Nodal Discontinuous Galerkin Method Research Articles

An efficient parallel scheme based on the nodal discontinuous Galerkin method for fluid flow simulations

An efficient parallel scheme based on the nodal discontinuous Galerkin finite element method (nodal-DGFEM) for the numerical solution of the partial differential equations governing fluid flow phenomena is discussed. The flow solver is demonstrated to perform numerical simulation of two-dimensional flow regimes on unstructured triangular grids. The parallel implementation serves to fulfill the requisition of the numerical method regarding high-performance computing resources. The distributed memory programming model with the domain decomposition approach is adopted. The message passing interface library is used for communication among the parallel processes, which are assigned domain-decomposed subproblems. The presented parallelization strategy accurately and efficiently tackles the communication of multi-node data on the element edges between the neighboring parallel processes. The efficacy and efficiency of the parallel solver are demonstrated through solving the well-known problem of non-viscous isentropic convecting vortex flow on parallel systems. The parallelization would extend the scope of the DGFEM by producing solutions in reasonable time frames.

Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. The new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method. As a result, the accuracy of the low-order method does not degrade when used with high-order target methods. The method is applied to both scalar conservation laws, for which the discrete maximum principle is naturally enforced, and to systems of conservation laws such as the Euler equations, for which positivity of density and a minimum principle for specific entropy are enforced. Numerical results are presented on a number of benchmark test cases.

Numerical simulation of the interaction of wave phase conjugation with bubble clouds

The interaction of the Wave Phase Conjugation (WPC) process with bubble cloud dynamics is investigated via numerical modeling in application to bubble size measurements. The WPC simulator by Modarreszadeh and Timofeev (Computers and Fluids 197, 104353) is used as a starting point. In the simulator, a modified version of the high-order Nodal Discontinuous Galerkin method is used for modeling of acoustic wave propagation in the active WPC conjugator and the surrounding linear or non-linear fluid in 2D/axisymmetric domains. The effects of bubble dynamics are incorporated with the help of induced volume fractions. The modified Keller–Miksis model is employed to simulate vibrations of bubbles subjected to acoustic waves. After thorough verification and validation of the numerical scheme and the physical model, the interaction of various bubble clouds, including poly-disperse ones, with the WPC process is examined with the help of some existing and newly suggested descriptive indicators, e.g., the amplification factor and the absolute and relative signal intensities. The results show that signals scattered from bubble-plane clouds, in which the bubbles are non-physically synced, are strong enough to affect the modulation process if the stimulation frequency is close to the natural frequency of bubbles. For more realistic bubble clouds, the modulation process is almost intact. However, conjugate waves do carry some signatures of bubble-cloud dynamics, e.g., signals with frequencies higher than the natural frequency of bubbles are significantly diminished. The study demonstrates that WPC-based techniques have a promising potential to be used for measuring bubble dimensions.

An efficient reconstruction algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method

High-energy-density (HED) hydrodynamics studies such as those relevant to inertial confinement fusion and astrophysics require highly disparate densities, temperatures, viscosities, and other diffusion parameters over relatively short spatial scales. This presents a challenge for high-order accurate methods to effectively resolve the hydrodynamics at these scales, particularly in the presence of highly disparate diffusion. A significant volume of engineering and physics applications use an unstructured discontinuous Galerkin (DG) method developed based on the finite element mesh generation and algorithmic framework. This work discusses the application of an affine reconstructed nodal DG method for unstructured grids of triangles. Solving the diffusion terms in the DG method is non-trivial due to the solution representations being piecewise continuous. Hence, the diffusive flux is not defined on the interface of elements. The proposed numerical approach reconstructs a smooth solution in a parallelogram that is enclosed by the quadrilateral formed by two adjacent triangle elements. The interface between these two triangles is the diagonal of the enclosed parallelogram. Similar to triangles, the mapping of parallelograms from a physical domain to a reference domain is an affine mapping, which is necessary for an accurate and efficient implementation of the numerical algorithm. Thus, all computations can still be performed on the reference domain, which promotes efficiency in computation and storage. This reconstruction does not make assumptions on choice of polynomial basis. Reconstructed DG algorithms have previously been developed for modal implementations of the convection–diffusion equations. However, to the best of the authors’ knowledge, this is the first practical guideline that has been proposed for applying the reconstructed algorithm on a nodal discontinuous Galerkin method with a focus on accuracy and efficiency. The algorithm is demonstrated on a number of benchmark cases as well as a challenging substantive problem in HED hydrodynamics with highly disparate diffusion parameters.

An arbitrary high-order discontinuous Galerkin method with local time-stepping for linear acoustic wave propagation.

This paper presents a numerical scheme of arbitrary order of accuracy in both space and time, based on the arbitrary high-order derivatives methodology, for transient acoustic simulations. The scheme combines the nodal discontinuous Galerkin method for the spatial discretization and the Taylor series integrator (TSI) for the time integration. The main idea of the TSI is a temporal Taylor series expansion of all unknown acoustic variables in which the time derivatives are replaced by spatial derivatives via the Cauchy-Kovalewski procedure. The computational cost for the time integration is linearly proportional to the order of accuracy. To increase the computational efficiency for simulations involving strongly varying mesh sizes or material properties, a local time-stepping (LTS) algorithm accompanying the arbitrary high-order derivatives discontinuous Galerkin (ADER-DG) scheme, which ensures correct communications between domains with different time step sizes, is proposed. A numerical stability analysis in terms of the maximum allowable time step sizes is performed. Based on numerical convergence analysis, we demonstrate that for nonuniform meshes, a consistent high-order accuracy in space and time is achieved using ADER-DG with LTS. An application to the sound propagation across a transmissive noise barrier validates the potential of the proposed method for practical problems demanding high accuracy.

Lagrange nodal discontinuous Galerkin method for fractional Navier-Stokes equations

This paper provides a Lagrange nodal discontinuous Galerkin method for solving the time-dependent incompressible space fractional Navier-Stokes equations numerically. The existence and uniqueness of weak solutions are obtained. By combining the Lagrange method in temporal discretization and the hybridized discontinuous Galerkin method in spatial direction, the fully discrete scheme is presented and the stability is proved rigorously. Furthermore, the error estimates for the L2-norm are derived in both the velocity and the pressure. Finally, some numerical experiments are given to illustrate the performance of the proposed method and validate the theoretical result.

Time-domain impedance boundary condition modeling with the discontinuous Galerkin method for room acoustics simulations.

The time-domain nodal discontinuous Galerkin (TD-DG) method is emerging as a potential wave-based method for three-dimensional (3D) room acoustics modeling, where high-order accuracy in the low frequency range, geometrical flexibility, and accurate modeling of boundary conditions are of critical importance. This paper presents a formulation of broadband time-domain impedance boundary conditions (TDIBCs) of locally-reacting surfaces in the framework of the TD-DG method. The formulation is based on the approximation of the plane-wave reflection coefficient at normal incidence in the frequency domain using a sum of template rational functions, which can be directly transformed to the time-domain. The coupling of the TDIBCs with the discontinuous Galerkin discretization is achieved through the characteristic waves of the upwind flux along the boundary, where a series of first-order auxiliary differential equations is time-integrated in a high-order way. To verify the performance of the formulation, various numerical tests of single reflection scenarios are shown to demonstrate the cost efficiency and memory-efficiency of high-order basis functions, among which a 3D application to an impedance boundary of rigidly backed glass-wool baffle for room acoustic purposes is presented.

Frequency-dependent transmission boundary condition in the acoustic time-domain nodal discontinuous Galerkin model

Accurate modeling of boundary conditions is of critical importance for acoustic simulations. Recently, the time-domain nodal discontinuous Galerkin (TD-DG) method has emerged as a potential wave-based method for acoustic modeling. Although the acoustic reflection behavior of various time-domain impedance boundaries has been studied extensively, the modeling of the sound transmission across a locally-reacting layer of impedance discontinuity is far less developed. This paper presents a formulation of broadband time-domain transmission boundary conditions for locally-reacting surfaces in the framework of the TD-DG method. The formulation simulates the acoustic wave behavior at each of the boundary nodes using the plane-wave theory. Through the multi-pole model representation of the transmission coefficient, various types of transmission layers can be simulated. One-dimensional numerical examples demonstrate the capability of the proposed formulation to accurately simulate the reflection and transmission characteristics of the limp wall and the porous layer, where quantitative error behavior against analytical results is presented. Furthermore, to demonstrate the applicability, two scenarios of two-dimensional acoustic environment are considered. One is the sound transmission between two rooms partitioned by a limp panel and the other is the sound propagation through a transmissive noise barrier. Comparison of the predicted results from the proposed method against the results from the frequency-domain finite element simulations further verifies the formulation.

Algebraic Multigrid Schemes for High-Order Nodal Discontinuous Galerkin Methods

We present algebraic multigrid (AMG) methods for the efficient solution of the linear system of equations stemming from high-order discontinuous Galerkin (DG) discretizations of second-order elliptic problems. For DG methods, standard multigrid approaches cannot be employed because of redundancy of the degrees of freedom associated to the same grid point. We present new aggregation procedures and test them in extensive two-dimensional numerical experiments to demonstrate that the proposed AMG method is uniformly convergent with respect to all of the discretization parameters, namely the mesh-size and the polynomial approximation degree.

A high‐order nodal discontinuous Galerkin method to solve preconditioned multiphase Euler/Navier‐Stokes equations for inviscid/viscous cavitating flows

SummaryIn this study, a high‐order accurate numerical method is applied and examined for the simulation of the inviscid/viscous cavitating flows by solving the preconditioned multiphase Euler/Navier‐Stokes equations on triangle elements. The formulation used here is based on the homogeneous equilibrium model considering the continuity and momentum equations together with the transport equation for the vapor phase with applying appropriate mass transfer terms for calculating the evaporation/condensation of the liquid/vapor phase. The spatial derivative terms in the resulting system of equations are discretized by the nodal discontinuous Galerkin method (NDGM) and an implicit dual‐time stepping method is used for the time integration. An artificial viscosity approach is implemented and assessed for capturing the steep discontinuities in the interface between the two phases. The accuracy and robustness of the proposed method in solving the preconditioned multiphase Euler/Navier‐Stokes equations are examined by the simulation of different two‐dimensional and axisymmetric cavitating flows. A sensitivity study is also performed to examine the effects of different numerical parameters on the accuracy and performance of the solution of the NDGM. Indications are that the solution methodology proposed and applied here is based on the NDGM with the implicit dual‐time stepping method and the artificial viscosity approach is accurate and robust for the simulation of the inviscid and viscous cavitating flows.

Numerical simulation of 1-D oil and water displacements in petroleum reservoirs using the correction procedure via reconstruction (CPR) method

The focus of this work is to investigate and to apply, for the first time, a very high-resolution correction procedure via reconstruction (CPR) numerical discretization technique for the hyperbolic saturation equation that describes 1-D oil-water displacement through heterogeneous porous media. The CPR method can achieve high-order accuracy via a compact stencil consisting of the current cell and its immediate neighbors; in addition, the CPR recovers simplified versions of nodal discontinuous Galerkin (NDG), spectral volume (SV), and spectral difference (SD) methods using an adequate polynomial reconstruction function, whose coefficients are preprocessed and stored. Indeed the CPR versions of NDG and SV/SD are highly efficient. In order to suppress numerical oscillations near shocks, which are typical from higher-order schemes, a hierarchical MLP (multi-dimensional limiting process) is used in the reconstruction stage. The integration in time is carried out using a third-order RK (Runge-Kutta) method. A number of 1-D two-phase flow benchmark problems are solved, to verify the accuracy, efficiency, and shock-capturing capability of the adopted methodology.

High-order numerical simulation of axisymmetric wave phase conjugation

In the present paper, a general physico-mathematical model of magneto-acoustic Wave Phase Conjugation (WPC) is proposed, which includes wave propagation both in solid zones (the conjugator itself and other solid regions, if present) and in fluid zones surrounding the conjugator. Acoustic waves in the solid zones are assumed to behave linearly while in the fluid zones the waves may be either linear or non-linear. The axisymmetric governing equations of the model are discretized on an unstructured triangular mesh with a modified version of the Nodal Discontinuous Galerkin (NDG) method, which is a compact, high-order technique based on non-collocated solution and flux bases. The third-order Strong Stability-Preserving (SSP) Runge-Kutta (RK) scheme is used for integration in time. To avoid any false reflections from the outer boundaries of the computational domain the Nearly Perfectly Matched Layer (NPML) technique is adopted. A comprehensive set of test problems addressing all facets of the proposed numerical model is used for verification via comparison with analytical solutions. After the accuracy, performance, and reliability of the proposed model are established, two WPC problems are numerically simulated for the linear and weakly non-linear regimes. The main WPC properties, such as retro-focusing and parametric resonance, are observed in the numerical experiments with good accuracy.

Simulation of elastic wave propagation across fractures using a nodal discontinuous Galerkin method—theory, implementation and validation

SUMMARY An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.

An implicit dual-time stepping high-order nodal discontinuous Galerkin method for solving incompressible flows on triangle elements

In this work, a high-order nodal discontinuous Galerkin method (NDGM) is developed and assessed for the simulation of 2D incompressible flows on triangle elements. The governing equations are the 2D incompressible Navier–Stokes equations with the artificial compressibility method. The discretization of the spatial derivatives in the resulting system of equations is made by the NDGM and the time integration is performed by applying the implicit dual-time stepping method. Three numerical fluxes, namely, the local Lax–Friedrich, Roe and AUSM+-up are formulated and applied to assess and compare their accuracy and performance in the simulation of incompressible flows using the NDGM. Several steady and unsteady incompressible flow problems are simulated to examine the accuracy and robustness of the proposed solution methodology and they are the Kovasznay, backward facing step, NACA0012 airfoil, circular cylinder and two side-by-side circular cylinders. Indications are that the NDGM applied for solving the incompressible Navier–Stokes equations with the artificial compressibility approach and the implicit dual-time stepping method is accurate and robust for the simulation of steady and unsteady incompressible flow problems.

Implementation of the Nodal Discontinuous Galerkin Method for the Plate Vibration Problem using Linear Elasticity Equations

This work presents a numerical solution of the forced plate vibration problem using the nodal discontinuous Galerkin (DG) method. The plate is modelled as a three-dimensional domain, and its vibration is governed by the linear elasticity equations. The nodal DG method discretises the spatial domain and computes the spatial derivatives of the equations element-wise, while the time integration is conducted using the Runge-Kutta method. This method is in particular of interest as it is very favourable to carry out the computation by a parallel implementation. Several aspects regarding the numerical implementation such as the plate boundary conditions, the point force excitation, and the upwind numerical flux are presented. The numerical results are validated for rectangular concrete plates with different sets of boundary conditions and thicknesses, by a comparison with the exact mobilities that are derived from the classical plate theory (CPT) and the first order shear deformation theory (FSDT). The plate thickness is varied to understand its effect regarding the comparison with the CPT. An excellent agreement between the numerical solution and the FSDT was found. The agreement with the CPT occurs only at the first couple of resonance frequencies, and as the plate is getting thinner. Furthermore, the numerical example is extended to an L-shaped concrete plate. The mobility is then compared with the mobilities obtained by the CPT, FSDT, and linear elasticity equations.

Discontinuous Galerkin discretizations of the Boltzmann–BGK equations for nearly incompressible flows: Semi-analytic time stepping and absorbing boundary layers

We present an efficient nodal discontinuous Galerkin method for approximating nearly incompressible flows using the Boltzmann equations. The equations are discretized with Hermite polynomials in velocity space yielding a first order conservation law. A stabilized unsplit perfectly matching layer (PML) formulation is introduced for the resulting nonlinear flow equations. The proposed PML equations exponentially absorb the difference between the nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic time discretization methods to improve the time step restrictions in small relaxation times. We also introduce a multirate semi-analytic Adams–Bashforth method which preserves efficiency in stiff regimes. Accuracy and performance of the method are tested using distinct cases including isothermal vortex, flow around square cylinder, and wall mounted square cylinder test cases.

Room acoustics modelling in the time-domain with the nodal discontinuous Galerkin method.

To solve the linear acoustic equations for room acoustic purposes, the performance of the time-domain nodal discontinuous Galerkin (DG) method is evaluated. A nodal DG method is used for the evaluation of the spatial derivatives, and for the time-integration an explicit multi-stage Runge-Kutta method is adopted. The scheme supports a high order approximation on unstructured meshes. To model frequency-independent real-valued impedance boundary conditions, a formulation based on the plane wave reflection coefficient is proposed. Semi-discrete stability of the scheme is analyzed using the energy method. The performance of the DG method is evaluated for four three-dimensional configurations. The first two cases concern sound propagations in free field and over a flat impedance ground surface. Results show that the solution converges with increasing DG polynomial order and the accuracy of the impedance boundary condition is independent on the incidence angle. The third configuration is a cuboid room with rigid boundaries, for which an analytical solution serves as the reference solution. Finally, DG results for a real room scenario are compared with experimental results. For both room scenarios, results show good agreements.

A nodal discontinuous Galerkin finite element method for the poroelastic wave equation

We use the nodal discontinuous Galerkin method with a Lax-Friedrich flux to model the wave propagation in transversely isotropic and poroelastic media. The effect of dissipation due to global fluid flow causes a stiff relaxation term, which is incorporated in the numerical scheme through an operator splitting approach. The well-posedness of the poroelastic system is proved by adopting an approach based on characteristic variables. An error analysis for a plane wave propagating in poroelastic media shows a convergence rate of O(hn+ 1). Computational experiments are shown for various combinations of homogeneous and heterogeneous poroelastic media.

High-order integral nodal discontinuous Gegenbauer-Galerkin method for solving viscous Burgers' equation

ABSTRACTWe present a high-order integral nodal discontinuous Galerkin (DG) method to solve Burgers' equation. The method lays the first stone of a novel class of integral nodal DG methods exhibiting exponential convergence rates in both spatial and temporal directions; thus, producing highly accurate approximations using a significantly small number of collocation points. This useful result is proven theoretically under some mild conditions. The paper also introduces the first rigorous rounding-error analysis for the Gegenbauer integration matrices proving their stability feature. Two useful strategies were proposed to significantly reduce the errors in certain special cases and to handle problems with relatively large time domains. Extensive numerical comparisons with other competitive numerical methods manifest the superior accuracy and efficiency of the proposed numerical method. The established numerical method is so accurate in general for sufficiently smooth solutions to the extent that exact, or nearly exact solutions can be achieved using relatively small collocation points as the viscosity parameter .

An Absorbing Boundary Condition Based on Perfectly Matched Layer Technique Combined with Discontinuous Galerkin Boltzmann Method for Low Mach Number Flow Noise

For flow noise simulations, the nonreflecting boundary condition (NRBC) is significant to confine the computational domain to a small domain. Lattice Boltzmann method (LBM) has advantages for noise because of its low dissipation, but is limited to the uniform grid. In this paper, an absorbing boundary condition (ABC) based on perfectly matched layer (PML) technique is introduced to LBM. Then PML stability is analyzed and a new strategy is developed to achieve robustness. Invoking the decoupling time integration, the underlying equation for streaming is solved with the nodal discontinuous Galerkin method. Benchmark acoustic problems were used to demonstrate the PML absorption. Moreover, PML parameters, long time behavior and inhomogeneous pseudo mean flow are discussed. The methodology appears to work very well and would be hoped for practical flow noise computation.