Discontinuous Galerkin Method Research Articles

On the Integrals of Functions Over Polyhedra

The discontinuous Galerkin method is highly flexible for handling complex meshes. One of the challenges of the method lies in the calculation of the coefficients of the elementary matrices. Indeed, these coefficients involve integrals of polynomials over the cells, and as soon as these cells have a complex shape, the integration becomes difficult. In this paper, we focus on how to perform such integration by reducing volume integrals over a cell to line integrals along the edges of the cell. In some cases, the integration can be done exactly, and in others, the line integration can be performed using quadrature formulas. We will provide a recipe, for example, in the case where the cell is an arbitrary polyhedron with flat faces, to compute the integral of a polynomial exactly. However, the method we propose also allows for the calculation of the integral of a function over a polyhedron with curved surfaces.

Nitsche extended finite element method of a Ventcel transmission problem with discontinuities at the interface

The objective of this work is to study a diffusion equation with non-standard transmission conditions, which include discontinuities and Ventcel boundary conditions at the interface. In order to handle jumps and means of the flux and the test-functions, we use broken Sobolev spaces. We present a Nitsche type finite element approach and compare it with a discontinuous Galerkin method. We establish consistency, stability and a priori error estimates and verify them numerically.

Local Discontinuous Galerkin Method for Solving Compound Nonlinear KdV‐Burgers Equations

ABSTRACTIn this work, an efficient local discontinuous Galerkin scheme is applied to numerically solve the nonlinear compound KdV‐Burgers equation. The numerical scheme utilizes a local discontinuous Galerkin discretization technique in the spatial direction coupled with a higher order strong‐stability‐preserving Runge–Kutta scheme in the temporal direction. The stability analysis of the implemented numerical scheme, along with a detailed error estimate for smooth solutions, are also established by carefully selecting the interface numerical fluxes. In addition, numerical simulations are carried out using several illustrative examples, and the results obtained are then compared with solutions acquired by the analytical ‐expansion method to validate the acceptable accuracy and plausibility of the proposed numerical technique. Also, both two‐dimensional and three‐dimensional graphical representations are presented to visually demonstrate the physical significance of the resulting traveling wave solutions.

An essentially oscillation-free reconstructed discontinuous Galerkin method for hyperbolic conservation laws

An essentially oscillation-free reconstructed discontinuous Galerkin method for hyperbolic conservation laws

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

An efficient discontinuous Galerkin method for the hydro-dynamically coupled phase-field vesicle membrane model

An efficient discontinuous Galerkin method for the hydro-dynamically coupled phase-field vesicle membrane model

A staggered discontinuous Galerkin method for solving S transport equation on arbitrary polygonal grids

A staggered discontinuous Galerkin method for solving S transport equation on arbitrary polygonal grids

A medius error analysis for the conforming discontinuous Galerkin finite element methods

Abstract In this paper, we derive an improved error estimate of a conforming discontinuous Galerkin (CDG) method for both second and fourth order elliptic problems, assuming only minimal regularity on the exact solutions. The result we established is called a medius error estimate since it relies on both a priori and a posteriori analysis. Compared with the standard interior penalty discontinuous Galerkin (IPDG) method, when choosing the standard DG norm, an additional term ‖∇u − ∇ w v h ‖0 is incorporated in the CDG formulation for second order elliptic equation, while for the case of fourth order equation, this term becomes ‖ Δ u − Δ w v h ‖ 0 ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{v}}_{h}{{\Vert}}_{0}$ . These terms disappear if we directly analyze the nonstandard error formulations ‖∇u − ∇ w u h ‖0 and ‖ Δ u − Δ w u h ‖ 0 ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{u}}_{h}{{\Vert}}_{0}$ . Extensive numerical results are also carried out to validate our theoretical findings.

An Algebraic Preconditioner for the Exactly Divergence‐Free Discontinuous Galerkin Method for Stokes

ABSTRACTWe present an algebraic preconditioner for the exactly divergence‐free discontinuous Galerkin (DG) discretization of Cockburn, Kanschat, and Schötzau [J. Sci. Comput., 31 (2007), pp. 61–73] and Wang and Ye [SIAM J. Numer. Anal., 45 (2007), pp. 1269–1286] for the Stokes problem. The exactly divergence‐free DG method uses finite elements that use an ‐conforming basis, thereby significantly complicating its solution by iterative methods. Several preconditioners for this Stokes discretization has been developed, but each is based on specialized solvers or decompositions. To avoid requiring custom solvers, we hybridize the ‐conforming finite element so that the velocity lives in a standard ‐DG space, and present a simple algebraic preconditioner for the extended hybridized system. The proposed preconditioner is optimal in mesh size , effective in 2d and 3d, and only relies on standard relaxation and algebraic multigrid methods available in many packages. Furthermore, the Schur complement approximation is robust in element order , although more AMG cycles are needed on the velocity block when increasing .

Low regularity error estimates of three pressure-robust discontinuous Galerkin methods for Stokes equations

Low regularity error estimates of three pressure-robust discontinuous Galerkin methods for Stokes equations

Development of a high-order global dynamical core using the discontinuous Galerkin method for an atmospheric large-eddy simulation (LES) and proposal of test cases: SCALE-DG v0.8.0

Development of a high-order global dynamical core using the discontinuous Galerkin method for an atmospheric large-eddy simulation (LES) and proposal of test cases: SCALE-DG v0.8.0

Analysis of Robust Hybridized Discontinuous Galerkin Methods for Viscoacoustic Wave Equations

In this paper we study hybridized discontinuous Galerkin methods for viscoacoustic wave equations with a general number of viscosity terms. For viscoacoustic equations rewritten as a first order symmetric hyperbolic system, we develop a hybridized local discontinuous Galerkin method which is robust for the number of viscosity terms. We show that the method satisfies a discrete energy estimate such that the implicit constants in the energy estimate are independent of the number of viscosity terms and the length of simulation time. Furthermore, we show that the sizes of reduced system after static condensation are independent of the number of viscosity terms. Optimal a priori error estimates with the Crank–Nicolson scheme are proved and numerical test results are included.

Numerical Simulations of Shock-driven Heavy Fluid Layer

The present study presents the numerical simulations for a shocked-heavy fluid layer with a stratified N2/SF6/N2 configuration. Simulations were conducted using a third-order modal discontinuous Galerkin method to solve the compressible two-component Euler equations. The results were validated against experimental data, confirming the accuracy of the computational approach. Dynamics of the heavy fluid layer were found to be strongly influenced by the shock Mach numbers Ms = 1.15, 1.25, 1.5. At a lower Mach number Ms = 1.15, the interface deformations remained smooth and relatively symmetric, with limited vorticity generation and weak perturbations. Baroclinic effects at this stage were minimal, and the instability growth remained linear. As the Mach number increased to Ms = 1.25, the interaction became nonlinear, leading to the formation of small-scaled vortex structures driven by moderate baroclinic effects. Interface mixing intensified as rotational motion increased. At the highest Mach number Ms = 1.5, the interface rapidly evolved into chaotic structures, characterized by significant vorticity amplification, vortex roll-up, and the onset of turbulence. The baroclinic vorticity, resulting from the misalignment of pressure and density gradients, dominated the vorticity production mechanism, particularly at higher Mach numbers. Quantitative analysis demonstrated that average vorticity, baroclinic vorticity, and enstrophy grew rapidly with increasing Mach numbers. Enstrophy, which quantifies turbulence intensity, exhibited pronounced growth at Ms = 1.5, marking the transition to turbulent mixing.

An edgewise iterative scheme for the discontinuous Galerkin method with Lagrange multiplier for Poisson’s equation

An edgewise iterative scheme for the discontinuous Galerkin method with Lagrange multiplier for Poisson’s equation

Realizability-preserving discontinuous Galerkin method for spectral two-moment radiation transport in special relativity

Realizability-preserving discontinuous Galerkin method for spectral two-moment radiation transport in special relativity

A modified artificial viscosity for shock-capturing and its application in discontinuous Galerkin methods

To improve the performance of artificial viscosity (AV) methods in compressible multi-scale flow simulations, we aim to develop a modified AV flux with both good shock-capturing capability and adjustable low-dissipation in vortex-dominated regions. First, representative quantities for shocks and vortices are selected and dissipation equations for these quantities based on four existing AV fluxes are derived. Second, the dissipation behavior of these AV fluxes is analyzed and compared based on theoretical analysis and numerical experiments. Third, a modified AV flux is proposed by combining the existing AV fluxes, which has shock-capturing capability similar to the Laplacian form AV flux and has adjustable low-dissipation in vortex-dominated regions. Cases including shock tubes, Kelvin–Helmholtz instability, shock wave/shear-layer interaction, shock wave/boundary-layer interaction, supersonic airfoil, and high pressure turbine vane are adopted to test and compare the performance of different AV fluxes. The results show that the modified AV flux simulates shocks with relatively low numerical oscillations and captures more flow details in vortex-dominated regions, which indicates that it is promising in compressible turbulence simulations.

Sapphire++: A particle transport code combining a spherical harmonic expansion and the discontinuous Galerkin method

Sapphire++: A particle transport code combining a spherical harmonic expansion and the discontinuous Galerkin method

A mixed discontinuous Galerkin method for the Biot equations

A mixed discontinuous Galerkin method for the Biot equations

A high-order discontinuous Galerkin method for compressible interfacial flows with consistent and conservative Phase Fields

A high-order discontinuous Galerkin method for compressible interfacial flows with consistent and conservative Phase Fields

The Runge–Kutta discontinuous Galerkin method with stage-dependent polynomial spaces for hyperbolic conservation laws

The Runge–Kutta discontinuous Galerkin method with stage-dependent polynomial spaces for hyperbolic conservation laws