Nodal Discontinuous Galerkin Method Research Articles

An open-source time-domain wave-based room acoustic software in Python based on the nodal discontinuous Galerkin method

In this work, we introduce an open-source implementation of a time-domain wave-based room acoustic modeling software package, named DG_RoomAcoustics. In this software, the linear acoustic equations are spatially discretized by the nodal discontinuous Galerkin method, and are integrated in time by either the explicit Runge-Kutta or the arbitrary high-order derivatives (ADER) integration schemes. Following the principles of object-oriented programming paradigm, the software is structured to ensure generic applicability and to facilitate future extensions with additional functionalities (e.g., different time integration schemes, boundary conditions). A comprehensive presentation of the physical and numerical aspects of the problem is provided. A detailed exposition of the code structure and components is presented, all of which are released under an open-source license, fostering community feedback and collaborative contributions for ongoing improvements. A brief overview of the current capabilities of the software is introduced. Future work regarding possible functionality extensions and performance optimizing are discussed.

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough for the sound waves to decay. The computational cost is lowered by strategically truncating the computational domain with an efficient boundary treatment. One commonly used boundary treatment is the perfectly matched layer (PML), which dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable, and an optimization procedure for the damping functions is proposed to enhance the performance of the formulation.

Nodal discontinuous Galerkin methods for Maxwell's equations in Lorentz-Kerr-Raman medium without nonlinear algebraic solver

The propagation of electromagnetic waves is modeled by time-dependent Maxwell's equations coupled with constitutive laws that describe the responses of the media. In this work, we consider a nonlinear model that describes the electromagnetic wave in an optical medium with the linear Lorentz effect and the cubic nonlinear instantaneous Kerr and delayed Raman effects. Mathematically this model obeys an energy conservative/dissipative law. Though there have been active efforts in designing numerical methods (e.g. of finite difference / finite element /discontinuous Galerkin type) to simulate this model, the methods proposed here are distinctive in that they are free of any nonlinear algebraic solvers. Moreover, in the absence of the Raman effect, our methods also enjoy a provable discrete energy law, and optimal a priori error estimates are further established when the exact solutions are sufficiently smooth. The key ingredients of the new methods include some novel treatment in time discretizations and nodal discontinuous Galerkin spatial discretization for the specific nonlinearities, and they also render a local nature of the methods and hence their suitability for parallel implementation with great efficiency. Numerical experiments are performed to illustrate the accuracy, stability, computational efficiency and parallel scalability of the proposed methods. We further apply the methods to simulate some physically relevant problems in one, two, and three dimensions.

A nodal discontinuous Galerkin method for wave propagation in coupled acoustic–elastic media

AbstractThe accurate numerical solution at an acoustic–elastic interface is important for offshore exploration. The solution requires careful implementation for the acoustic–elastic boundary conditions. In this work, we leverage a nodal discontinuous Galerkin method, in which the unstructured uniform triangular meshes are used for the model meshing and an explicit upwind numerical flux derived from the Riemann problem is adopted to handle the boundary conditions at the acoustic–elastic interface. Several numerical results are provided to assess the accuracy and convergence properties of this method. The convergence analysis is carried out in the coupled model with a flat interface, and the accuracy of the proposed method is verified in the curved interface coupled model. Finally, a more complex model with a salt dome, inspired by real geophysical applications, is carried out in this study. The numerical results demonstrate that the proposed nodal discontinuous Galerkin method is effective and accurate for dealing with the coupled acoustic–elastic media with complex geometries.

A leap-frog nodal discontinuous Galerkin method for Maxwell polynomial chaos Debye model

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method.

Semi-Lagrangian nodal discontinuous Galerkin method for the BGK model

Semi-Lagrangian nodal discontinuous Galerkin method for the BGK model

Efficient Implementation of Modern Entropy Stable and Kinetic Energy Preserving Discontinuous Galerkin Methods for Conservation Laws

Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entropy stability. While these techniques increase the robustness of DG methods significantly, they are also computationally more demanding than standard weak form nodal DG methods. We present several implementation techniques to improve the efficiency of flux differencing DG methods that use tensor product quadrilateral or hexahedral elements, in 2D or 3D, respectively. Focus is mostly given to CPUs and DG methods for the compressible Euler equations, although these techniques are generally also useful for other physical systems, including the compressible Navier-Stokes and magnetohydrodynamics equations. We present results using two open source codes, Trixi.jl written in Julia and FLUXO written in Fortran, to demonstrate that our proposed implementation techniques are applicable to different code bases and programming languages.

Discontinuous Galerkin method for vibration of structures with piecewise constant material properties

In this work, the vibrations of a structure excited by an impact source are modelled using the time domain nodal discontinuous Galerkin (DG) method, which solves linear elasticity equations. A scaled lightweight wooden floor (LWF) structure which consists of components that differ in their mechanical properties is taken as a case study. The Rankine-Hugoniot jump conditions for piece-wise constant material properties are used to obtain accurate numerical fluxes in the DG method, and their detailed derivation is the main contribution of this work. Free boundary conditions are applied on the surface of the structures, and constant viscous damping force is added to the model to have vibrational energy losses. To validate the numerical results, the mobility of the structure is calculated and compared with experimental data. The agreement is good regarding the natural frequencies, with a maximum absolute difference of less than 11 Hz in the frequency range below 200 Hz. The adopted damping approach is shown to be insufficient to represent a broad frequency range.

A linear low effort stabilization method for the Euler equations using discontinuous Galerkin methods

SummaryWe present a novel and simple yet intuitive approach to the stabilization problem for the numerically solved Euler equations with gravity source term relying on a low‐order nodal Discontinuous Galerkin Method (DGM). Instead of assuming isothermal or polytropic solutions, we only take a hydrostatic balance as a given property of the flow and use the hydrostatic equation to calculate a hydrostatic pressure reconstruction that replaces the gravity source term. We compare two environments that both solve the Euler equations using the DGM: deal.II and StormFlash. We utilize StormFlash as it allows for the use of the novel stabilization method. Without stabilization, StormFlash does not yield results that resemble correct physical behavior while the results with stabilization for StormFlash, as well as deal.II model the fluid flow more accurately. Convergence rates for deal.II do not match the expected order while the convergence rates for StormFlash with the stabilization scheme (with the exceptions for the L errors for momentum) meet the expectation. The results from StormFlash with stabilization also fit reference solutions from the literature much better than those from deal.II. We conclude that this novel scheme is a low cost approach to stabilize the Euler equations while not limiting the flow in any way other than it being in hydrostatic balance.

High order positivity-preserving nodal discontinuous Galerkin methods for anisotropic diffusion problems

In this paper, we develop second and third order accurate positivity-preserving (PP) nodal discontinuous Galerkin (DG) methods for one and two dimensional anisotropic diffusion problems. The key idea is to first represent the cell average of its numerical approximation as a weighted summation of Gaussian quadrature point values used in the updating of nodal DG methods, and then transform these Gaussian quadrature point values to some other special chosen point values. We prove that by taking parameters in the definition of nodal DG methods appropriately, together with a suitable time stability condition, the cell averages can be kept positive. A polynomial scaling limiter is then applied to obtain positive numerical approximations on the whole cell without sacrificing accuracy. Stability analysis without the PP limiter is also rigorously established. Numerical experiments are performed to demonstrate desired orders of accuracy, PP and good performances of our proposed approach.

A new Poisson-type equation applicable to the three-dimensional non-hydrostatic model in the framework of the discontinuous Galerkin method

In this study, a new Poisson-type equation for non-hydrostatic pressure in terrain-following σ-coordinates is proposed. The resulting equation consists of a standard elliptic operator and other first-order operators for non-hydrostatic pressure. Compared with the original Poisson-type equation, the treatment of the boundary condition at the lateral boundary can be simplified. Numerical tests indicate that the reformulated Poisson-type equation shows better convergence properties than the original equation when the quadrature-free nodal discontinuous Galerkin method is adopted for numerical discretization. Additionally, for the standing wave case, the numerical error of the three-dimensional non-hydrostatic model with the reformulated Poisson equation is smaller than that with the original equation, the numerical results of the solitary wave case indicate that the dissipation of the three-dimensional non-hydrostatic model is small, and transformation of wave over a circular shoal indicates that the developed model can properly simulate combined wave refraction and diffraction.

A Mixed‐Flux‐Based Nodal Discontinuous Galerkin Method for 3D Dynamic Rupture Modeling

AbstractNumerical simulation of rupture dynamics provides critical insights for understanding earthquake physics, while the complex geometry and heterogeneous material properties of natural faults make numerical method development challenging. The discontinuous Galerkin (DG) method is suitable for handling these complexities. In the DG method, the fault boundary conditions can be conveniently imposed through the upwind flux by solving a Riemann problem based on a velocity‐strain elastodynamic equation. However, the universal adoption of upwind flux can cause spatial oscillations in cases where elements on adjacent sides of the fault surface are not nearly symmetric. Here we propose a nodal DG method with an upwind/central mixed‐flux scheme to solve the spatial oscillation problem and thus reduce the dependence on mesh quality. Our method can achieve high scalability in parallel computing under different orders of accuracy. We verify the new method by comparing simulation results with those from other methods on a series of published benchmark problems with complex fault geometries, heterogeneous materials, off‐fault plasticity, roughness, thermal pressurization, and various versions of fault friction laws. Finally, we demonstrate that our method can be applied to simulate the dynamic rupture process of the 2008 Mw 7.9 Wenchuan earthquake along realistic fault geometry, including branches, step‐overs and a previously neglected sub‐parallel fault segment, offering a high potential to systematically explore the influences of fault zone properties on earthquake rupture dynamics.

A modified equation analysis for immersed boundary methods based on volume penalization: Applications to linear advection–diffusion equations and high-order discontinuous Galerkin schemes

The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection–diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immersed boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection–diffusion equations.

A mathematical model for the corneal transparency problem

AbstractUnderstanding the physical basis of corneal transparency has been a subject of interest amongst physicists, basic scientists and ophthalmologists. Impairment of corneal clarity is a significant cause of visual morbidity worldwide. Thus, it is essential to understand the mechanisms behind corneal transparency and how the alterations due to corneal pathologies affect vision. We use Maxwell’s equations to model light propagation in ocular tissues and a nodal discontinuous Galerkin method combined with an explicit Runge-Kutta method to simulate light propagation in normal and pathological corneas. Our simulation results illustrate that an increase in the diameter of some fibres causes an increase in backscattering. Thus, these may represent some of the physical changes in the cornea that might result in loss of transparency and visual morbidity.

ISI TAŞINIMI İÇİN YAPISAL OLMAYAN AĞLARDA BİR HIZLANDIRILMIŞ SÜREKSİZ GALERKİN METDU: FORMÜLASYONU VE DOĞRULANMASI

We present a GPU-accelerated method for large scale, coupled incompressible fluid flow and heat transfer problems. A high-order, nodal discontinuous Galerkin method is utilized to discretize governing equations on unstructured triangular meshes. A semi-implicit scheme with explicit treatment of the advective terms and implicit treatment of the split Stokes operators are used for time discretization. The pressure system is solved with a conjugate gradient method together with a fully GPU-accelerated multigrid preconditioner. The code is built on scalable libParanumal solver which is a library of high-performance kernels for high-order discretizations. Performance portability is achieved by using the open concurrent compute abstraction, OCCA. A set of numerical experiments including free and mixed convection problems indicate that our approach experimentally reaches design order of accuracy.

A high-order nodal discontinuous Galerkin method for simulation of three-dimensional non-cavitating/cavitating flows

In this study, the nodal discontinuous Galerkin method is formulated in three-dimensions and applied to simulate three-dimensional non-cavitating/cavitating flows. For this aim, the three-dimensional preconditioned Navier-Stokes equations based on the artificial compressibility approach considering appropriate source terms to model cavitating phenomena are used. The spatial derivative terms in the resulting equations are discretized by utilizing the nodal discontinuous Galerkin method on tetrahedral elements and the derivative of the solution vector with respect to the artificial time is discretized by applying an explicit time integration method. An artificial viscosity method is formulated in three-dimensions and applied to properly capture the discontinuities in the cavitating flow-field. Here, different non-cavitating/cavitating flow problems including non-cavitating flow in a 3:1:1 driven cavity, non-cavitating flow between two concentric rotating spheres, cavitating flow over a sphere and cavitating flow over a wavy cylinder are solved to examine the accuracy and robustness of the proposed solution method. These flow problems are selected to properly assess the numerical method in the presence of straight and curved boundaries and for different Reynolds and cavitation numbers. From linear to cubic polynomial degrees are used in the simulations and the numerical results are compared with the available analytical and numerical results. Indications are that the present numerical method based on the NDGM formulated in three-dimensions can accurately simulate the three-dimensional non-cavitating/cavitating flows. • Formulating a nodal discontinuous Galerkin method (NDGM) in 3D to simulate noncavitating/cavitating flows. • Implementing an artificial viscosity approach in 3D for capturing the steep discontinuities in the cavity interface. • Examining the accuracy and robustness of the solution method by computing different 3D noncavitating/cavitating flow problems. • Considering the 3D NDGM proposed to assess other solution methods for computing noncavitating/cavitating flows.

Development of a Discontinuous Galerkin Ionosphere‐Plasmasphere Model

AbstractThe objective of this study is to explore the application of high‐order numerical methods in ionosphere‐plasmasphere modeling. Specifically, the nodal discontinuous Galerkin method is chosen to solve the multifluid dynamical equations along the magnetic field lines. A general curvilinear magnetic field‐line‐following coordinate system is also used in the model. Numerical simulations with different combinations of number of elements (K) and polynomial orders (N) show the converging results, indicating the robustness of the algorithms and implementation. The model also captures the dawn terminator effect very well in the He+ field.

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

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 polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

Energy Stable Nodal Discontinuous Galerkin Methods for Nonlinear Maxwell’s Equations in Multi-dimensions

Energy Stable Nodal Discontinuous Galerkin Methods for Nonlinear Maxwell’s Equations in Multi-dimensions

An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part II: Subcell finite volume shock capturing

The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment.Hennemann et al. (2020) [25] recently presented an entropy stable shock-capturing strategy for DGSEM discretizations of the Euler equations that blends the DGSEM scheme with a subcell first-order finite volume (FV) method. Our first contribution is the extension of the method of Hennemann et al. to systems with non-conservative terms, such as the GLM-MHD equations. In our approach, the advective and non-conservative terms of the equations are discretized with a hybrid FV/DGSEM scheme, whereas the visco-resistive terms are discretized only with the high-order DGSEM method. We prove that the extended method is semi-discretely entropy stable on three-dimensional unstructured curvilinear meshes. Our second contribution is the derivation and analysis of a second entropy stable shock-capturing method that provides enhanced resolution by using a subcell reconstruction procedure that is carefully built to ensure entropy stability.We provide a numerical verification of the properties of the hybrid FV/DGSEM schemes on curvilinear meshes and show their robustness and accuracy with common benchmark cases, such as the Orszag-Tang vortex and the GEM (Geospace Environmental Modeling) reconnection challenge. Finally, we simulate a space physics application: the interaction of Jupiter's magnetic field with the plasma torus generated by the moon Io.