Abstract Summation-by-parts (SBP) operators have a number of properties that make them
an attractive option for higher-order spatial discretizations of partial differential equations. In …

This article reviews several unstructured grid-based high-order methods for the
compressible Euler and Navier–Stokes equations. We treat the spatial and temporal …

Completely revised text focuses on use of spectral methods to solve boundary value,
eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational …

The aim of this book is to teach you the essentials of spectral collocation methods with the
aid of 40 short MATLAB® programs, or “M-files.”* The programs are available online at …

This paper shows that the discontinuous Galerkin collocation spectral element method with
Gauss--Lobatto points (DGSEM-GL) satisfies the discrete summation-by-parts (SBP) …

We present a convergent high-order accurate scheme for the solution of linear conservation
laws in geometrically complex domains. As our main example we include a detailed …

Emerging as the mathematical expression of principles of conservation, conservation laws
have proven themselves to provide effective and accurate predictive models of our physical …

We establish local well-posedness in the Sobolev space Hs with any s> 3 2 for an integrable
nonlinearly dispersive wave equation arising as a model for shallow water waves known as …

Recently a new high-order formulation for 1D conservation laws was developed by Huynh
using the idea of “flux reconstruction”. The formulation was capable of unifying several …

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably
stable, polynomial-based spectral collocation element methods of arbitrary order for the …