Related papers: Strongly consistent low-dissipation WENO schemes f…
The first paper of this series presents a discretely entropy stable discontinuous Galerkin (DG) method for the resistive magnetohydrodynamics (MHD) equations on three-dimensional curvilinear unstructured hexahedral meshes. Compared to other…
In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular, we demonstrate that these schemes provide essential…
A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general…
Grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the…
The high-order numerical solution of the non-linear shallow water equations (and of hyperbolic systems in general) is susceptible to unphysical Gibbs oscillations that form in the proximity of strong gradients. The solution to this problem…
High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a…
We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass,…
This article investigates matrix-free higher-order discontinuous Galerkin discretizations of the Navier--Stokes equations for incompressible flows with variable viscosity. The viscosity field may be prescribed analytically or governed by a…
Alternative finite difference Weighted Essentially Non-Oscillatory (AFD-WENO) schemes allow us to very efficiently update hyperbolic systems even in complex geometries. Recent innovations in AFD-WENO methods allow us to treat hyperbolic…
A systematic numerical study on weak Galerkin (WG) finite element method for second order linear parabolic problems is presented by allowing polynomial approximations with various degrees for each local element. Convergence of both…
In this paper, we propose a class of non-oscillatory, entropy-stable discontinuous Galerkin (NOES-DG) schemes for solving hyperbolic conservation laws. By incorporating a specific form of artificial viscosity, our new scheme directly…
This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite…
In this paper, we present a novel nodal integration scheme for meshfree Galerkin methods that draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field…
In this paper, we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By using the standard Brezzi theory on mixed methods, we carefully define…
We propose a high-order finite difference weighted ENO (WENO) method for the ideal magnetohydrodynamics (MHD) equations. The proposed method is single-stage, single-step, maintains a discrete divergence-free condition on the magnetic field,…
We study a numerical method for convection diffusion equations, in the regime of small viscosity. It can be described as an exponentially fitted conforming Petrov-Galerkin method. We identify norms for which we have both continuity and an…
A numerical dynamical low-rank approximation (DLRA) scheme for the solution of the Vlasov-Poisson equation is presented. Based on the formulation of the DLRA equations as Friedrichs' systems in a continuous setting, it combines recently…
We give an a posteriori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain…
We present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of…
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The…