shown that the LG operators are significantly more accurate on comparable grids. 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 generalizes previous SS work to include the possibility of p-refinement at non-conforming interfaces. A generalization of existing entropy stability machinery is developed to accommodate the nuances of fully multi-dimensional summation-by-parts (SBP) operators. The entropy stability of the compressible Euler equations on non-conforming interfaces is demonstrated using the newly developed LG operators and multi-dimensional interface interpolation operators

Entropy Stable Staggered Grid Spectral Collocation for the Burgers' and Compressible Navier-Stokes Equations

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for Burgers' and the compressible Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [1, 2], 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 semi-discrete operators discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality for both Burgers' and 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 to implement. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinearly stability proof for the compressible Navier-Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators)

SBP operators for CPR methods: Master's thesis

Summation-by-parts (SBP) operators have been used in the finite difference framework, providing means to prove conservation and discrete stability by the energy method, predominantly for linear (or linearised) equations. Recently, there have been some approaches to generalise the notion of SBP operators and to apply these ideas to other methods. The correction procedure via reconstruction (CPR), also known as flux reconstruction (FR) or lifting collocation penalty (LCP), is a unifying framework of high order methods for conservation laws, recovering some discontinuous Galerkin, spectral difference and spectral volume methods. Using a reformulation of CPR methods relying on SBP operators and simultaneous approximation terms (SATs), conservation and stability are investigated, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Extensions of SBP methods with diagonal-norm operators to Burgers’ equation are possible by a skew-symmetric form and the introduction of additional correction terms. An analytical setting allowing a generalised notion of SBP methods including modal bases is described and applied to Burgers’ equation, resulting in an extension of the previously mentioned skew-symmetric form. Finally, an extension of the results to multiple space dimensions is presented