@Article{CSIAM-AM-1-1,
author = {Tianheng, Chen and Shu, Chi-Wang},
title = {Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2020},
volume = {1},
number = {1},
pages = {1--52},
abstract = {<p style="text-align: justify;">In this paper, we will build a roadmap for the growing literature of high
order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying
to elucidate the motivations and emphasize the contributions. Compared to the classic
DG method which is only provably stable for the square entropy, these DG methods
can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm,
such that the discrete operators collocated at the quadrature points should satisfy the
SBP property. The construction is relatively easy for quadrature rules with collocated
surface nodes. We use the flux differencing technique to ensure entropy balance within
elements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules is
achieved through careful modifications of SATs.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.2020-0003},
url = {https://global-sci.com/article/82366/review-of-entropy-stable-discontinuous-galerkin-methods-for-systems-of-conservation-laws-on-unstructured-simplex-meshes}
}