Efficient Discontinuous Galerkin Research Articles

Conservation and stability in a discontinuous Galerkin method for the vector invariant spherical shallow water equations

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, absolute vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.

Novel Efficient Discontinuous Galerkin Time-Domain Modeling of Dispersive Chiral Metamaterials via Media Homogenization

Novel Efficient Discontinuous Galerkin Time-Domain Modeling of Dispersive Chiral Metamaterials via Media Homogenization

An efficient discontinuous Galerkin - mixed finite element model for variable density flow in fractured porous media

Modeling variable density flow (VDF) in fractured porous media is computationally challenging. The challenges are mainly related to: (i) the high permeability contrast between the matrix and the fractures and, (ii) the nonlinearity induced by density variations. Due to their local mass conservation property, cell-centered methods, such as finite volumes (FV), mixed finite elements (MFE) or Discontinuous Galerkin (DG) methods are well suited for modeling mass transfer in highly heterogeneous domains. When applied for fractured media, these methods require small grid cells next to fractures because of the cross-flow equilibrium assumption (i.e. the pressure and concentration in the fracture and in the adjacent matrix grid-cells are assumed the same). To avoid this constraint, an efficient model is developed in this work using advanced cell-centered numerical methods for VDF in fractured porous media with cross-flow equilibrium assumed only across the fractures. The new model uses the hybrid MFE method for the flow discretization in the matrix and in the fracture continua. Mass transport in the matrix is modeled using a monotonic upwind MFE scheme. Advection-dominated transport in fractures is discretized with the DG method, which is well adapted for hyperbolic equations. The new model ensures continuity of pressure, concentration, fluid mass flux, advective and dispersive contaminant fluxes at matrix-fracture interfaces as well as at the intersection of several fractures. The time discretization is performed using high-order adaptive time integration techniques via the method of lines (MOL). The developed model is first validated by comparison against a 2D-2D model for a test problem dealing with linear flow and transport in a 2D domain involving a “+”-shaped barrier/fracture network. Then, the MFE-DG model is used for the simulation of the Henry saltwater intrusion problem and the results are validated by comparison against the semi-analytical solution in the case of unfractured and fractured aquifers. Finally, the simulation of a transport problem with variable viscosity in a fractured heterogeneous domain points out the superiority of the new model in terms of accuracy and efficiency when compared to a standard finite element model.

Simulation of the nonlinear Kerr and Raman effect with a parallel local time-stepping DGTD solver.

In this paper, an efficient discontinuous Galerkin time-domain (DGTD) method is proposed to solve Maxwell's equations for nonlinear Kerr or Raman media. Based on our previous work, an arbitrary high-order derivatives DGTD method with a local time-stepping scheme is introduced for simulating dynamic optical responses in nonlinear dispersive media such that the nonlinear effects do not impose constraints on the stability conditions for linear subdomains. Therefore, the scheme enables the simulations in the nonlinear and linear media regions with independent time-stepping increments, which greatly improves the efficiency of the time-domain analysis. Moreover, by applying an iteration solution scheme, the proposed method preserves the intrinsic local features, which is favorable for the realization of highly parallelized algorithms. Numerical examples demonstrate the accuracy and the efficiency of our proposed method. We believe the proposed method provides an effective tool for numerical analysis of nonlinear optical phenomena.

Generalized Periodic Boundary Conditions for DGTD Analysis of Arbitrary Skewed Periodic Structures

An efficient discontinuous Galerkin time-domain (DGTD) method with an implementation of generalized periodic boundary conditions (PBCs) is proposed to analyze the electromagnetic scattering from arbitrary skewed periodic structures. The transformed field variable approach and the discontinuous Galerkin technique with nonconformal mesh are presented to implement the generalized PBCs for arbitrary skewed periodic structures under both normally and obliquely incident illuminations. The arbitrary high-order time-stepping scheme, which retains the DGTD feature of high-order accuracy and breaks the Butcher barrier, is extended to a transformed version of Maxwell’s equations introduced by the generalized PBCs implementation. The proposed method enables an efficient modeling of arbitrary skewed arrays with a fixed unit-cell mesh. Numerical examples for skewed periodic structures, such as an infinite gold film, 1-D and 2-D staggered dipole frequency-selective surfaces (FSSs), mechanically reconfigurable FSSs, and skewed nanohole arrays, are presented to demonstrate the accuracy and applicability of the proposed method.

An Efficient Discontinuous Galerkin Method Using a Tetrahedral Mesh for 3D Seismic Wave Modeling

ABSTRACTThe discontinuous Galerkin (DG) method is a numerical algorithm that is widely used in various fields. It has high accuracy and low numerical dispersion with advantages of easy handling boundary conditions and good parallel performance. In this study, we develop an efficient parallel weighted Runge–Kutta discontinuous Galerkin (WRKDG) method on unstructured meshes for solving 3D seismic wave equations. The DG method we use is based on the first-order formulation of a hyperbolic system with an explicit weighted Runge–Kutta time discretization. We describe the numerical scheme and parallel implementation in detail, and analyze the stability condition and numerical dispersion and dissipation. We carry out a convergence test on unstructured meshes and investigate the parallel efficiency of the implementation of the WRKDG method. The speedup curve shows that the method has good parallel performance. Finally, we present several numerical simulation examples, including acoustic and elastic wave propagations in isotropic and anisotropic media. Numerical results further verify the effectiveness of the WRKDG method in solving 3D wave propagation problems.

Efficient discontinuous Galerkin scheme for analyzing nanostructured photoconductive devices.

Incorporation of plasmonic nanostructures in the design of photoconductive devices (PCDs) has significantly improved their optical-to-terahertz conversion efficiency. However, this improvement comes at the cost of increased complexity for the design and simulation of these devices. Indeed, accurate and efficient modeling of multiphysics processes and intricate device geometries of nanostructured PCDs is challenging due to the high computational cost resulting from multiple characteristic scales in time and space. In this work, a discontinuous Galerkin (DG)-based unit-cell scheme for efficient simulation of PCDs with periodic nanostructures is proposed. The scheme considers two physical stages of the device and models them using two coupled systems: a system of Poisson and drift-diffusion equations describing the nonequilibrium steady state, and a system of Maxwell and drift-diffusion equations describing the transient stage. A "potential-drop" boundary condition is enforced on the opposing boundaries of the unit cell to mimic the effect of the bias voltage. Periodic boundary conditions are used for carrier densities and electromagnetic fields. The unit-cell model described by these coupled equations and boundary conditions is discretized using DG methods. Numerical results demonstrate that the proposed DG-based unit-cell scheme has the same accuracy in predicting the THz photocurrent as the DG framework that takes into account the whole device, while it significantly reduces the computational cost.

An Efficient Discontinuous Galerkin Method for Cavity Design

Cavities are strong scattering parts for an aircraft. A discontinuous Galerkin (DG) method is presented to efficiently optimize cavity scattering. The DG method is shown to have ability of only remeshing the optimized parts and not remeshing the whole cavity in the computing process of scattering from the whole cavity. An effective and simple preconditioner, called distance sparse preconditioner (DSP), is applied to the DG method, different from the conventional block-diagonal preconditioner (BDP) and sparse approximate inverse preconditioner (SAI). The former one is usually recommended in the DG methods and the latter one is generally used in method of moment (MoM). Numerical results show that the DSP is more efficient than the BDP and SAI in the DG method for the cavity analysis. The effects of cavity opening shape, area, and transition shape are studied for scattering from cavities. Some interesting and useful numerical results are presented for cavity design.

An Exponential-Based DGTD Method for Modeling 3-D Plasma-Surrounded Hypersonic Vehicles

Plasma surrounding a hypersonic vehicle can lead to the absorption of incoming radar radiation or the interruption of communication. Therefore, the development of hypersonic vehicle relies on a clear understanding of the strong interaction between the surrounding plasma and the incident electromagnetic wave. However, because of the presence of complex geometrical features and heterogeneous media, these applications are often multiscale. The modeling of such applications in 3-D is extremely challenging for traditional numerical methods. In this work, an efficient discontinuous Galerkin time-domain (DGTD) method for the 3-D modeling of plasma-surrounded hypersonic vehicles is presented. To overcome the grid-induced stiffness problem arising from the modeling of such multiscale applications with dispersive media, and to improve the modeling efficiency, a high-order exponential-based time integration for the time-domain Maxwell’s equations discretized by DG schemes with a general flux formulation is proposed. This time integration has excellent stability properties in the locally refined part of the mesh and hence enables the usage of larger time steps compared with the explicit time marching scheme. Additionally, the exponential-based time integration is developed to adapt an anisotropic perfectly matched layer for the modeling of open-domain problems. Finally, numerical experiments are presented to investigate the accuracy and convergence properties and to demonstrate the well-modeling capability of the proposed method.

Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations

This work presents and compares efficient implementations of high-order discontinuous Galerkin methods: a modal matrix-free discontinuous Galerkin (DG) method, a hybridizable discontinuous Galerkin (HDG) method, and a primal formulation of HDG, applied to the implicit solution of unsteady compressible flows. The matrix-free implementation allows for a reduction of the memory footprint of the solver when dealing with implicit time-accurate discretizations. HDG reduces the number of globally-coupled degrees of freedom relative to DG, at high order, by statically condensing element-interior degrees of freedom from the system in favor of face unknowns. The primal formulation further reduces the element-interior degrees of freedom by eliminating the gradient as a separate unknown. This paper introduces a p-multigrid preconditioner implementation for these discretizations and presents results for various flow problems. Benefits of the p-multigrid strategy relative to simpler, less expensive, preconditioners are observed for stiff systems, such as those arising from low-Mach number flows at high-order approximation. The p-multigrid preconditioner also shows excellent scalability for parallel computations. Additional savings in both speed and memory occur with a matrix-free/reduced version of the preconditioner.

Bernstein-Bézier weight-adjusted discontinuous Galerkin methods for wave propagation in heterogeneous media

This paper presents an efficient discontinuous Galerkin method to simulate wave propagation in heterogeneous media with sub-cell variations. This method is based on a weight-adjusted discontinuous Galerkin method (WADG), which achieves high order accuracy for arbitrary heterogeneous media [1]. However, the computational cost of WADG grows rapidly with the order of approximation. In this work, we propose a Bernstein-Bézier weight-adjusted discontinuous Galerkin method (BBWADG) to address this cost. By approximating sub-cell heterogeneities by a fixed degree polynomial, the main steps of WADG can be expressed as polynomial multiplication and L2 projection, which we carry out using fast Bernstein algorithms. The proposed approach reduces the overall computational complexity from O(N2d) to O(Nd+1) in d dimensions. Numerical experiments illustrate the accuracy of the proposed approach, and computational experiments for a GPU implementation of BBWADG verify that this theoretical complexity is achieved in practice.

Discontinuous Galerkin time domain analysis of electromagnetic scattering from dispersive periodic nanostructures at oblique incidence.

An efficient discontinuous Galerkin time domain (DGTD) method with a generalized dispersive material (GDM) model and periodic boundary conditions (PBCs), hereto referred to as DGTD-GDM-PBCs, is proposed to analyze the electromagnetic scattering from dispersive periodic nanostructures. The GDM model is utilized to achieve a robust and accurate universal model for arbitrary dispersive materials. Using a transformed field variable technique, PBCs are introduced to efficiently truncate the computational domain in the periodic directions for both normally and obliquely incident illumination cases. Based on the transformed Maxwell's equations with PBCs, the formulation of the DGTD method with a GDM model is derived. Furthermore, a Runge-Kutta time-stepping scheme is proposed to update the semi-discrete transformed Maxwell's equations and auxiliary differential equations (ADEs) with high order accuracy. Numerical examples for periodic nanostructures with dispersive elements, such as reflection and transmission of a thin film, surface plasmon at the interfaces of a metallic hole array structure, and absorption properties of a dual-band infrared absorber are presented to demonstrate the accuracy and capability of the proposed method.

Efficient, Accurate and Energy Stable Discontinuous Galerkin Methods for Phase Field Models of Two-Phase Incompressible Flows

Efficient, Accurate and Energy Stable Discontinuous Galerkin Methods for Phase Field Models of Two-Phase Incompressible Flows

Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows

We present a robust and accurate discretization approach for incompressible turbulent flows based on high-order discontinuous Galerkin methods. The DG discretization of the incompressible Navier–Stokes equations uses the local Lax–Friedrichs flux for the convective term, the symmetric interior penalty method for the viscous term, and central fluxes for the velocity–pressure coupling terms. Stability of the discretization approach for under-resolved, turbulent flow problems is realized by a purely numerical stabilization approach. Consistent penalty terms that enforce the incompressibility constraint as well as inter-element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under-resolved regime. The penalty parameters are derived by means of dimensional analysis using penalty factors of order 1. Applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows. The proposed approach is applicable independently of the solution strategy used to solve the incompressible Navier–Stokes equations, i.e., it can be used for both projection-type solution methods as well as monolithic solution approaches. Since our approach is based on consistent penalty terms, it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems. Robustness and accuracy are verified for the Orr–Sommerfeld stability problem, the Taylor–Green vortex problem, and turbulent channel flow. Moreover, the accuracy of high-order discretizations as compared to low-order discretizations is investigated for these flow problems. A comparison to state-of-the-art computational approaches for large-eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers.

A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier–Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad–div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at Reτ=180 as well as 590.

An efficient discontinuous Galerkin method for two-phase flow modeling by conservative velocity projection

Purpose – The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection. Design/methodology/approach – The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations. Findings – The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements. Research limitations/implications – The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible. Practical implications – The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media. Originality/value – The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.

An efficient discontinuous Galerkin finite element method with nested domain decomposition for simulations of microresistivity imaging

A discontinuous Galerkin finite element method is employed to study the responses of microresistivity imaging tools used in the oil and gas exploration industry. The multiscale structure of an imaging problem is decomposed into several nested subdomains based on its geometric characteristics. Each subdomain is discretized independently, and numerical flux is used to couple all subdomains together. The nested domain decomposition scheme will lead to a block tridiagonal linear system, and the block Thomas algorithm is utilized here to eliminate the subdomain based iteration in the step of solving the linear system. Numerical results demonstrate the validity and efficiency of this method.

Predictor-Corrector LU-SGS Discontinuous Galerkin Finite Element Method for Conservation Laws

Efficient implicit predictor-corrector LU-SGS discontinuous Galerkin (DG) approach for compressible Euler equations on unstructured grids is investigated by adding the error compensation of high-order term. The original LU-SGS and GMRES schemes for DG method are discussed. Van Albada limiter is employed to make the scheme monotone. The numerical experiments performed for the transonic inviscid flows around NACA0012 airfoil, RAE2822 airfoil, and ONERA M6 wing indicate that the present algorithm has the advantages of low storage requirements and high convergence acceleration. The computational efficiency is close to that of GMRES scheme, nearly 2.1 times greater than that of LU-SGS scheme on unstructured grids for 2D cases, and almost 5.5 times greater than that of RK4 on unstructured grids for 3D cases.

A memory efficient discontinuous galerkin finite-element time-domain scheme for simulations of finite periodic structures

A discontinuous Galerkin finite-element time-domain (DG-FETD) scheme with nonconforming interface meshes is proposed for transient simulations of finite periodic structures. A finite periodic structure is divided into an array of identical building blocks. Each building block is decomposed into a volume part and six surface parts, and they are computed and stored separately. By doing this one building block can represent the whole discretized structure, and the repeat of meshes and system matrices can be exploited to the maximum extent. This DG-FETD scheme costs much less memory compared to conventional methods in simulating finite periodic structures. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:1929–1933, 2014

Mixed interior penalty discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions

AbstractThis article is concerned with developing efficient discontinuous Galerkin methods for approximating viscosity (and classical) solutions of fully nonlinear second‐order elliptic and parabolic partial differential equations (PDEs) including the Monge–Ampère equation and the Hamilton–Jacobi–Bellman equation. A general framework for constructing interior penalty discontinuous Galerkin (IP‐DG) methods for these PDEs is presented. The key idea is to introduce multiple discrete Hessians for the viscosity solution as a means to characterize the behavior of the function. The PDE is rewritten in a mixed form composed of a single nonlinear equation paired with a system of linear equations that defines multiple Hessian approximations. To form the single nonlinear equation, the nonlinear PDE operator is replaced by the projection of a numerical operator into the discontinuous Galerkin test space. The numerical operator uses the multiple Hessian approximations to form a numerical moment which fulfills consistency and g‐monotonicity requirements of the framework. The numerical moment will be used to design solvers that will be shown to help the IP‐DG methods select the “correct” solution that corresponds to the unique viscosity solution. Numerical experiments are also presented to gauge the effectiveness and accuracy of the proposed mixed IP‐DG methods.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1538–1557, 2014