Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods
<|reference_start|>Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods: There is a pressing demand for robust, high-order baseline schemes for conservation laws that minimize reliance on supplementary stabilization. In this work, we respond to this demand by developing new baseline schemes within a nodal discontinuous Galerkin (DG) framework, utilizing upwind summation-by-parts (USBP) operators and flux vector splittings. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our method encompasses a broader class of USBP operators, not limited to equidistant grid points. This approach facilitates the development of novel USBP operators on Legendre--Gauss--Lobatto (LGL) points, which are suited for nodal discontinuous Galerkin (DG) methods. The resulting DG-USBP operators combine the strengths of traditional summation-by-parts (SBP) schemes with the benefits of upwind discretizations, including inherent dissipation mechanisms. Through numerical experiments, ranging from one-dimensional convergence tests to multi-dimensional curvilinear and under-resolved flow simulations, we find that DG-USBP operators, when integrated with flux vector splitting methods, foster more robust baseline schemes without excessive artificial dissipation.<|reference_end|>
@article{glaubitz2024generalized,
    title={Generalized upwind summation-by-parts operators and their application to
  nodal discontinuous Galerkin methods},
    author={Jan Glaubitz, Hendrik Ranocha, Andrew R. Winters, Michael
  Schlottke-Lakemper, Philipp "Offner, Gregor Gassner},
    journal={arXiv preprint arXiv:2406.14557},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.14557},
    primaryClass={math.NA cs.NA}
}