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 entropy-stable discontinuous Galerkin methods for the compressible Euler and
Navier-Stokes equations require the positivity of thermodynamic quantities in order to …

In this work, we present a novel numerical discretization of a variable pressure multilayer
shallow water model. The model can be written as a hyperbolic PDE system and allows the …

Provably stable flux reconstruction (FR) schemes are derived for partial differential
equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction …

Entropy stable schemes ensure that physically meaningful numerical solutions also satisfy a
semi-discrete entropy inequality under appropriate boundary conditions. In this work, we …

Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated
volume and facet nodes, known as diagonal-E operators, are attractive for entropy-stable …

This work introduces a general strategy to develop well-balanced high-order Discontinuous
Galerkin (DG) numerical schemes for systems of balance laws. The essence of our …

We demonstrate that the streamline‐upwind/Petrov–Galerkin (SUPG) formulation enhanced
with YZβ discontinuity‐capturing, that is, the SUPG‐YZβ formulation, is an efficient and …

We present an entropy stable nodal discontinuous Galerkin spectral element method
(DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes …

High order entropy stable schemes provide improved robustness for computational
simulations of fluid flows. However, additional stabilization and positivity preserving limiting …