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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…
The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a…
The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using…
In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the…
The long-time evolution of extreme mass-ratio inspiral systems requires minimal phase and dispersion errors to accurately compute far-field waveforms, while high accuracy is essential near the smaller black hole (modeled as a Dirac delta…
We investigate the numerical performance of a Discontinuous Galerkin (DG) hydrodynamics implementation when applied to the problem of driven, isothermal supersonic turbulence. While the high-order element-based spectral approach of DG is…
We present a discontinuous finite element method for the shallow water equations which exploits high-resolution realistic bathymetry data without any regularity assumption, also in the case of high-order discretizations. We prove a number…
In this paper we propose a new spatially high order accurate semi-implicit discontinuous Galerkin (DG) method for the solution of the two dimensional incompressible Navier-Stokes equations on staggered unstructured curved meshes. While the…
We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with…
In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order…
Fourier continuation is an approach used to create periodic extensions of non-periodic functions in order to obtain highly-accurate Fourier expansions. These methods have been used in PDE-solvers and have demonstrated high-order convergence…
We present a new high-order accurate Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate material dynamics (for e.g., gasses, fluids, and solids) with up to fourth-order accuracy on cubic meshes. The variables, such as…
This paper proposes novel high-order accurate discontinuous Galerkin (DG) schemes for the one- and two-dimensional ten-moment Gaussian closure equations with source terms defined by a known potential function. Our DG schemes exhibit the…
Discontinuous Galerkin (DG) methods are widely adopted to discretize the radiation transport equation (RTE) with diffusive scalings. One of the most important advantages of the DG methods for RTE is their asymptotic preserving (AP)…
In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using a subcell flux corrections and convex limiting. These methods are…
Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…
We present the recent development of hybridizable and embedded discontinuous Galerkin (DG) methods for wave propagation problems in fluids, solids, and electromagnetism. In each of these areas, we describe the methods, discuss their main…
We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical…
We present the two-dimensional unstructured grids extension of the a posteriori local subcell correction of discontinuous Galerkin (DG) schemes introduced in [52]. The technique is based on the reformulation of DG scheme as a finite volume…
In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve linear elasticity problem with arbitrary order discontinuous finite element spaces in $d$-dimension ($d=2,3$). This method uses polynomials of degree $k+1$ for the…