scholarly journals Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties

2020 ◽  
Vol 82 (2) ◽  
Author(s):  
Jason E. Hicken
Keyword(s):  
High Order ◽  
Summation By Parts ◽  
Entropy Stable

High-order accurate entropy-stable discontinuous collocated Galerkin methods with the summation-by-parts property for compressible CFD frameworks: Scalable SSDC algorithms and flow solver

2021 ◽  
Vol 424 ◽  
pp. 109844 ◽  
Author(s):  
Matteo Parsani ◽  
Radouan Boukharfane ◽  
Irving Reyna Nolasco ◽  
David C. Del Rey Fernández ◽  
Stefano Zampini ◽  
...  
Keyword(s):  
High Order ◽  
Galerkin Methods ◽  
Summation By Parts ◽  
Flow Solver ◽  
Entropy Stable

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

2022 ◽  
Vol 90 (3) ◽  
Author(s):  
Gregor J. Gassner ◽  
Magnus Svärd ◽  
Florian J. Hindenlang
Keyword(s):  
Growth Rate ◽  
Density Wave ◽  
High Order ◽  
Energy Stability ◽  
Collocation Methods ◽  
Local Energy ◽  
Summation By Parts ◽  
Non Linear ◽  
Entropy Stable ◽  
Split Form

AbstractThe 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.

scholarly journals Entropy stable essentially nonoscillatory methods based on RBF reconstruction

2019 ◽  
Vol 53 (3) ◽  
pp. 925-958 ◽  
Author(s):  
Jan S. Hesthaven ◽  
Fabian Mönkeberg
Keyword(s):  
Radial Basis Functions ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Basis Functions ◽  
One Dimensional ◽  
Hyperbolic Conservation ◽  
Radial Basis ◽  
Sign Property ◽  
Entropy Stable ◽  
Essentially Nonoscillatory

To solve hyperbolic conservation laws we propose to use high-order essentially nonoscillatory methods based on radial basis functions. We introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Thus, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. The main contribution is the construction of an algorithm and a smoothness indicator that ensures an interpolation function which fulfills the sign-property on general one dimensional grids.

Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements

2018 ◽  
Vol 356 ◽  
pp. 410-438 ◽  
Author(s):  
Jared Crean ◽  
Jason E. Hicken ◽  
David C. Del Rey Fernández ◽  
David W. Zingg ◽  
Mark H. Carpenter
Keyword(s):  
Euler Equations ◽  
Summation By Parts ◽  
Curved Elements ◽  
Entropy Stable
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