Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes (CSIAM Transactions on Applied Mathematics, 2020)
In this paper, we will build a roadmap for the growing literature of highorder quadrature-based entropy stable discontinuous Galerkin (DG) methods, tryingto elucidate the motivations and emphasize the contributions. Compared to the classicDG method which is only provably stable for the square entropy, these DG methodscan 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 theSBP property. The construction is relatively easy for quadrature rules with collocatedsurface nodes. We use the flux differencing technique to ensure entropy balance withinelements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules isachieved through careful modifications of SATs.