Nonlinearly-Stable High-Order Methods on Simplices with Improved Efficiency

David Zingg

Slide at 12:40

Entropy-Stable Schemes for Unstructured Grids (Crean et al. 2018)
Extended the work of Fisher and Carpenter to general curved elements building on the multidimensional SBP operators of Hicken et al. (2016)
Entropy-conservative scheme gives (with periodic boundary conditions):
Entropy-stable scheme provides (through dissipative interior penalties):
1. an order hP approximation to the differential form of the Euler equations;
2. discretely conservative of p, pu, pv, and e, and; 3. entropy stable, in that the entropy is non-increasing in time.
Caveat: requires positivity of thermodynamic quantities

References

1.
J. Crean et al. (2017) Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. Journal of Computational Physics
2.
J. E. Hicken et al. (2016) Multidimensional Summation-by-Parts Operators: General Theory and Application to Simplex Elements. SIAM Journal on Scientific Computing

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Summary (AI generated)

Bounding the entropy provides an L2 bound on the solution, provided that the thermodynamic variables remain positive. However, ensuring the positivity of these variables is challenging. When we consider non-smooth solutions, the problem becomes an inequality.

The foundational work of Fisher and Carpenter established a connection between the SBP property and two-point flux functions, leading to the development of provably entropy stable schemes. This work was further advanced by Jared Crean, a student of Jason Higgins at Rensselaer, who extended these concepts to simplices.