Nonlinearly-Stable High-Order Methods on Simplices with Improved Efficiency - presented by David Zingg

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

David Zingg

DZ
Slide at 14:22
Operator based on collapsed coordinates Spectral-element formulations in collapsed coordinates are well established for continuous and discontinuous Galerkin methods - `improves efficiency at high polynomial degrees (Moxey et al., 2020), used, for example, in Nektar++ Our contribution has been to extend this approach to efficient operators with the SBP property such that entropy stability can be proven (Montoya and Zingg, 2024 X 2)
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References
  • 1.
    D. Moxey et al. (2019) Nektar++: Enhancing the capability and application of high-fidelity spectral/hp element methods. Computer Physics Communications
  • 2.
    T. Montoya and D. W. Zingg (2024) Efficient entropy-stable discontinuous spectral-element methods using tensor-product summation-by-parts operators on triangles and tetrahedra. Journal of Computational Physics
  • 3.
    T. Montoya and D. W. Zingg (2024) Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra. SIAM Journal on Scientific Computing
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Summary (AI generated)

In this discussion, I would like to reference a well-known concept: Nektar++, which utilizes collapsed coordinates. Our contribution is to extend this framework to incorporate the Summation by Parts (SBP) property, facilitating more straightforward entropy stability proofs. It is important to note that the concepts of spectral element formulations and collapsed coordinates are established ideas, not original to our work, and they contribute to the efficiency of the code. Furthermore, the advantages of this approach become increasingly significant at higher polynomial degrees.

To clarify, the collapsed coordinate transformation is precisely what its name suggests.