Stability Issues of Entropy-Stable and/or Split-form High-order Schemes

The focus of the present research is on the analysis of local energy stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we further demonstrate numerically that such schemes cause exponential growth of errors during the simulation. Furthermore, we encounter a similar abnormal behaviour for the compressible Euler equations, for a smooth exact solution of a density wave. Finally, for the same case, we demonstrate numerically that other commonly known split-forms, such as the Kennedy and Gruber splitting, are also locally energy unstable.

Development and analysis of entropy stable no-slip wall boundary conditions for the Eulerian model for viscous and heat conducting compressible flows

Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Phys A Stat Mech Appl 506:350–375, 2018). The spatial discretization is based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier–Stokes models. The numerical results obtained with the two models are compared, and similarities and differences are then highlighted. However, the differences are very small and probably smaller than what the current experimental technology allows to measure.

Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.