In this paper we will review a recent emerging paradigm shift in the construction and
analysis of high order Discontinuous Galerkin methods applied to approximate solutions of …

High order methods based on diagonal-norm summation by parts operators can be shown
to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of …

We present and analyze an entropy-stable semi-discretization of the Euler equations based
on high-order summation-by-parts (SBP) operators. In particular, we consider general …

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 …

The construction of high order entropy stable collocation schemes on quadrilateral and
hexahedral elements has relied on the use of Gauss--Legendre--Lobatto collocation points …

Reduced order models of nonlinear conservation laws in fluid dynamics do not typically
inherit stability properties of the full order model. We introduce projection-based hyper …

We present a new class of efficient and robust discontinuous spectral-element methods of
arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular …

Abstract Summation-by-parts (SBP) operators allow us to systematically develop energy-
stable and high-order accurate numerical methods for time-dependent differential equations …

The paper presents high-order accurate, energy-, and entropy-stable discretizations
constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble …

The book focuses on stability and approximation results concerning recent numerical
methods for the numerical solution of hyperbolic conservation laws. The work begins with a …